[
    {
        "id": "hamilton-easy-graph208",
        "image": "data/hamilton/graph_visual/easy/graph208.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (3,8) (9,0) (2,3) (6,7) (4,8) (1,7) (9,2) (6,2) (4,2) (3,6) (6,1) (5,1) (4,9) (3,5) (3,7) (8,5) (4,7)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->9->2->3->8->4->7->6->1->5."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph3",
        "image": "data/hamilton/graph_visual/easy/graph3.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (3,4) (1,2) (1,3) (2,0) (1,0) (3,2)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->1->3->4."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph296",
        "image": "data/hamilton/graph_visual/easy/graph296.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (6,7) (3,9) (6,1) (0,1) (9,1) (2,7) (6,8) (0,8) (4,8) (2,6) (0,3) (2,4) (9,5) (4,0) (8,2) (8,1) (0,5) (5,6) (4,6) (6,9) (0,9) (3,8) (7,9) (0,7) (1,2) (3,6)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->1->6->7->2->4->8->3->9->5."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph91",
        "image": "data/hamilton/graph_visual/easy/graph91.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (4,0) (3,6) (0,2) (1,5) (3,5) (1,3) (7,3) (0,6) (0,1) (1,4) (7,6) (4,2) (2,3) (3,4) (5,2) (7,5)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->4->1->5->2->3->6->7."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph38",
        "image": "data/hamilton/graph_visual/easy/graph38.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (4,1) (3,5) (6,5) (0,1) (3,0) (2,4) (0,4) (0,6) (2,3) (2,1) (6,2) (3,4)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->1->4->2->3->5->6."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph259",
        "image": "data/hamilton/graph_visual/easy/graph259.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (5,1) (7,5) (2,8) (6,5) (1,6) (4,7) (7,9) (9,2) (6,2) (4,6) (0,9) (1,7) (4,2) (2,0) (8,4) (1,2) (3,0) (0,7) (2,3) (9,3) (6,7) (5,4)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->9->7->5->1->6->4->8->2->3."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph10",
        "image": "data/hamilton/graph_visual/easy/graph10.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,4) (5,1) (3,1) (2,4) (5,3) (1,2) (4,1)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->4->2->1->5->3."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph172",
        "image": "data/hamilton/graph_visual/easy/graph172.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (8,6) (0,1) (6,0) (0,5) (4,5) (3,4) (1,6) (0,4) (8,5) (3,2) (6,7) (4,6) (0,3) (8,2) (6,5) (1,8) (2,1) (3,1) (4,8) (3,7) (1,5) (5,2) (6,2) (7,0) (8,7) (7,2) (7,1) (4,1)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->1->6->8->5->4->3->2->7."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph15",
        "image": "data/hamilton/graph_visual/easy/graph15.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (2,1) (3,1) (4,1) (0,1) (5,3) (2,3) (4,5)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->1->2->3->5->4."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph267",
        "image": "data/hamilton/graph_visual/easy/graph267.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (0,9) (4,0) (2,4) (3,9) (5,7) (7,0) (5,1) (6,4) (1,8) (3,7) (6,0) (9,2) (6,5) (2,7) (6,2) (2,8) (1,0) (8,6) (7,4) (9,6)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->9->3->7->5->1->8->2->4->6."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph247",
        "image": "data/hamilton/graph_visual/easy/graph247.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (0,6) (7,9) (2,1) (3,9) (7,4) (6,8) (0,1) (8,5) (0,3) (6,4) (0,2) (4,1) (1,5) (7,2) (5,0) (5,2) (2,6) (8,2) (1,6) (2,3) (9,1) (6,3) (7,5) (0,8) (9,2) (9,8) (9,5) (7,0) (7,6) (7,3) (9,0) (2,4)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->6->8->5->1->2->3->9->7->4."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph250",
        "image": "data/hamilton/graph_visual/easy/graph250.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (6,2) (8,4) (5,7) (7,4) (6,5) (1,8) (1,9) (2,4) (6,7) (4,5) (8,9) (5,1) (6,9) (4,0) (6,8) (2,0) (9,5) (3,6) (2,7) (6,1) (4,1)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->4->8->1->9->5->7->2->6->3."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph273",
        "image": "data/hamilton/graph_visual/easy/graph273.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (7,8) (9,8) (6,8) (1,7) (2,0) (0,5) (8,2) (8,4) (7,3) (1,4) (5,9) (1,0) (3,4) (5,4) (5,8) (1,3) (9,6) (4,9) (1,9) (0,7) (7,2) (4,2) (2,5) (3,2) (3,0) (9,3) (8,1) (7,6) (6,5)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->8->7->1->4->3->9->5->6."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph149",
        "image": "data/hamilton/graph_visual/easy/graph149.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (6,7) (1,2) (8,5) (4,6) (5,7) (4,1) (3,0) (4,3) (3,5) (1,6) (3,8) (8,7) (2,7) (6,2) (3,1) (2,3) (0,2) (0,6) (6,8) (4,5) (2,8) (5,2) (7,0) (6,3) (2,4)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->4->6->7->5->8->2->1."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph113",
        "image": "data/hamilton/graph_visual/easy/graph113.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (2,4) (4,8) (3,0) (2,1) (1,0) (3,8) (3,7) (5,6) (2,3) (3,6) (5,4) (6,2) (1,7)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->8->4->5->6->2->1->7."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph48",
        "image": "data/hamilton/graph_visual/easy/graph48.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (4,2) (6,2) (6,4) (5,2) (1,6) (0,3) (1,2) (3,2) (3,4) (3,5) (6,3) (1,3)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->4->6->1->2->5."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph52",
        "image": "data/hamilton/graph_visual/easy/graph52.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (2,6) (3,6) (1,0) (3,5) (3,2) (3,0) (5,4) (2,1) (5,1) (6,1) (3,1) (0,2) (4,3)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->1->2->6->3->5->4."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph255",
        "image": "data/hamilton/graph_visual/easy/graph255.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (8,9) (7,3) (5,2) (4,8) (3,8) (4,7) (6,4) (6,9) (3,6) (7,8) (0,7) (6,1) (6,5) (3,4) (8,5) (1,4) (6,8) (0,1) (3,5) (9,7) (7,6) (8,0) (3,2) (6,0) (5,1) (3,0) (7,1) (6,2) (9,0) (9,2)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->7->3->8->9->6->4->1->5->2."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph18",
        "image": "data/hamilton/graph_visual/easy/graph18.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,1) (2,3) (2,5) (1,2) (0,2) (1,4) (3,1) (4,0) (0,5) (5,3) (5,4)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->1->2->3->5->4."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph28",
        "image": "data/hamilton/graph_visual/easy/graph28.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (5,4) (0,6) (1,2) (4,2) (0,4) (1,3) (6,4) (4,1) (6,5)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->6->5->4->2->1->3."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph204",
        "image": "data/hamilton/graph_visual/easy/graph204.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (8,5) (3,2) (4,2) (3,6) (8,6) (4,6) (0,3) (6,0) (5,2) (8,7) (1,2) (4,0) (5,9) (7,3) (3,4) (9,2) (6,7) (1,4) (1,6)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->2->4->1->6->7->8->5->9."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph244",
        "image": "data/hamilton/graph_visual/easy/graph244.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (9,2) (5,2) (0,7) (7,2) (1,0) (9,7) (6,2) (7,8) (2,1) (0,3) (3,1) (4,2) (1,7) (8,6) (5,6) (8,3) (6,4) (6,3) (8,0) (1,6) (4,8) (7,4) (9,3) (9,4) (6,9) (8,2) (7,5)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->7->2->9->4->8->3->1->6->5."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph253",
        "image": "data/hamilton/graph_visual/easy/graph253.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (6,2) (4,1) (0,8) (5,1) (1,6) (8,1) (4,7) (3,9) (0,4) (9,8) (3,8) (4,3) (1,2) (4,2) (4,9) (7,5) (4,6) (6,7) (2,3) (0,7) (2,0) (6,0) (9,5) (4,5) (2,9)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->8->1->4->7->5->9->3->2->6."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph232",
        "image": "data/hamilton/graph_visual/easy/graph232.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (5,9) (4,1) (6,4) (6,3) (1,5) (8,9) (3,7) (2,6) (9,7) (5,0) (0,8) (4,2) (9,4) (9,6) (4,0) (2,7) (4,3) (0,3) (2,3)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->5->1->4->6->3->2->7->9->8."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph7",
        "image": "data/hamilton/graph_visual/easy/graph7.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (3,1) (2,4) (1,2) (4,0) (3,2)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->4->2->1->3."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph176",
        "image": "data/hamilton/graph_visual/easy/graph176.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (4,6) (0,1) (6,5) (8,2) (5,7) (6,1) (4,1) (3,1) (8,4) (6,2) (4,3) (7,6) (1,2) (3,2) (2,0) (0,3) (4,5) (5,2)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->1->6->7->5->4->8->2->3."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph299",
        "image": "data/hamilton/graph_visual/easy/graph299.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (7,9) (5,4) (2,9) (0,2) (8,1) (3,8) (5,9) (7,0) (0,1) (5,7) (4,3) (2,5) (4,7) (1,2) (8,0) (3,0) (6,3) (4,1) (4,0) (1,9) (5,1) (5,0) (2,6) (9,4) (6,1) (3,2) (8,9)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->9->7->5->4->3->8->1->6."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph70",
        "image": "data/hamilton/graph_visual/easy/graph70.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (4,7) (0,2) (4,1) (5,6) (7,3) (6,2) (6,7) (5,2) (7,1) (1,3) (0,4)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->5->6->7->4->1->3."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph243",
        "image": "data/hamilton/graph_visual/easy/graph243.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (5,3) (9,4) (3,0) (8,7) (6,1) (4,6) (4,2) (8,1) (6,2) (4,5) (4,7) (2,0) (9,5)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->5->9->4->2->6->1->8->7."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph62",
        "image": "data/hamilton/graph_visual/easy/graph62.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (0,2) (5,1) (4,1) (7,4) (1,3) (2,1) (7,5) (3,7) (2,6) (6,4)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->6->4->1->5->7->3."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph245",
        "image": "data/hamilton/graph_visual/easy/graph245.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (1,7) (0,4) (8,2) (1,8) (9,7) (5,1) (5,2) (2,1) (9,8) (7,4) (1,9) (9,2) (1,4) (1,6) (0,9) (5,4) (3,7) (8,4) (7,8) (0,2) (7,5) (9,6) (5,9) (0,6) (3,6) (2,7) (6,8)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->4->7->1->8->2->5->9->6->3."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph55",
        "image": "data/hamilton/graph_visual/easy/graph55.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (4,5) (3,0) (3,2) (6,5) (2,5) (4,3) (5,1) (4,2) (6,0) (5,0) (6,3) (6,1) (4,0)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->2->4->5->6->1."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph205",
        "image": "data/hamilton/graph_visual/easy/graph205.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (1,9) (8,0) (3,8) (0,6) (9,4) (4,8) (0,1) (9,5) (5,2) (8,5) (2,8) (7,1) (6,7) (4,3) (9,7) (7,4) (7,2) (5,3)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->8->2->5->3->4->9->1->7->6."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph249",
        "image": "data/hamilton/graph_visual/easy/graph249.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (0,3) (3,8) (4,6) (8,1) (8,7) (5,9) (0,4) (1,5) (6,9) (7,0) (8,4) (9,8) (2,6) (5,8) (3,9) (6,0) (0,2) (6,7) (2,9) (2,4) (9,7)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->8->1->5->9->2->4->6->7."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph73",
        "image": "data/hamilton/graph_visual/easy/graph73.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (4,1) (0,7) (4,6) (4,5) (6,7) (4,2) (4,0) (0,2) (5,2) (1,7) (1,3) (5,7) (3,6) (6,2) (6,5)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->7->6->3->1->4->5->2."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph57",
        "image": "data/hamilton/graph_visual/easy/graph57.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (3,4) (4,1) (6,1) (0,2) (0,7) (4,7) (3,1) (7,3) (1,2) (0,4) (5,3) (7,6) (6,3) (6,5)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->1->4->3->7->6->5."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph189",
        "image": "data/hamilton/graph_visual/easy/graph189.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (8,9) (0,1) (8,0) (6,8) (4,5) (1,4) (6,9) (7,4) (6,5) (3,6) (3,1) (3,4) (3,7) (2,9) (2,7) (3,8) (5,8) (6,7) (5,9) (4,2) (2,3) (6,2) (0,5)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->1->4->5->6->8->9->2->7->3."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph63",
        "image": "data/hamilton/graph_visual/easy/graph63.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (3,2) (0,3) (4,1) (4,6) (2,0) (4,7) (5,2) (6,2) (4,0) (3,5) (5,6) (1,2)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->5->6->2->1->4->7."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph215",
        "image": "data/hamilton/graph_visual/easy/graph215.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (7,4) (5,6) (9,7) (9,1) (3,7) (4,9) (6,9) (2,5) (9,8) (3,6) (7,0) (4,2) (1,7) (0,8)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->8->9->1->7->4->2->5->6->3."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph107",
        "image": "data/hamilton/graph_visual/easy/graph107.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (0,4) (2,1) (8,1) (2,6) (7,4) (6,4) (1,5) (6,8) (2,4) (7,8) (0,8) (3,4) (1,4) (7,5) (6,1) (3,0) (3,6) (3,7)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->4->2->1->8->6->3->7->5."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph20",
        "image": "data/hamilton/graph_visual/easy/graph20.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (5,1) (4,1) (0,3) (2,3) (1,3) (3,5) (0,4) (2,4) (1,2) (0,2)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->2->4->1->5."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph146",
        "image": "data/hamilton/graph_visual/easy/graph146.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (7,5) (5,0) (4,7) (2,0) (3,0) (4,8) (7,2) (4,6) (6,7) (3,8) (4,2) (3,2) (4,3) (5,4) (6,2) (7,0) (4,0) (5,3) (8,5) (6,1)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->5->7->4->8->3->2->6->1."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph89",
        "image": "data/hamilton/graph_visual/easy/graph89.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (5,3) (6,5) (1,4) (6,0) (2,7) (4,3) (5,7) (4,0) (0,7) (7,4) (6,7) (7,1) (6,3) (7,3) (5,0) (0,1) (6,4) (2,3)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->6->5->3->4->1->7->2."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph17",
        "image": "data/hamilton/graph_visual/easy/graph17.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (3,1) (4,5) (5,0) (4,0) (2,1) (1,0) (5,1) (2,3) (2,4) (4,1)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->5->4->2->1->3."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph46",
        "image": "data/hamilton/graph_visual/easy/graph46.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (4,3) (6,5) (1,5) (3,0) (2,4) (1,0) (2,5) (0,4) (2,3) (2,0) (6,2)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->4->2->6->5->1."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph163",
        "image": "data/hamilton/graph_visual/easy/graph163.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (0,3) (1,3) (4,7) (7,8) (2,8) (8,4) (3,2) (1,8) (0,4) (8,3) (6,5) (6,0) (4,5) (1,7) (7,6) (8,5) (1,4) (6,2) (7,3) (5,0) (2,1) (0,1) (7,0)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->1->8->7->4->5->6->2."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph56",
        "image": "data/hamilton/graph_visual/easy/graph56.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (7,1) (7,4) (5,0) (1,0) (7,0) (3,0) (5,2) (2,4) (6,4) (0,4) (7,6) (6,3) (1,4) (4,5) (3,2) (1,5)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->5->2->4->1->7->6->3."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph221",
        "image": "data/hamilton/graph_visual/easy/graph221.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (9,6) (5,6) (0,9) (2,7) (4,9) (4,5) (9,7) (5,0) (2,9) (1,5) (3,5) (6,0) (7,0) (7,5) (4,3) (8,7) (1,6) (4,6) (9,8) (3,8) (2,0) (2,3) (1,3) (7,3)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->9->6->1->5->4->3->8->7->2."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph284",
        "image": "data/hamilton/graph_visual/easy/graph284.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (2,4) (1,5) (1,3) (6,8) (8,2) (4,9) (2,5) (1,7) (4,6) (8,5) (2,0) (6,7) (5,0) (4,7) (3,9) (1,8) (4,0) (4,5) (8,7) (8,4) (0,9) (8,3) (0,3) (2,7) (9,2) (7,0) (0,6) (2,6) (6,5) (3,6)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->4->9->3->1->5->8->6->7."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph77",
        "image": "data/hamilton/graph_visual/easy/graph77.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (5,2) (5,4) (0,6) (2,3) (3,7) (7,6) (4,0) (7,1) (7,2) (2,0) (4,6) (5,7)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->6->4->5->2->3->7->1."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph101",
        "image": "data/hamilton/graph_visual/easy/graph101.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (1,3) (6,3) (5,0) (1,4) (0,7) (5,4) (2,3) (6,2) (5,3) (7,2) (3,0) (6,5) (4,6) (6,7)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->5->4->1->3->6->2->7."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph93",
        "image": "data/hamilton/graph_visual/easy/graph93.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (3,2) (2,1) (6,2) (2,5) (7,0) (7,2) (3,5) (6,1) (6,3) (1,0) (0,4) (6,5) (5,7) (1,4) (4,2) (5,0)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->7->2->3->5->6->1->4."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph194",
        "image": "data/hamilton/graph_visual/easy/graph194.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (1,4) (4,8) (9,1) (5,7) (9,7) (3,1) (7,2) (7,8) (9,0) (2,1) (6,2) (1,6) (0,2) (8,0) (0,7) (5,3) (8,2)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->9->7->5->3->1->4->8->2->6."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph119",
        "image": "data/hamilton/graph_visual/easy/graph119.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (6,4) (4,0) (2,0) (3,4) (3,1) (1,7) (6,5) (2,6) (5,2) (5,8) (1,6) (5,4) (7,5) (5,3)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->6->4->3->1->7->5->8."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph87",
        "image": "data/hamilton/graph_visual/easy/graph87.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,1) (7,5) (4,3) (2,4) (6,3) (7,6) (6,4) (3,5) (6,5) (3,0) (1,5) (4,0) (4,7) (1,7) (0,2) (5,4) (7,2) (5,0) (3,1) (5,2)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->4->2->1->5->7->6."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph6",
        "image": "data/hamilton/graph_visual/easy/graph6.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (2,1) (0,1) (1,3) (2,4) (0,3) (4,0) (4,3) (2,0) (4,1)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->1->2->4->3."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph226",
        "image": "data/hamilton/graph_visual/easy/graph226.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (5,7) (1,0) (4,7) (2,6) (0,7) (3,4) (1,6) (7,3) (4,9) (1,8) (9,8) (3,1) (8,0) (7,1) (5,4)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->8->9->4->5->7->3->1->6->2."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph138",
        "image": "data/hamilton/graph_visual/easy/graph138.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (3,0) (1,6) (0,2) (4,1) (6,3) (1,2) (7,6) (5,8) (5,7) (7,3) (0,1) (2,3) (8,4) (1,7)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->6->7->5->8->4->1->2."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph59",
        "image": "data/hamilton/graph_visual/easy/graph59.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,0) (4,7) (6,2) (6,3) (4,1) (4,3) (5,2) (3,7) (6,1) (4,5) (1,2) (2,7)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->5->4->7->3->6->1."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph54",
        "image": "data/hamilton/graph_visual/easy/graph54.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (5,1) (1,4) (3,1) (4,0) (3,2) (5,2) (5,4) (1,0) (6,5) (3,5)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->4->1->3->2->5->6."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph130",
        "image": "data/hamilton/graph_visual/easy/graph130.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (1,6) (1,7) (1,3) (6,2) (1,4) (1,8) (3,7) (5,0) (2,8) (4,7) (6,0) (8,5) (7,8) (4,6) (7,0) (2,4) (8,0) (0,4)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->5->8->1->6->2->4->7->3."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph82",
        "image": "data/hamilton/graph_visual/easy/graph82.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,1) (2,4) (3,6) (4,5) (3,5) (6,1) (5,7) (0,2) (7,0) (4,3) (0,1) (1,5) (7,2) (6,5) (7,3) (6,2) (0,5) (0,6)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->1->6->3->4->5->7."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph298",
        "image": "data/hamilton/graph_visual/easy/graph298.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (0,4) (1,3) (1,7) (3,9) (0,9) (4,9) (5,4) (8,6) (2,5) (2,1) (4,6) (8,4) (7,2) (1,4) (0,8) (8,7) (8,9) (1,8) (2,3) (5,0) (6,3) (0,7) (2,4) (9,5) (8,3) (7,4) (7,9)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->4->9->3->6->8->7->1->2->5."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph102",
        "image": "data/hamilton/graph_visual/easy/graph102.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (6,0) (5,4) (0,2) (4,0) (4,6) (3,5) (2,1) (4,7) (3,0) (3,7) (1,4) (5,6) (6,3) (2,3)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->6->5->4->7->3->2->1."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph35",
        "image": "data/hamilton/graph_visual/easy/graph35.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (6,1) (0,3) (3,2) (1,2) (4,5) (6,5) (6,0) (5,2) (6,2)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->2->1->6->5->4."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph230",
        "image": "data/hamilton/graph_visual/easy/graph230.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (6,5) (4,8) (4,5) (8,2) (9,2) (4,3) (3,0) (9,1) (9,7) (9,8) (6,8) (9,4) (7,0) (7,2) (5,0) (5,1) (3,2)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->4->8->2->7->9->1->5->6."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph136",
        "image": "data/hamilton/graph_visual/easy/graph136.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (2,6) (8,6) (3,0) (7,1) (8,7) (2,5) (6,5) (5,4) (5,0) (7,4) (4,2) (4,6) (1,4) (2,8) (5,8) (3,1) (2,7) (7,3)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->1->7->8->6->2->5->4."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph180",
        "image": "data/hamilton/graph_visual/easy/graph180.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (5,0) (4,8) (2,3) (5,4) (1,6) (0,8) (2,4) (8,2) (1,8) (3,4) (7,4) (1,7) (5,6) (6,2) (3,6) (8,5) (6,7) (2,0) (2,7) (3,1) (4,0) (7,8) (5,1)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->5->4->8->2->3->6->1->7."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph276",
        "image": "data/hamilton/graph_visual/easy/graph276.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (3,2) (8,7) (8,3) (2,7) (9,6) (1,4) (3,5) (8,5) (0,8) (1,7) (0,3) (1,5) (2,5) (3,9) (0,1) (0,7) (2,1) (4,3) (5,0) (3,6) (9,8) (6,2) (5,4)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->8->7->2->5->1->4->3->9->6."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph33",
        "image": "data/hamilton/graph_visual/easy/graph33.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (3,1) (4,5) (5,2) (0,4) (2,4) (3,2) (3,5) (6,5) (3,4) (0,6)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->6->5->4->2->3->1."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph195",
        "image": "data/hamilton/graph_visual/easy/graph195.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (8,3) (1,6) (2,9) (7,0) (2,8) (1,5) (8,7) (2,6) (4,2) (1,9) (2,7) (9,4) (3,4) (3,9) (7,9) (6,5) (0,8) (6,4) (1,0) (5,3) (9,0)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->7->8->3->4->2->9->1->6->5."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph58",
        "image": "data/hamilton/graph_visual/easy/graph58.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (3,6) (0,7) (6,4) (3,4) (0,1) (2,3) (6,1) (4,0) (3,7) (2,1) (7,5)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->4->6->1->2->3->7->5."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph241",
        "image": "data/hamilton/graph_visual/easy/graph241.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (1,4) (6,9) (2,6) (8,7) (7,0) (0,8) (2,9) (2,4) (8,9) (1,6) (0,4) (5,6) (6,8) (1,0) (3,8) (1,8) (0,9) (4,7) (5,7) (0,5)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->7->5->6->9->2->4->1->8->3."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph264",
        "image": "data/hamilton/graph_visual/easy/graph264.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (3,0) (4,5) (5,2) (0,7) (9,2) (2,1) (1,3) (2,7) (1,8) (4,2) (4,0) (1,4) (0,5) (8,3) (6,2) (9,8) (6,3) (9,4) (3,9) (8,5) (5,7) (5,1) (1,6) (0,2) (4,3)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->1->8->9->4->5->7->2->6."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph85",
        "image": "data/hamilton/graph_visual/easy/graph85.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (0,6) (5,6) (3,0) (6,3) (1,5) (4,0) (6,4) (2,5) (1,2) (4,7) (7,0) (4,2) (3,4) (3,1) (2,6)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->6->5->2->1->3->4->7."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph211",
        "image": "data/hamilton/graph_visual/easy/graph211.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (5,9) (7,2) (1,9) (6,1) (7,4) (6,3) (7,9) (4,6) (2,1) (7,3) (9,2) (8,2) (3,4) (6,0) (7,1) (0,5)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->5->9->1->6->3->4->7->2->8."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph278",
        "image": "data/hamilton/graph_visual/easy/graph278.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (4,6) (8,0) (0,4) (2,5) (4,9) (1,0) (7,5) (4,8) (8,9) (6,7) (8,3) (3,5) (9,0) (3,1) (1,5) (2,6) (8,5) (1,7) (4,5) (0,3) (8,1) (7,2) (1,4) (2,9) (5,9) (7,3) (6,3) (5,0) (9,7) (1,9) (7,4) (0,2)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->8->4->6->7->5->2->9->1->3."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph294",
        "image": "data/hamilton/graph_visual/easy/graph294.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (8,3) (8,9) (4,8) (1,3) (7,8) (5,0) (3,5) (5,2) (7,5) (7,9) (1,7) (1,0) (7,4) (9,6) (5,4) (0,7) (2,4) (3,7) (2,8) (5,1) (4,1) (8,5) (3,4)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->5->3->8->2->4->1->7->9->6."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph183",
        "image": "data/hamilton/graph_visual/easy/graph183.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (0,6) (2,1) (7,2) (0,5) (5,7) (8,4) (8,2) (7,0) (5,4) (1,4) (5,3) (5,6) (5,8) (3,2) (3,7) (3,1) (5,2) (3,8) (0,8) (3,6) (6,8) (1,8) (5,1) (7,4)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->6->5->7->2->1->4->8->3."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph125",
        "image": "data/hamilton/graph_visual/easy/graph125.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (1,2) (0,1) (3,6) (5,1) (1,4) (6,0) (2,8) (2,7) (4,2) (3,2) (3,7) (5,4) (2,5) (1,7) (8,3)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->6->3->7->1->5->4->2->8."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph31",
        "image": "data/hamilton/graph_visual/easy/graph31.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (6,3) (5,4) (5,1) (0,5) (6,4) (3,4) (1,6) (2,3) (0,1)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->5->1->6->4->3->2."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph167",
        "image": "data/hamilton/graph_visual/easy/graph167.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (5,4) (5,8) (7,1) (3,2) (6,3) (4,2) (4,7) (4,6) (3,7) (5,0) (0,8) (3,5) (8,3) (4,1) (1,8) (2,0) (4,3) (6,0) (1,5) (8,7)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->5->8->1->7->4->2->3->6."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph184",
        "image": "data/hamilton/graph_visual/easy/graph184.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (5,8) (0,6) (4,2) (1,8) (5,7) (2,0) (6,3) (6,1) (8,0) (6,8) (4,3) (7,4) (6,2) (7,2) (4,1) (5,0) (3,8) (5,2) (6,7) (0,4) (2,8)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->6->3->4->2->7->5->8->1."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph266",
        "image": "data/hamilton/graph_visual/easy/graph266.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (9,4) (3,1) (2,7) (0,9) (5,0) (3,5) (2,8) (6,2) (1,0) (4,3) (7,1) (3,7) (0,3) (5,8) (1,2) (6,8) (4,1) (6,3) (9,1) (9,6) (6,1) (1,8) (1,5) (9,2) (8,7) (9,7) (0,6) (6,4)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->9->4->3->1->7->2->6->8->5."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph74",
        "image": "data/hamilton/graph_visual/easy/graph74.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,0) (7,2) (6,7) (2,1) (6,1) (4,3) (5,4) (3,1) (5,0) (7,5) (0,4) (6,4) (7,0) (4,2) (4,7) (5,3)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->7->6->1->3->4->5."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph94",
        "image": "data/hamilton/graph_visual/easy/graph94.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (7,4) (3,4) (0,7) (7,3) (5,1) (3,1) (6,0) (1,2) (7,6) (5,0) (6,5) (6,4) (6,3) (2,0) (5,4) (7,1) (2,7)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->7->4->3->6->5->1->2."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph213",
        "image": "data/hamilton/graph_visual/easy/graph213.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (9,4) (6,9) (0,9) (5,8) (9,3) (3,4) (2,1) (4,6) (3,1) (9,7) (2,5) (3,0) (9,8) (6,5) (2,8) (8,3) (3,6)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->1->2->8->5->6->4->9->7."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph128",
        "image": "data/hamilton/graph_visual/easy/graph128.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (0,5) (8,0) (0,1) (4,6) (0,7) (3,5) (2,5) (6,7) (2,0) (0,6) (1,4) (2,8) (8,7) (2,1) (7,1) (3,7)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->5->3->7->6->4->1->2->8."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph24",
        "image": "data/hamilton/graph_visual/easy/graph24.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (0,6) (3,1) (0,3) (2,3) (6,3) (0,2) (6,2) (1,4) (3,4) (6,1) (5,2)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->6->1->4->3->2->5."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph16",
        "image": "data/hamilton/graph_visual/easy/graph16.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (4,1) (1,5) (2,3) (4,2) (0,3) (5,4) (0,1) (3,5) (0,2)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->2->4->1->5."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph277",
        "image": "data/hamilton/graph_visual/easy/graph277.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (4,5) (4,0) (2,6) (7,8) (1,7) (1,4) (6,0) (8,4) (3,2) (1,6) (8,5) (3,8) (7,6) (9,2) (9,8) (0,7) (0,8) (1,5) (1,8) (5,7) (6,3) (2,7) (2,5) (1,3) (9,6) (4,6) (3,0)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->4->5->8->7->1->6->3->2->9."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph154",
        "image": "data/hamilton/graph_visual/easy/graph154.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (0,7) (8,5) (2,0) (0,8) (3,4) (2,8) (1,5) (4,5) (6,4) (8,6) (0,1) (5,6) (2,1) (7,1) (8,7) (0,4) (3,2) (5,7) (8,4) (8,3) (1,8) (7,4) (6,1)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->7->1->5->8->2->3->4->6."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph27",
        "image": "data/hamilton/graph_visual/easy/graph27.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (2,3) (6,5) (1,3) (0,4) (2,5) (4,3) (5,3) (6,4)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->4->6->5->2->3->1."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph217",
        "image": "data/hamilton/graph_visual/easy/graph217.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (6,5) (1,9) (0,6) (4,9) (8,3) (4,7) (7,1) (7,2) (2,8) (2,0) (2,6) (4,0) (9,0) (6,8) (1,6) (8,1) (2,4) (3,9) (1,4) (5,0) (1,3)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->7->4->9->1->3->8->6->5."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph144",
        "image": "data/hamilton/graph_visual/easy/graph144.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (3,1) (3,7) (2,4) (8,6) (7,6) (3,8) (7,4) (0,1) (0,3) (2,8) (5,8) (8,4) (2,1) (5,0) (6,3)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->1->3->6->7->4->2->8->5."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph206",
        "image": "data/hamilton/graph_visual/easy/graph206.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (8,4) (6,9) (6,8) (4,3) (1,8) (7,6) (0,5) (6,5) (4,7) (4,2) (9,4) (5,8) (3,1) (3,5) (9,7) (6,2) (8,3) (3,7) (1,6) (0,8) (8,2) (0,9)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->5->6->9->4->7->3->1->8->2."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph290",
        "image": "data/hamilton/graph_visual/easy/graph290.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (3,7) (4,0) (0,5) (8,3) (0,2) (5,8) (5,4) (2,9) (9,8) (3,6) (5,9) (7,4) (9,6) (2,5) (2,7) (5,1) (8,6) (3,1) (3,5) (9,0) (7,9) (2,1) (6,7) (4,9) (4,2) (7,5) (6,1) (7,1) (1,8) (8,7) (4,3)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->4->5->8->3->7->2->9->6->1."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph150",
        "image": "data/hamilton/graph_visual/easy/graph150.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (7,6) (0,1) (2,8) (0,8) (1,4) (0,5) (2,5) (6,8) (0,6) (3,6) (7,2) (8,1) (5,6) (6,2) (7,4) (0,3) (4,2) (3,8) (1,7) (8,4) (3,1)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->1->4->7->6->3->8->2->5."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph229",
        "image": "data/hamilton/graph_visual/easy/graph229.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (1,7) (2,7) (0,7) (5,2) (0,6) (9,3) (9,5) (0,2) (9,4) (5,0) (6,2) (9,8) (7,3) (6,7) (4,0) (2,8) (7,9) (5,3) (7,5) (1,6) (8,7) (3,2) (8,0)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->6->1->7->3->5->2->8->9->4."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph104",
        "image": "data/hamilton/graph_visual/easy/graph104.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (1,6) (5,6) (1,4) (5,1) (6,0) (3,0) (7,4) (8,3) (4,5) (5,2) (0,7) (4,6) (7,8) (6,8) (5,7) (1,0)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->8->7->4->1->6->5->2."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph88",
        "image": "data/hamilton/graph_visual/easy/graph88.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (4,7) (1,3) (0,2) (1,2) (7,2) (5,7) (6,1) (3,2) (5,6) (2,4) (6,7) (2,6) (2,5) (0,3)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->3->1->6->5->7->4."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph8",
        "image": "data/hamilton/graph_visual/easy/graph8.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (3,0) (0,1) (4,3) (2,3) (0,5) (4,1) (5,4) (2,1) (5,2) (4,2)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->4->1->2->5."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph92",
        "image": "data/hamilton/graph_visual/easy/graph92.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (7,4) (6,5) (0,7) (2,6) (4,0) (5,3) (0,3) (2,5) (1,0) (1,6) (0,6) (4,3) (2,3) (5,4) (6,3) (1,4) (6,7) (4,6)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->7->4->3->5->2->6->1."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph231",
        "image": "data/hamilton/graph_visual/easy/graph231.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (9,8) (7,2) (5,2) (6,2) (3,0) (0,9) (1,8) (2,8) (5,6) (1,5) (5,0) (7,8) (8,5) (0,1) (9,4) (3,1) (2,3) (6,3) (7,9) (5,9) (6,0) (4,2)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->1->8->7->2->6->5->9->4."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph47",
        "image": "data/hamilton/graph_visual/easy/graph47.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (2,4) (5,2) (1,0) (1,3) (5,3) (5,4) (5,0) (5,1) (6,1) (1,4) (2,0) (1,2)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->4->5->3->1->6."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph185",
        "image": "data/hamilton/graph_visual/easy/graph185.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (6,8) (5,2) (4,2) (8,4) (2,0) (7,6) (1,2) (8,2) (6,2) (6,0) (8,3) (6,4) (0,3) (5,3) (5,1) (1,0) (2,3) (4,7) (1,3) (0,5) (8,1)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->5->3->1->8->6->7->4."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph157",
        "image": "data/hamilton/graph_visual/easy/graph157.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (1,8) (5,2) (8,2) (1,5) (2,6) (1,7) (4,6) (3,0) (0,5) (7,6) (5,4) (6,3) (4,8) (3,8) (1,6) (5,7) (5,6) (1,2) (7,2) (8,7) (1,4) (1,0) (3,5) (2,3)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->6->2->5->1->7->8->4."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph75",
        "image": "data/hamilton/graph_visual/easy/graph75.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (0,2) (6,4) (7,0) (6,2) (1,4) (1,3) (5,0) (1,5) (7,1) (2,1) (2,7) (6,5) (3,2)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->7->2->3->1->4->6->5."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph72",
        "image": "data/hamilton/graph_visual/easy/graph72.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,0) (6,2) (5,4) (6,0) (1,4) (3,6) (5,6) (1,0) (7,5) (7,1) (4,6) (2,4)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->4->1->7->5->6->3."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph156",
        "image": "data/hamilton/graph_visual/easy/graph156.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (4,1) (3,6) (2,6) (5,2) (1,7) (2,4) (5,6) (7,5) (1,5) (8,5) (0,2) (5,3) (0,8) (8,6) (2,8) (0,4) (5,4) (0,3)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->4->1->7->5->8->6->3."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph39",
        "image": "data/hamilton/graph_visual/easy/graph39.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (6,2) (4,2) (1,4) (2,3) (3,6) (3,4) (6,5) (0,2) (3,0) (0,1) (1,6) (0,4) (2,5)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->3->4->1->6->5."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph236",
        "image": "data/hamilton/graph_visual/easy/graph236.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (1,7) (4,8) (0,7) (6,3) (3,9) (7,9) (0,8) (8,1) (2,9) (0,3) (9,0) (1,4) (2,5) (8,2) (9,6) (8,6) (2,4) (2,7)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->7->1->4->8->6->3->9->2->5."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph271",
        "image": "data/hamilton/graph_visual/easy/graph271.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (7,1) (8,0) (1,2) (0,2) (3,0) (6,9) (8,7) (5,6) (0,9) (3,2) (7,4) (1,5) (6,0) (9,8) (6,7) (2,4) (8,2) (5,0) (8,5) (3,8) (4,5) (9,1) (0,1) (6,8) (2,9) (9,4) (5,7) (8,4) (4,1)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->8->7->1->5->6->9->4->2->3."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph79",
        "image": "data/hamilton/graph_visual/easy/graph79.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (0,3) (2,3) (5,6) (6,7) (0,7) (3,1) (0,6) (2,6) (4,0) (5,7) (1,4) (6,3) (5,0) (1,2)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->7->5->6->2->3->1->4."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph262",
        "image": "data/hamilton/graph_visual/easy/graph262.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (8,3) (1,9) (0,7) (6,9) (7,5) (9,7) (1,8) (3,1) (2,1) (4,3) (8,2) (0,5) (9,4) (3,7) (5,3) (2,5) (0,4) (4,1) (0,2) (4,2) (6,1) (8,9) (6,4) (8,7) (5,6) (5,4)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->7->5->3->8->1->9->6->4->2."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph261",
        "image": "data/hamilton/graph_visual/easy/graph261.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (2,8) (8,4) (4,6) (9,7) (2,7) (6,5) (8,3) (8,9) (6,3) (5,8) (4,2) (9,3) (1,6) (1,8) (3,0) (7,3) (5,3) (2,6) (9,0) (7,4) (9,5) (1,0) (3,2) (6,0)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->8->2->4->7->9->5->6->1."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph181",
        "image": "data/hamilton/graph_visual/easy/graph181.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (0,5) (0,4) (8,6) (2,8) (5,7) (5,3) (7,8) (1,5) (3,1) (3,8) (7,3) (2,0) (0,3) (2,3) (3,6) (8,0) (5,8) (7,4) (4,6) (0,7) (4,2) (2,5) (6,2) (4,1)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->5->7->8->6->3->1->4->2."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph5",
        "image": "data/hamilton/graph_visual/easy/graph5.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (1,2) (3,2) (3,0) (3,4) (1,0)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->1->2->3->4."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph95",
        "image": "data/hamilton/graph_visual/easy/graph95.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (7,2) (1,5) (4,2) (6,4) (7,0) (3,4) (1,7) (1,2) (0,1) (5,6) (5,3) (6,2) (3,1) (0,5) (6,1) (3,0)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->7->2->4->6->5->1->3."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph272",
        "image": "data/hamilton/graph_visual/easy/graph272.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (9,1) (6,1) (3,9) (4,1) (0,2) (8,7) (5,2) (4,3) (2,7) (5,4) (5,9) (8,6) (0,6) (4,7) (5,8) (8,1) (6,3) (9,6) (7,0) (6,7) (8,9) (4,0) (2,8) (0,8)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->5->4->1->9->3->6->8->7."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph83",
        "image": "data/hamilton/graph_visual/easy/graph83.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (0,5) (7,1) (1,5) (7,2) (7,5) (6,5) (1,3) (6,1) (6,3) (0,6) (5,2) (5,4) (0,1) (4,7) (2,3)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->5->1->6->3->2->7->4."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph114",
        "image": "data/hamilton/graph_visual/easy/graph114.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (0,1) (4,7) (2,1) (8,3) (3,4) (1,4) (0,6) (7,5) (0,5) (8,7) (6,4) (7,3) (2,5)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->1->2->5->7->8->3->4->6."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph179",
        "image": "data/hamilton/graph_visual/easy/graph179.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (7,3) (4,1) (5,2) (7,6) (7,8) (6,4) (2,1) (6,3) (3,2) (8,1) (0,8) (2,6) (2,8) (5,1) (5,8) (1,3) (5,0) (2,4)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->8->7->3->6->4->1->2->5."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph225",
        "image": "data/hamilton/graph_visual/easy/graph225.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (7,6) (7,0) (1,4) (3,1) (5,2) (2,0) (8,9) (3,7) (1,2) (1,7) (5,4) (5,3) (8,1) (8,5) (0,4) (2,4) (2,7) (4,6) (6,8) (5,7) (7,8) (5,6)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->7->6->4->2->5->3->1->8->9."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph67",
        "image": "data/hamilton/graph_visual/easy/graph67.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (0,3) (5,1) (0,6) (5,6) (6,4) (0,2) (3,7) (1,2) (5,7) (7,6) (2,4)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->7->5->1->2->4->6."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph115",
        "image": "data/hamilton/graph_visual/easy/graph115.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (6,1) (3,6) (1,4) (7,1) (5,8) (6,8) (3,8) (3,1) (3,7) (4,6) (6,5) (6,2) (0,2) (1,5) (3,2) (3,5) (7,4)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->6->1->4->7->3->8->5."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph110",
        "image": "data/hamilton/graph_visual/easy/graph110.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (7,8) (3,2) (5,6) (7,5) (4,0) (4,3) (1,5) (2,0) (6,0) (7,4) (7,0) (4,6) (1,0) (6,2) (5,2) (7,3) (6,8)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->4->3->2->6->8->7->5->1."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph201",
        "image": "data/hamilton/graph_visual/easy/graph201.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (1,7) (1,8) (3,5) (7,0) (0,9) (4,6) (0,2) (6,1) (9,1) (5,7) (9,2) (6,8) (0,8) (0,4) (3,0) (7,4) (7,8) (8,2) (5,6) (6,2) (8,9) (1,4)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->7->4->1->8->9->2->6->5->3."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph166",
        "image": "data/hamilton/graph_visual/easy/graph166.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (4,3) (4,1) (1,0) (8,1) (3,7) (0,6) (5,2) (2,8) (7,1) (8,0) (6,5) (3,8) (7,8) (3,6) (5,1) (4,5) (2,7) (2,6) (2,3) (0,3) (0,7)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->1->4->3->7->8->2->5->6."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph42",
        "image": "data/hamilton/graph_visual/easy/graph42.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (6,1) (5,1) (2,5) (5,3) (4,0) (5,0) (4,1) (0,1) (3,2) (4,2) (5,6)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->4->1->6->5->2->3."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph212",
        "image": "data/hamilton/graph_visual/easy/graph212.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (4,9) (0,6) (4,0) (6,8) (8,5) (3,4) (7,4) (9,0) (7,0) (1,2) (1,4) (2,8) (0,1) (3,8) (3,5) (5,2) (3,6) (1,9) (8,1) (5,1) (9,8)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->6->8->3->5->2->1->9->4->7."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph293",
        "image": "data/hamilton/graph_visual/easy/graph293.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (7,0) (2,5) (7,9) (9,6) (5,6) (4,8) (6,1) (0,3) (5,3) (8,7) (3,4) (4,0) (0,5) (8,1) (2,7) (3,9) (6,8) (7,3) (4,1) (1,7) (1,9) (0,6) (0,1) (0,2)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->7->9->6->1->8->4->3->5->2."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph239",
        "image": "data/hamilton/graph_visual/easy/graph239.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (9,0) (0,5) (1,9) (4,2) (5,9) (0,2) (8,3) (2,5) (9,2) (0,3) (3,2) (7,6) (3,5) (6,8) (7,5) (4,3) (6,4) (5,8)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->5->7->6->8->3->4->2->9->1."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph19",
        "image": "data/hamilton/graph_visual/easy/graph19.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (2,0) (0,1) (4,5) (4,3) (4,2) (5,3) (4,1) (2,5)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->5->3->4->1."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph190",
        "image": "data/hamilton/graph_visual/easy/graph190.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (7,3) (4,5) (8,2) (4,2) (0,1) (4,6) (4,9) (5,2) (2,6) (2,7) (1,4) (8,3) (9,7) (5,8) (7,0) (4,7) (7,5) (5,0)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->1->4->6->2->5->8->3->7->9."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph30",
        "image": "data/hamilton/graph_visual/easy/graph30.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (0,6) (1,6) (5,1) (2,6) (3,4) (4,6) (4,5) (4,0) (0,2) (5,3) (6,5)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->4->3->5->1->6->2."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph69",
        "image": "data/hamilton/graph_visual/easy/graph69.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (1,6) (3,5) (5,0) (0,6) (2,1) (4,3) (6,5) (1,0) (2,6) (0,4) (7,6) (1,4) (3,2) (5,7)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->5->3->4->1->2->6->7."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph111",
        "image": "data/hamilton/graph_visual/easy/graph111.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (5,0) (6,3) (1,5) (2,5) (5,7) (7,8) (1,8) (3,8) (8,5) (8,2) (7,4) (4,6) (1,0)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->1->5->2->8->7->4->6->3."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph98",
        "image": "data/hamilton/graph_visual/easy/graph98.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (5,3) (1,5) (0,7) (7,1) (3,6) (5,7) (7,4) (4,6) (3,2) (6,7) (4,2) (0,4) (2,0) (1,0) (5,4) (3,4) (5,2) (2,1) (6,0)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->7->1->5->3->6->4->2."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph37",
        "image": "data/hamilton/graph_visual/easy/graph37.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (3,5) (2,5) (3,1) (6,1) (0,1) (6,2) (6,4)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->1->3->5->2->6->4."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph260",
        "image": "data/hamilton/graph_visual/easy/graph260.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (7,5) (4,2) (3,0) (4,1) (9,7) (6,8) (9,0) (7,1) (8,0) (4,9) (6,9) (3,1) (3,8) (6,2) (6,3) (1,2) (4,7) (6,4) (1,8) (3,9) (9,8) (4,3) (0,4) (3,5) (3,7) (0,1) (7,8) (5,8) (0,2)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->1->4->2->6->8->9->7->5."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph41",
        "image": "data/hamilton/graph_visual/easy/graph41.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (3,4) (3,2) (6,4) (5,4) (4,0) (2,6) (1,0) (4,1) (1,3) (5,0) (0,2) (6,0) (0,3) (3,6)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->1->3->2->6->4->5."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph131",
        "image": "data/hamilton/graph_visual/easy/graph131.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (3,0) (5,6) (5,4) (6,3) (1,4) (2,6) (3,8) (8,5) (8,6) (2,7) (6,1) (7,8) (0,8)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->6->2->7->8->5->4->1."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph160",
        "image": "data/hamilton/graph_visual/easy/graph160.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (2,3) (8,5) (8,2) (2,5) (8,3) (5,1) (2,7) (6,0) (3,4) (7,6) (6,5) (3,7) (3,0) (6,2) (0,8) (4,8) (2,0) (0,1) (7,8) (4,0) (1,7) (0,7) (1,6) (1,8)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->6->7->2->3->4->8->5->1."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph263",
        "image": "data/hamilton/graph_visual/easy/graph263.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (5,0) (2,8) (4,8) (0,8) (6,8) (7,0) (4,1) (4,6) (3,0) (3,9) (7,6) (0,6) (7,8) (8,5) (2,1) (4,0) (7,9) (1,9) (4,3) (0,1) (2,0) (2,7) (5,2) (3,6) (5,1) (2,9) (3,8) (2,6) (5,6) (5,4)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->5->8->2->1->4->6->7->9->3."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph186",
        "image": "data/hamilton/graph_visual/easy/graph186.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (7,1) (3,6) (4,2) (4,7) (6,2) (7,8) (6,0) (3,5) (8,5) (1,3) (1,0) (7,3) (5,0) (5,1) (2,0) (1,6) (2,3) (0,8) (5,7) (4,3) (4,5) (6,4) (0,3)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->6->3->1->7->8->5->4->2."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph71",
        "image": "data/hamilton/graph_visual/easy/graph71.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (4,3) (6,7) (1,6) (3,0) (3,1) (1,0) (7,5) (4,7) (2,6) (5,6) (0,2)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->1->3->4->7->5->6->2."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph76",
        "image": "data/hamilton/graph_visual/easy/graph76.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (7,1) (5,2) (0,2) (3,4) (6,0) (2,3) (0,4) (7,3) (0,7) (6,5) (7,6)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->4->3->2->5->6->7->1."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph200",
        "image": "data/hamilton/graph_visual/easy/graph200.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (9,8) (5,7) (6,7) (7,2) (3,1) (4,3) (1,0) (8,4) (2,0) (3,8) (4,1) (5,8) (7,1) (6,1) (9,0) (6,2) (5,3) (3,7) (6,0) (1,9) (4,2) (0,4) (8,0) (3,9) (3,2) (5,1)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->1->3->4->2->6->7->5->8->9."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph117",
        "image": "data/hamilton/graph_visual/easy/graph117.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (5,0) (8,7) (5,1) (1,0) (6,0) (6,1) (3,4) (1,7) (0,7) (1,8) (4,5) (2,7) (2,4) (5,2) (5,3)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->5->3->4->2->7->8->1->6."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph21",
        "image": "data/hamilton/graph_visual/easy/graph21.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,1) (4,1) (3,2) (4,3) (0,3) (0,4) (4,2) (1,5) (5,4)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->1->5->4->3->2."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph223",
        "image": "data/hamilton/graph_visual/easy/graph223.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (9,5) (3,9) (5,6) (5,8) (2,0) (7,4) (1,7) (6,9) (6,1) (4,5) (3,0) (0,5) (8,4) (9,8) (7,2) (4,1) (4,2) (6,7) (3,5) (1,0) (8,0) (2,9) (1,2)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->7->4->8->5->3->9->6->1."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph234",
        "image": "data/hamilton/graph_visual/easy/graph234.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (2,0) (4,8) (2,1) (9,0) (3,9) (3,4) (6,7) (6,5) (8,0) (7,8) (1,4) (7,1) (4,0) (8,3) (5,9) (1,6)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->1->4->8->7->6->5->9->3."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph242",
        "image": "data/hamilton/graph_visual/easy/graph242.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (0,1) (4,9) (4,2) (1,7) (6,9) (0,2) (7,8) (1,5) (5,3) (3,9) (2,3) (6,5) (7,4) (2,6) (5,2) (3,7) (2,7) (6,1) (4,0) (7,5) (9,7) (6,7) (0,7)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->1->5->3->9->4->2->6->7->8."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph135",
        "image": "data/hamilton/graph_visual/easy/graph135.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (1,3) (1,8) (2,1) (8,6) (4,0) (0,6) (7,6) (6,4) (5,6) (3,7) (2,3) (3,8) (8,5) (0,3) (2,6) (2,4)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->4->6->7->3->2->1->8->5."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph268",
        "image": "data/hamilton/graph_visual/easy/graph268.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (6,7) (7,3) (8,7) (9,3) (2,7) (2,1) (9,8) (3,5) (0,9) (8,1) (5,6) (5,2) (4,8) (9,2) (9,5) (7,4) (4,9) (8,3) (5,0) (0,7) (9,7) (0,2) (1,9) (3,1) (4,6)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->9->3->7->6->5->2->1->8->4."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph197",
        "image": "data/hamilton/graph_visual/easy/graph197.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (5,1) (9,2) (6,4) (6,2) (6,1) (5,3) (7,8) (8,3) (1,8) (4,3) (3,1) (3,6) (4,1) (1,7) (2,0) (8,2) (0,9)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->9->2->6->4->3->5->1->8->7."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph238",
        "image": "data/hamilton/graph_visual/easy/graph238.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (9,5) (6,4) (4,8) (1,7) (6,9) (1,6) (9,3) (2,8) (6,3) (2,4) (8,9) (6,7) (4,9) (7,3) (9,0) (1,5) (4,1) (5,3) (9,7) (8,1)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->9->5->1->7->3->6->4->8->2."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph240",
        "image": "data/hamilton/graph_visual/easy/graph240.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (2,9) (2,8) (4,3) (3,0) (0,9) (5,1) (7,2) (9,8) (7,6) (4,7) (1,2) (3,9) (0,6) (6,1) (1,7) (7,3)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->6->7->4->3->9->8->2->1->5."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph43",
        "image": "data/hamilton/graph_visual/easy/graph43.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,2) (5,1) (1,0) (0,2) (6,3) (2,5) (1,3) (3,2) (6,2) (4,3)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->1->5->2->6->3->4."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph23",
        "image": "data/hamilton/graph_visual/easy/graph23.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (3,0) (1,3) (1,2) (0,2) (3,2) (2,4) (4,3) (5,4)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->1->2->4->5."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph170",
        "image": "data/hamilton/graph_visual/easy/graph170.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (7,0) (5,7) (5,1) (3,8) (2,5) (0,1) (2,1) (0,5) (4,2) (5,8) (7,4) (3,1) (6,2) (2,3) (5,4) (0,8) (3,6) (3,4) (1,4) (2,0) (6,7) (8,6) (1,6)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->7->5->1->2->4->3->8->6."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph193",
        "image": "data/hamilton/graph_visual/easy/graph193.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (0,2) (0,5) (1,4) (8,2) (8,5) (0,6) (9,3) (1,6) (0,9) (9,6) (5,1) (8,0) (3,6) (9,7) (9,1) (0,3) (2,3) (6,8) (3,8) (4,3)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->8->5->1->4->3->6->9->7."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph177",
        "image": "data/hamilton/graph_visual/easy/graph177.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (7,5) (4,2) (1,0) (8,0) (1,3) (6,5) (8,4) (2,5) (7,8) (3,5) (6,8) (3,6) (8,5) (2,7) (7,1) (0,4) (1,8) (2,3) (2,0) (0,7) (4,1) (2,6)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->1->3->5->7->8->4->2->6."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph152",
        "image": "data/hamilton/graph_visual/easy/graph152.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (2,4) (5,2) (4,6) (0,3) (8,6) (6,3) (5,3) (1,8) (2,0) (8,5) (2,6) (0,1) (0,8) (0,6) (8,2) (7,0) (8,7) (1,2) (1,4) (3,4) (4,0)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->6->4->1->2->5->8->7."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph49",
        "image": "data/hamilton/graph_visual/easy/graph49.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (2,5) (1,6) (2,6) (2,3) (4,0) (2,1) (4,1) (5,3) (4,6) (3,1) (4,5) (5,0)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->4->1->6->2->5->3."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph133",
        "image": "data/hamilton/graph_visual/easy/graph133.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (6,5) (7,4) (0,6) (0,5) (0,1) (1,7) (4,3) (2,7) (1,3) (2,4) (6,8) (5,1) (5,2)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->1->3->4->7->2->5->6->8."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph142",
        "image": "data/hamilton/graph_visual/easy/graph142.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (6,8) (1,7) (5,7) (3,2) (5,1) (7,6) (7,4) (6,5) (2,8) (0,2) (4,1) (5,3) (0,5) (3,0) (2,7) (4,3)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->3->5->1->4->7->6->8."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph269",
        "image": "data/hamilton/graph_visual/easy/graph269.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (1,4) (8,0) (3,1) (4,8) (7,9) (9,5) (3,8) (3,0) (7,3) (3,2) (9,8) (3,6) (8,2) (4,5) (9,6) (1,2) (4,9) (7,0) (3,4) (3,5) (0,1) (5,6) (9,2) (7,1) (5,8) (6,2)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->8->4->1->3->7->9->5->6->2."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph2",
        "image": "data/hamilton/graph_visual/easy/graph2.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,3) (3,2) (2,0) (1,4) (4,2)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->3->4->1."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph191",
        "image": "data/hamilton/graph_visual/easy/graph191.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (7,2) (3,8) (8,9) (5,7) (2,4) (4,7) (2,6) (4,0) (1,7) (5,3) (3,2) (6,3) (9,4) (5,4) (9,5) (0,8) (3,4)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->4->2->6->3->8->9->5->7->1."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph1",
        "image": "data/hamilton/graph_visual/easy/graph1.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (2,1) (4,1) (1,3) (3,4) (0,2)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->1->4->3."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph65",
        "image": "data/hamilton/graph_visual/easy/graph65.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (0,7) (3,6) (2,0) (7,1) (3,4) (1,0) (2,3) (0,6) (4,5) (2,7) (2,1) (6,7) (7,4)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->1->7->6->3->4->5."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph209",
        "image": "data/hamilton/graph_visual/easy/graph209.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (5,9) (7,8) (7,3) (9,6) (8,5) (8,4) (9,2) (6,2) (0,5) (1,3) (0,9) (0,8) (1,7) (3,0) (4,6) (9,3) (0,2) (1,6) (6,5) (1,4) (1,2) (5,1) (0,6) (4,2)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->5->9->6->2->1->3->7->8->4."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph12",
        "image": "data/hamilton/graph_visual/easy/graph12.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (3,5) (4,3) (1,3) (4,5) (4,2) (4,0) (2,5)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->4->2->5->3->1."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph148",
        "image": "data/hamilton/graph_visual/easy/graph148.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (0,6) (2,6) (1,5) (0,2) (0,7) (2,8) (0,4) (4,8) (4,1) (1,8) (3,1) (5,4) (4,6) (5,7) (7,4) (0,8) (3,7)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->6->2->8->4->1->5->7->3."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph123",
        "image": "data/hamilton/graph_visual/easy/graph123.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (8,6) (0,8) (8,4) (0,2) (5,6) (7,3) (7,8) (2,4) (6,7) (5,1) (4,1) (8,1) (3,4) (7,4) (0,3) (3,6) (0,6)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->8->7->3->6->5->1->4->2."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph13",
        "image": "data/hamilton/graph_visual/easy/graph13.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (2,5) (1,2) (4,3) (3,1) (0,1) (5,3) (1,5) (4,0) (4,5) (2,3) (4,1)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->1->2->5->3->4."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph100",
        "image": "data/hamilton/graph_visual/easy/graph100.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (3,5) (5,6) (2,0) (4,3) (6,1) (6,7) (4,1) (2,4) (2,6) (1,0) (5,2) (4,5) (7,0) (2,3) (3,0)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->5->3->4->1->6->7."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph40",
        "image": "data/hamilton/graph_visual/easy/graph40.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (0,6) (5,6) (3,4) (1,5) (4,6) (1,3) (6,2) (5,0) (0,3) (2,3) (4,5) (3,6) (1,4) (3,5) (1,2)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->6->5->1->4->3->2."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph139",
        "image": "data/hamilton/graph_visual/easy/graph139.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (2,1) (7,1) (6,7) (8,0) (6,3) (8,5) (8,4) (0,5) (2,5) (7,5) (2,0) (0,4) (6,2) (3,1) (7,0) (7,2) (5,3) (5,1)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->1->7->6->3->5->8->4."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph224",
        "image": "data/hamilton/graph_visual/easy/graph224.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (4,2) (5,4) (2,0) (8,3) (7,6) (3,5) (6,8) (5,6) (7,8) (1,6) (7,9) (1,8) (2,6) (5,7) (8,9) (2,7) (2,5) (3,4) (1,4) (2,9) (4,0) (9,0)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->4->5->3->8->1->6->7->9."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph112",
        "image": "data/hamilton/graph_visual/easy/graph112.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (7,0) (8,5) (4,3) (0,8) (7,8) (0,1) (5,0) (4,6) (2,6) (1,5) (4,5) (2,5) (8,6) (3,8) (4,0) (7,6) (2,7) (8,2)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->7->8->3->4->6->2->5->1."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph198",
        "image": "data/hamilton/graph_visual/easy/graph198.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (1,8) (3,9) (6,1) (9,5) (2,4) (4,9) (4,8) (6,3) (4,3) (5,7) (6,5) (0,2) (6,7) (2,5) (0,9) (8,7) (9,2)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->4->9->3->6->1->8->7->5."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph143",
        "image": "data/hamilton/graph_visual/easy/graph143.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (3,7) (8,6) (8,2) (6,1) (1,4) (0,3) (6,3) (6,2) (5,1) (5,4) (6,0) (5,6) (2,7) (3,5) (7,6) (5,8)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->7->2->8->6->1->4->5."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph265",
        "image": "data/hamilton/graph_visual/easy/graph265.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (1,4) (8,3) (5,6) (8,7) (1,2) (4,3) (4,8) (5,4) (6,0) (9,0) (1,7) (1,9) (6,3) (2,4) (8,0) (4,9) (8,1) (0,2) (6,7) (7,3) (9,3) (1,3) (3,5) (4,6) (0,1) (2,6) (3,0) (5,2) (9,2)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->6->5->4->1->2->9->3->8->7."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph78",
        "image": "data/hamilton/graph_visual/easy/graph78.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (5,6) (3,0) (5,7) (7,0) (3,1) (0,2) (3,2) (4,2) (2,5) (4,1) (1,7) (2,6)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->1->4->2->6->5->7."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph126",
        "image": "data/hamilton/graph_visual/easy/graph126.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (7,3) (8,7) (4,2) (5,8) (1,3) (5,0) (2,1) (7,4) (6,0) (6,7) (5,6) (4,6) (2,0)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->5->8->7->3->1->2->4->6."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph106",
        "image": "data/hamilton/graph_visual/easy/graph106.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (7,0) (5,8) (8,3) (4,7) (3,2) (3,6) (3,5) (0,5) (2,4) (2,6) (8,2) (7,8) (0,2) (3,0) (6,5) (0,4) (4,1)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->7->8->5->3->6->2->4->1."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph137",
        "image": "data/hamilton/graph_visual/easy/graph137.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (6,1) (7,6) (8,3) (2,1) (2,4) (3,1) (2,6) (1,5) (5,0) (7,2) (8,7) (4,5) (8,4) (0,3) (4,7) (6,0) (8,2) (7,5)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->5->1->6->7->2->4->8->3."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph90",
        "image": "data/hamilton/graph_visual/easy/graph90.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (3,2) (3,6) (2,7) (2,6) (1,7) (1,0) (7,0) (1,6) (5,3) (0,6) (7,6) (4,6) (2,5) (5,4)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->1->7->2->3->6->4->5."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph53",
        "image": "data/hamilton/graph_visual/easy/graph53.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (3,0) (6,2) (2,4) (6,4) (3,2) (1,0) (2,5) (3,5) (5,6) (1,5) (4,5) (0,4) (6,0) (5,0) (4,1)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->2->6->4->5->1."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph228",
        "image": "data/hamilton/graph_visual/easy/graph228.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (1,6) (5,2) (0,2) (7,2) (1,2) (4,5) (1,0) (7,1) (6,8) (5,8) (5,0) (1,5) (9,8) (1,8) (6,3) (3,1) (6,4)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->7->1->3->6->4->5->8->9."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph282",
        "image": "data/hamilton/graph_visual/easy/graph282.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (9,4) (0,2) (8,9) (3,9) (3,2) (7,8) (1,5) (3,4) (6,2) (7,6) (4,5) (7,9) (5,6) (9,6) (0,6) (1,8) (3,6) (4,7) (7,0) (1,6) (1,4) (5,8) (0,3) (4,6) (1,9) (4,2) (3,7) (9,0) (5,9) (1,2)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->3->9->4->5->1->8->7->6."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph64",
        "image": "data/hamilton/graph_visual/easy/graph64.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,6) (5,6) (2,3) (7,5) (5,2) (7,0) (1,3) (4,7) (3,0) (1,0)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->1->3->2->6->5->7->4."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph175",
        "image": "data/hamilton/graph_visual/easy/graph175.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (4,5) (3,5) (1,7) (3,8) (6,4) (7,4) (2,4) (7,0) (0,8) (0,4) (5,0) (6,8) (0,2) (7,6) (4,1)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->5->3->8->6->7->1->4->2."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph61",
        "image": "data/hamilton/graph_visual/easy/graph61.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (4,1) (3,2) (7,6) (1,6) (3,6) (5,0) (2,0) (0,4) (4,5) (7,4) (3,1) (7,3) (7,1) (2,1)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->5->4->1->6->7->3->2."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph174",
        "image": "data/hamilton/graph_visual/easy/graph174.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (4,1) (1,8) (4,8) (1,7) (7,8) (2,1) (7,6) (3,2) (3,0) (5,6) (6,2) (6,8) (0,7) (3,6) (8,5) (7,2) (3,8) (2,0) (5,3) (3,1) (7,5) (3,7) (0,6) (0,8)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->2->1->4->8->7->6->5."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph44",
        "image": "data/hamilton/graph_visual/easy/graph44.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (2,6) (1,4) (3,6) (1,6) (3,0) (2,4) (3,5) (4,6) (5,4) (3,2) (1,3) (2,0) (0,4) (5,6) (1,5)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->6->2->4->1->5."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph203",
        "image": "data/hamilton/graph_visual/easy/graph203.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (0,3) (4,2) (6,3) (7,1) (0,5) (5,4) (1,2) (9,3) (5,1) (4,6) (8,0) (2,9) (5,9) (3,8) (9,4) (4,3) (9,1)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->8->3->6->4->2->9->5->1->7."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph188",
        "image": "data/hamilton/graph_visual/easy/graph188.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (3,8) (2,8) (3,5) (1,7) (7,8) (4,9) (0,3) (0,2) (9,7) (2,9) (6,8) (3,7) (6,9) (7,4) (2,1) (4,8)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->1->7->4->9->6->8->3->5."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph214",
        "image": "data/hamilton/graph_visual/easy/graph214.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (4,1) (9,0) (1,8) (0,7) (2,9) (2,7) (6,9) (7,1) (5,4) (5,8) (7,8) (9,8) (5,7) (4,6) (9,4) (3,4) (2,3) (5,1) (0,4) (8,3) (0,3) (6,2) (3,5) (3,1) (9,3)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->9->2->7->1->8->5->3->4->6."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph140",
        "image": "data/hamilton/graph_visual/easy/graph140.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (5,7) (5,0) (6,1) (1,4) (1,3) (1,5) (0,6) (7,4) (6,5) (7,0) (5,8) (4,5) (6,2) (5,3) (4,3) (8,7) (3,0) (8,4)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->5->7->8->4->3->1->6->2."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph9",
        "image": "data/hamilton/graph_visual/easy/graph9.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (3,5) (5,2) (0,4) (2,0) (2,4) (1,3) (0,1)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->4->2->5->3->1."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph248",
        "image": "data/hamilton/graph_visual/easy/graph248.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (3,2) (4,3) (0,3) (3,7) (4,1) (8,2) (9,8) (5,3) (6,2) (4,6) (9,5) (8,5) (6,3) (9,3) (9,4) (1,0) (2,9) (4,0) (9,7) (7,1) (7,4) (7,8) (8,3) (8,6) (5,7)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->2->8->9->5->7->1->4->6."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph237",
        "image": "data/hamilton/graph_visual/easy/graph237.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (8,0) (7,1) (5,9) (2,9) (2,5) (7,5) (8,1) (6,7) (0,6) (4,8) (0,3) (7,4) (8,5) (2,8) (8,7) (5,3) (2,3) (4,0) (9,0) (0,5) (1,9)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->6->7->1->9->5->3->2->8->4."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph218",
        "image": "data/hamilton/graph_visual/easy/graph218.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (1,4) (1,2) (7,1) (8,4) (9,5) (2,6) (0,2) (0,3) (9,7) (2,5) (0,9) (6,9) (4,9) (5,4) (8,7) (5,8) (3,6) (5,6)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->1->4->8->7->9->5->6->3."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph192",
        "image": "data/hamilton/graph_visual/easy/graph192.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (3,4) (1,0) (8,4) (1,3) (5,7) (1,9) (5,9) (3,5) (3,7) (2,3) (3,8) (0,8) (9,2) (8,6) (4,2) (0,6)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->1->3->7->5->9->2->4->8->6."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph207",
        "image": "data/hamilton/graph_visual/easy/graph207.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (1,9) (0,6) (7,8) (5,9) (6,2) (0,3) (9,6) (2,0) (2,3) (5,8) (3,7) (7,0) (7,4) (2,5) (7,2) (8,9) (7,9) (4,8)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->6->2->3->7->4->8->5->9->1."
            }
        ]
    },
    {
        "id": "hamilton-easy-graph51",
        "image": "data/hamilton/graph_visual/easy/graph51.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (2,3) (4,2) (4,3) (4,6) (6,1) (4,0) (2,0) (1,5) (2,5) (1,4) (0,5)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->4->3->2->5->1->6."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph429",
        "image": "data/hamilton/graph_visual/hard/graph429.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 17, and the edges are: (0,9) (6,16) (15,3) (3,14) (6,9) (5,15) (5,11) (14,16) (5,13) (4,11) (7,4) (9,16) (8,7) (9,15) (1,0) (12,15) (6,7) (6,1) (4,16) (10,14) (8,9) (6,4) (5,0) (2,15) (1,11) (9,11) (11,17) (13,9) (0,6) (11,12) (3,2) (7,13) (2,11) (6,3) (7,10) (2,9) (5,2) (8,6) (10,16) (16,2) (17,15) (14,12) (16,0) (11,13) (0,10) (4,17) (3,7) (13,8) (14,11) (10,8) (6,14) (8,16) (13,12) (6,13) (12,17)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->9->6->16->14->3->15->12->17->4->7->10->8->13->5->2->11->1."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph555",
        "image": "data/hamilton/graph_visual/hard/graph555.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 19, and the edges are: (10,1) (18,5) (5,9) (9,15) (8,19) (4,10) (14,7) (0,14) (14,8) (4,9) (11,15) (11,7) (9,0) (3,18) (10,12) (9,12) (5,16) (4,12) (15,3) (9,3) (8,10) (9,14) (5,14) (17,3) (0,1) (14,15) (13,3) (9,8) (19,14) (18,17) (2,13) (7,5) (16,18) (3,4) (16,6) (8,3) (2,10) (11,1) (0,17) (4,11) (3,0) (9,10) (14,17) (1,7) (13,18) (4,0) (16,9) (7,4) (6,10) (18,0) (11,17) (3,19) (0,10) (15,17) (1,14) (19,12) (7,8) (5,19) (12,2) (13,0) (18,1) (16,15) (17,12) (16,19) (14,13) (19,6) (14,10) (5,6) (2,0) (16,14) (3,12)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->14->7->11->15->9->5->16->6->19->8->10->1->18->17->3->13->2->12->4."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph5",
        "image": "data/hamilton/graph_visual/hard/graph5.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 10, and the edges are: (6,2) (9,6) (0,2) (7,3) (9,1) (8,2) (5,7) (0,3) (7,0) (3,10) (4,5) (9,10) (8,9) (7,2) (4,3) (2,4) (3,2) (1,10) (8,6) (9,4) (1,5) (6,0) (7,8)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->6->9->1->10->3->4->5->7->8."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph345",
        "image": "data/hamilton/graph_visual/hard/graph345.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (14,12) (11,8) (0,2) (3,0) (6,11) (12,8) (4,6) (9,2) (10,1) (13,9) (5,0) (14,0) (2,13) (5,15) (4,7) (4,14) (6,10) (5,8) (8,7) (9,1) (3,6) (1,13) (1,15) (3,13) (14,11) (12,3) (13,14) (7,11) (11,1) (15,3) (10,12) (0,6) (6,2) (10,11) (8,1) (9,3) (11,13) (14,3) (10,9) (14,7) (1,2) (3,2) (5,2) (3,8) (9,8) (13,15) (11,4) (1,7) (9,5) (12,13) (7,10) (10,8) (0,8) (5,14) (3,4) (3,7) (11,12) (12,9) (10,14) (7,13) (1,0) (15,12) (2,14) (14,6) (14,15) (14,8) (1,3) (2,15) (0,13) (12,7) (8,13) (13,4) (7,9) (15,11) (10,3) (9,0) (0,15) (5,11) (6,1)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->9->13->1->10->6->11->8->12->14->4->7->3->15->5."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph575",
        "image": "data/hamilton/graph_visual/hard/graph575.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 19, and the edges are: (3,14) (18,13) (10,9) (11,1) (12,8) (16,17) (11,10) (1,9) (19,15) (0,2) (11,7) (17,12) (5,12) (12,6) (16,2) (12,14) (18,1) (7,4) (18,5) (2,6) (10,19) (2,12) (8,2) (11,3) (1,10) (5,19) (2,11) (17,15) (15,7) (4,11) (12,3) (3,15) (17,2) (14,8) (9,17) (1,2) (3,18) (18,12) (12,4) (5,14) (14,10) (0,13) (19,9) (10,16) (9,7) (9,2) (16,8) (3,7) (18,8) (11,8) (6,7) (0,15) (9,14) (0,9) (8,17) (1,3) (8,5) (16,18) (6,16) (10,12) (0,10) (19,3) (3,4) (1,6) (6,0) (11,16) (14,13) (1,17) (14,1) (18,19) (9,18) (7,16) (0,4) (16,19) (0,7) (18,0) (16,14) (10,3) (17,0) (16,9) (2,7) (6,3) (7,5) (12,15) (8,9) (8,4) (6,4) (7,18) (15,16) (15,14) (5,0) (8,6) (0,12) (5,3) (4,9) (7,8) (16,1) (12,11) (12,1) (19,13) (13,6) (2,13) (4,14) (8,10) (12,7) (14,17) (12,13) (10,18) (1,0) (4,13) (5,6) (17,5) (10,17) (16,3) (2,18)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->16->17->12->8->14->3->11->1->9->10->19->15->7->4->6->13->18->5."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph72",
        "image": "data/hamilton/graph_visual/hard/graph72.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 11, and the edges are: (7,4) (9,4) (10,9) (5,11) (3,1) (8,1) (7,9) (9,3) (1,0) (3,7) (3,10) (6,10) (9,6) (7,2) (5,8) (11,2) (8,4) (11,10) (5,0) (5,1) (10,0) (2,9) (7,0) (1,4) (3,2) (6,3) (4,5) (8,7) (1,6) (5,3)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->1->3->9->4->5->8->7->2->11->10->6."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph186",
        "image": "data/hamilton/graph_visual/hard/graph186.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 13, and the edges are: (6,5) (12,11) (6,2) (4,8) (7,8) (3,4) (8,12) (12,3) (11,7) (5,2) (8,0) (3,5) (3,13) (3,0) (4,0) (13,1) (9,2) (5,4) (5,9) (12,4) (6,4) (2,8) (6,9) (1,2) (6,8) (3,8) (5,13) (10,9) (13,4) (1,12) (3,2) (1,8) (3,11) (0,11) (7,12) (12,0) (5,10)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->8->4->3->13->5->10->9->6->2->1->12->11->7."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph152",
        "image": "data/hamilton/graph_visual/hard/graph152.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 12, and the edges are: (5,10) (3,9) (12,0) (12,6) (1,8) (12,3) (11,1) (1,2) (0,7) (6,0) (3,7) (4,0) (3,11) (0,3) (12,4) (5,1) (9,4) (2,0) (10,9) (9,6) (11,7) (2,8) (2,11) (4,1) (6,5) (2,3) (8,7) (7,4) (0,8) (2,4) (1,12) (2,7) (7,9) (5,4) (9,11) (0,11) (10,12) (9,0) (11,8) (11,10) (11,4) (9,12) (6,2) (6,1) (10,3)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->12->6->9->3->7->11->1->8->2->4->5->10."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph494",
        "image": "data/hamilton/graph_visual/hard/graph494.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (10,17) (5,9) (13,0) (1,5) (12,16) (13,11) (1,8) (12,6) (17,4) (0,6) (12,13) (16,5) (0,12) (18,12) (11,14) (1,11) (4,1) (14,0) (12,14) (3,17) (9,4) (9,2) (18,4) (6,2) (12,11) (10,15) (15,5) (3,1) (6,13) (0,15) (14,17) (6,10) (10,14) (12,1) (15,11) (1,13) (10,11) (11,9) (12,2) (14,7) (6,15) (5,12) (0,18) (1,0) (6,11) (14,4) (7,12) (2,1) (8,9) (7,9) (18,8) (5,2) (0,4) (5,7) (7,0) (9,1) (3,13) (1,14) (17,16) (4,8) (0,17) (12,8) (2,10)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->13->11->14->12->16->5->15->10->6->2->1->3->17->4->18->8->9->7."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph314",
        "image": "data/hamilton/graph_visual/hard/graph314.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (11,1) (1,2) (7,1) (4,8) (1,13) (0,3) (4,10) (13,4) (10,3) (15,13) (15,4) (10,5) (14,11) (11,15) (11,9) (10,1) (13,11) (15,2) (5,8) (12,10) (6,1) (8,15) (11,8) (9,15) (2,4) (8,9) (1,9) (11,0) (5,7) (2,0) (10,9) (9,6) (3,14) (12,1) (6,8) (6,3) (1,4) (9,3) (9,14) (12,9) (4,0) (5,1) (12,13) (1,15) (4,7) (7,11) (9,7) (3,8) (11,12) (2,9) (12,6)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->10->4->8->5->7->1->11->14->9->6->12->13->15->2."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph394",
        "image": "data/hamilton/graph_visual/hard/graph394.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 16, and the edges are: (12,3) (15,4) (16,4) (0,3) (15,16) (13,16) (13,11) (14,7) (8,1) (14,15) (1,5) (5,12) (16,0) (10,11) (3,1) (2,8) (8,0) (16,11) (3,6) (11,8) (1,12) (15,8) (0,5) (9,7) (9,10) (3,13) (14,1) (16,3) (7,12) (4,13) (6,12) (1,16) (10,13) (9,16) (14,0) (16,10) (16,8) (4,7) (7,8) (5,13) (15,7) (3,9) (13,0) (4,9) (2,5) (3,7) (2,14) (10,2) (8,14) (6,11) (6,2) (6,15) (3,8) (6,14) (2,0) (2,1) (4,8) (5,4) (12,10) (6,9) (10,0) (6,0) (12,14) (2,12) (11,0) (10,14) (5,3) (11,2) (14,5) (5,16) (0,9) (2,13) (15,10) (7,1) (2,3)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->12->5->1->8->2->14->7->9->10->11->13->16->4->15->6."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph221",
        "image": "data/hamilton/graph_visual/hard/graph221.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 13, and the edges are: (11,3) (6,1) (12,3) (5,1) (9,10) (6,9) (10,2) (8,4) (2,5) (9,5) (10,0) (5,12) (9,1) (4,2) (3,8) (13,5) (10,12) (12,2) (3,9) (3,13) (11,9) (7,2) (4,9) (3,2) (6,10) (13,8) (11,4) (8,6) (8,7) (10,1) (9,2) (4,0) (3,6) (0,9) (1,7) (5,3) (13,6) (8,0) (5,11) (12,4) (0,3) (9,13) (1,3) (11,10) (5,0) (13,11) (13,0) (7,12) (12,0) (13,10) (7,11) (1,13) (7,4)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->10->9->6->1->5->2->4->8->3->12->7->11->13."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph59",
        "image": "data/hamilton/graph_visual/hard/graph59.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 10, and the edges are: (1,6) (0,4) (0,2) (5,9) (8,0) (10,0) (5,7) (0,3) (4,5) (6,10) (0,1) (4,2) (8,9) (6,0) (5,1) (7,4) (2,5) (9,0) (3,8) (2,6) (8,2) (4,6) (2,9) (3,5) (6,9) (9,3) (2,1) (1,9) (10,8) (6,5) (3,4) (1,3) (10,7) (10,1) (8,1) (0,5) (10,2) (8,4) (9,10) (5,8) (10,3)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->4->5->9->8->3->1->6->2->10->7."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph256",
        "image": "data/hamilton/graph_visual/hard/graph256.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 14, and the edges are: (3,7) (8,10) (0,9) (1,9) (5,4) (9,8) (14,0) (4,9) (5,7) (12,1) (11,8) (12,6) (3,1) (4,10) (14,4) (7,11) (2,11) (9,5) (0,4) (6,8) (10,3) (6,13) (10,7) (11,0) (14,8) (3,4) (3,9) (3,11) (10,5) (8,1) (13,5) (7,9)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->9->1->12->6->13->5->4->14->8->10->3->7->11->2."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph144",
        "image": "data/hamilton/graph_visual/hard/graph144.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 12, and the edges are: (10,4) (9,6) (9,12) (4,5) (2,0) (2,12) (1,3) (5,11) (8,1) (2,7) (12,1) (5,0) (9,2) (9,10) (4,2) (6,3) (0,8) (12,8) (12,11) (8,3) (3,4) (3,2) (12,6) (12,4) (1,5) (9,11) (5,12) (4,1) (7,0) (6,4) (9,1) (2,10) (7,10) (12,3)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->12->9->11->5->1->8->3->6->4->10->7."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph307",
        "image": "data/hamilton/graph_visual/hard/graph307.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (13,15) (6,7) (15,1) (6,0) (15,5) (3,15) (13,14) (8,12) (4,15) (0,15) (8,15) (9,5) (2,7) (9,11) (7,8) (7,13) (4,2) (11,10) (4,7) (10,14) (6,1) (3,12) (15,6) (6,11) (4,9) (3,13) (8,13) (4,12) (7,9) (12,5) (12,2) (4,6) (7,0) (14,4) (4,0) (12,13) (12,15) (5,13) (6,12)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->6->7->2->4->14->10->11->9->5->12->8->13->3->15->1."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph497",
        "image": "data/hamilton/graph_visual/hard/graph497.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (4,3) (12,15) (11,8) (1,9) (18,13) (17,13) (6,0) (7,15) (15,9) (6,9) (14,5) (2,5) (7,17) (12,9) (2,6) (5,0) (5,12) (12,6) (8,18) (7,4) (6,18) (7,12) (9,10) (1,13) (3,12) (7,11) (3,11) (10,14) (6,3) (17,11) (6,16) (10,1) (0,1) (15,6) (15,0) (13,7) (2,12) (11,6) (15,14) (13,8) (7,18) (3,10) (9,5) (5,17) (14,6) (2,13) (8,0) (9,2) (1,5) (11,5) (6,7) (1,2) (9,7) (14,18) (2,16) (17,3) (17,9) (16,7) (1,11) (0,17) (2,8) (2,7) (16,4) (14,9) (8,14) (12,4) (4,14) (11,14) (14,13) (10,16) (4,13) (4,1) (7,8) (4,17) (3,2) (13,15) (1,18) (16,0) (15,18) (8,10) (18,16) (16,11) (0,13) (6,4) (0,3) (7,10) (16,13) (12,17)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->6->9->1->13->18->8->11->7->15->12->5->14->10->3->17->4->16->2."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph197",
        "image": "data/hamilton/graph_visual/hard/graph197.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 13, and the edges are: (3,6) (3,12) (4,3) (13,7) (1,7) (11,7) (5,11) (9,5) (10,11) (8,6) (7,8) (9,11) (4,0) (8,12) (11,6) (9,2) (0,2) (8,4) (8,0) (6,1) (1,4) (10,1) (13,0) (4,10) (11,4) (2,13) (12,7) (2,4) (13,4) (13,11) (13,1) (10,12) (3,8) (3,1) (12,9) (5,6) (2,7)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->4->3->6->8->7->1->10->12->9->5->11->13->2."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph170",
        "image": "data/hamilton/graph_visual/hard/graph170.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 12, and the edges are: (6,9) (2,8) (2,7) (6,2) (5,3) (10,9) (6,12) (6,8) (12,0) (8,11) (6,3) (11,10) (5,2) (7,4) (8,9) (3,9) (2,0) (12,9) (3,8) (4,12) (7,1) (7,9) (8,4) (12,7) (8,12) (11,1) (4,3) (11,2) (2,3) (4,10) (1,0) (5,10) (2,4) (7,5) (8,7) (0,5) (7,0) (7,11) (10,3) (3,7) (10,12) (12,3) (11,6) (3,11) (11,9) (5,12) (4,11)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->12->6->9->10->11->8->2->5->3->4->7->1."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph581",
        "image": "data/hamilton/graph_visual/hard/graph581.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 19, and the edges are: (1,11) (15,2) (15,5) (10,4) (16,19) (15,16) (14,12) (1,9) (8,5) (15,9) (14,8) (6,12) (5,2) (13,4) (0,4) (18,0) (3,17) (1,8) (12,4) (15,7) (14,2) (6,19) (18,7) (1,4) (1,17) (7,8) (0,6) (10,7) (18,1) (1,13) (7,19) (3,0) (0,11) (2,7) (18,6) (4,18) (9,14) (8,6) (15,10) (2,3) (7,13) (2,11) (9,11) (13,17) (19,14) (19,1) (6,11) (10,2) (15,4) (15,13) (4,3) (5,6) (16,10) (9,8) (3,18) (17,0) (12,16) (3,7) (17,10) (2,17) (19,3) (18,15) (3,11) (10,8) (18,5) (18,17) (17,12) (11,12) (2,16) (15,11) (18,2) (3,13) (4,7) (10,0) (9,0) (6,7) (16,8) (11,13) (8,11) (19,0) (9,4) (0,13) (13,10) (9,12) (19,17) (5,14) (17,11) (1,16) (13,14) (8,0) (4,17) (15,17) (1,10) (14,0) (11,4) (6,10) (6,17) (12,19) (19,13) (14,7) (7,12) (5,4) (5,17) (18,11) (3,9) (18,9) (8,13) (16,6) (16,11) (4,2) (15,19) (18,14) (19,10) (8,2) (16,14) (0,5) (3,1) (6,4) (3,5) (11,19) (14,4)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->4->10->7->15->2->5->8->14->12->6->19->16->1->11->9->3->18->17->13."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph111",
        "image": "data/hamilton/graph_visual/hard/graph111.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 11, and the edges are: (3,9) (11,10) (10,9) (6,10) (8,7) (1,2) (3,1) (5,10) (9,8) (5,4) (11,2) (8,10) (8,4) (8,2) (4,1) (9,5) (4,9) (0,6) (0,2) (1,5) (2,3) (7,4) (3,6) (4,2) (11,6) (1,10) (9,0) (10,3) (6,5) (9,1) (0,5) (11,3)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->6->10->11->2->1->3->9->8->7->4->5."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph293",
        "image": "data/hamilton/graph_visual/hard/graph293.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 14, and the edges are: (13,0) (0,2) (3,11) (12,3) (5,0) (0,8) (12,9) (11,9) (2,12) (7,3) (13,1) (5,10) (8,9) (5,11) (12,7) (9,0) (7,4) (11,8) (8,5) (6,14) (1,10) (8,14) (8,10) (6,1) (5,9) (7,13) (3,9) (8,4) (2,10) (10,0) (1,4) (4,2) (2,1) (9,10) (6,3) (6,2) (7,1) (8,12) (4,11) (0,1) (8,3) (10,3) (7,0) (6,4) (14,10) (11,14) (2,14) (2,13) (14,7) (1,12) (13,4) (9,6) (4,3) (10,4) (4,5) (9,13) (11,7) (12,13) (12,4) (14,0) (5,6) (1,11) (13,8) (11,6) (5,3) (2,7) (10,6) (6,12) (5,2) (13,11) (6,8) (0,11) (7,5)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->13->1->10->5->11->3->12->9->8->14->6->2->4->7."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph569",
        "image": "data/hamilton/graph_visual/hard/graph569.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 19, and the edges are: (6,18) (7,17) (4,5) (14,12) (0,7) (2,1) (19,5) (2,0) (18,11) (7,10) (12,18) (8,18) (12,17) (16,12) (9,14) (0,19) (17,15) (8,11) (0,8) (9,8) (7,1) (11,7) (4,11) (17,0) (18,7) (9,0) (6,9) (19,17) (11,13) (5,16) (7,9) (18,19) (8,16) (6,0) (12,6) (4,13) (8,10) (7,19) (15,4) (14,8) (4,19) (8,12) (4,1) (3,1) (2,13) (11,15) (6,2) (15,3) (0,13) (9,2) (6,17) (12,5) (5,17) (18,0) (0,1) (0,16) (13,15) (16,14) (1,9) (10,13) (5,10) (13,6) (8,3) (3,9) (4,9) (2,14) (12,2) (2,17) (10,11) (16,9) (0,5) (1,10) (13,1) (2,16) (17,4) (0,3) (14,0) (10,19) (12,10)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->7->17->12->14->9->8->18->6->2->1->3->15->4->11->13->10->19->5->16."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph125",
        "image": "data/hamilton/graph_visual/hard/graph125.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 12, and the edges are: (1,0) (9,3) (7,3) (0,10) (12,9) (2,7) (4,7) (2,0) (2,8) (10,8) (8,7) (6,5) (6,2) (4,8) (7,0) (6,8) (9,8) (1,2) (3,10) (7,5) (0,8) (4,6) (5,3) (10,4) (9,5) (3,4) (8,5) (0,11) (6,11) (6,1) (3,0)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->11->6->1->2->7->3->10->4->8->5->9->12."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph189",
        "image": "data/hamilton/graph_visual/hard/graph189.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 13, and the edges are: (12,13) (2,11) (4,9) (10,2) (11,12) (2,5) (13,1) (9,12) (6,11) (3,1) (6,13) (7,12) (8,5) (5,3) (9,11) (6,7) (3,13) (8,13) (11,1) (5,1) (6,5) (5,0) (5,4) (10,13) (8,10) (0,13) (10,9) (2,9) (10,0) (6,8) (3,12) (2,1) (9,6) (7,1) (0,2) (6,3) (7,13) (2,6) (12,10) (8,1)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->5->2->11->12->13->3->1->7->6->8->10->9->4."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph426",
        "image": "data/hamilton/graph_visual/hard/graph426.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 17, and the edges are: (13,4) (12,1) (14,13) (3,16) (12,0) (11,17) (14,1) (12,15) (0,3) (11,6) (13,6) (15,7) (17,12) (0,9) (9,14) (6,4) (12,14) (4,15) (2,14) (13,0) (1,16) (4,1) (5,9) (11,8) (4,9) (7,6) (17,4) (6,0) (2,13) (16,15) (5,0) (14,8) (9,15) (2,10) (2,3) (10,14) (8,2) (8,0) (6,8) (16,10) (5,13) (16,7) (17,15) (8,15) (13,7) (8,12) (6,1) (13,16) (15,2) (8,7) (11,2) (16,0) (11,9) (9,1) (11,5) (11,7) (5,8) (7,5) (13,15) (10,4) (11,4) (12,6) (6,5) (1,10) (9,6) (13,17) (11,16) (3,6) (9,2) (1,17)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->12->1->14->13->4->6->11->17->15->7->8->5->9->2->10->16->3."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph132",
        "image": "data/hamilton/graph_visual/hard/graph132.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 12, and the edges are: (4,10) (8,10) (2,9) (1,9) (6,3) (6,9) (8,0) (10,0) (12,9) (0,7) (2,7) (8,11) (2,8) (9,10) (4,12) (1,12) (2,3) (9,5) (2,5) (4,2) (9,11) (0,9) (12,8) (11,6) (12,10) (5,12) (4,11) (3,5) (6,12) (5,0) (3,4) (7,5)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->8->10->4->12->1->9->2->7->5->3->6->11."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph162",
        "image": "data/hamilton/graph_visual/hard/graph162.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 12, and the edges are: (6,11) (7,10) (5,10) (10,12) (5,3) (12,8) (9,1) (7,0) (1,8) (0,8) (10,0) (8,5) (11,9) (5,0) (2,0) (0,1) (11,7) (3,6) (10,9) (5,4) (3,2) (1,5) (10,11) (11,1) (4,12) (12,9) (5,7) (12,11) (1,2) (0,9) (3,11) (9,4) (2,4) (12,6) (10,3) (0,4) (0,11) (3,9) (7,9) (8,11) (5,9) (7,2) (3,4) (0,3)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->7->10->5->3->6->11->9->1->8->12->4->2."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph286",
        "image": "data/hamilton/graph_visual/hard/graph286.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 14, and the edges are: (9,8) (11,1) (9,11) (14,10) (8,14) (10,13) (8,0) (9,4) (7,0) (10,9) (3,0) (13,5) (8,1) (5,10) (8,7) (7,2) (11,2) (3,5) (14,1) (5,2) (3,2) (10,0) (1,4) (12,10) (3,1) (5,8) (12,6) (9,7) (9,12) (5,9) (6,0) (2,10) (12,11) (11,5) (4,7) (0,4) (10,1) (6,3) (12,1) (11,6) (11,3) (8,2) (10,8) (10,3) (3,7) (1,5) (13,8) (12,7) (4,8) (0,5) (12,13) (8,11) (6,8) (2,6) (1,13) (14,2) (4,6) (1,7) (8,12) (7,14) (3,9) (14,6) (4,3) (13,14)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->8->9->11->1->14->10->13->5->3->2->7->4->6->12."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph281",
        "image": "data/hamilton/graph_visual/hard/graph281.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 14, and the edges are: (13,1) (1,3) (7,10) (4,7) (10,0) (9,8) (12,14) (0,12) (5,10) (4,9) (2,0) (8,7) (2,5) (1,0) (4,10) (4,6) (1,2) (14,2) (13,4) (1,8) (0,14) (1,11) (0,4) (10,2) (2,11) (14,10) (8,2) (6,13) (9,12) (11,5) (12,2) (4,12) (5,7) (1,5) (5,9) (14,3) (9,3) (8,3) (2,7) (14,6) (13,0) (11,4) (13,11) (12,7) (5,12) (9,14) (11,7) (5,3) (0,9) (6,11) (4,1) (6,10) (2,6) (14,4) (10,11) (14,5) (11,3) (3,12) (11,12) (0,5) (5,6) (14,7) (8,5) (9,1) (8,13) (6,9) (4,3) (8,12)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->10->7->4->9->8->1->13->6->14->12->2->5->11->3."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph395",
        "image": "data/hamilton/graph_visual/hard/graph395.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 16, and the edges are: (11,5) (0,4) (12,4) (2,10) (8,1) (12,8) (16,0) (6,13) (8,10) (7,16) (9,6) (16,8) (14,6) (15,13) (14,12) (9,11) (2,14) (5,14) (2,8) (13,10) (0,11) (12,0) (15,5) (3,4) (15,11) (14,7) (3,2) (7,15) (12,7) (5,13) (6,11) (9,13) (16,9) (5,16) (13,0) (16,13) (6,1) (3,16) (4,10) (10,1) (9,10) (12,1) (3,6) (13,4) (0,10) (11,7) (15,2) (3,10) (8,4) (3,15) (14,1) (15,10) (14,13) (3,0) (1,2) (5,9) (0,1) (12,2) (7,9) (2,16) (12,15) (15,16) (14,8) (13,1) (13,7) (13,3) (6,12) (10,11) (2,4) (5,4) (4,6) (16,4) (6,0) (5,3) (14,11)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->4->12->8->1->6->13->15->5->11->9->16->7->14->2->10->3."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph560",
        "image": "data/hamilton/graph_visual/hard/graph560.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 19, and the edges are: (13,8) (1,3) (8,16) (16,12) (15,5) (0,6) (15,18) (8,14) (11,1) (2,15) (4,5) (13,19) (19,11) (15,3) (17,15) (12,15) (5,16) (18,19) (12,7) (5,8) (2,19) (8,3) (12,1) (6,17) (14,13) (19,4) (5,3) (6,1) (3,12) (10,3) (17,2) (2,6) (6,15) (13,5) (8,0) (18,12) (4,14) (5,2) (6,7) (1,9) (9,10) (16,11) (11,4) (6,8) (5,7) (9,18) (16,6) (2,7) (2,18) (12,5) (2,16) (9,11) (11,0) (7,15) (5,17) (9,13) (16,13) (10,15) (12,19) (13,10) (12,9) (12,14) (6,12) (11,18) (10,8) (19,10) (7,14) (16,1) (7,17) (0,12) (18,8) (18,7) (2,13)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->6->17->15->5->4->19->13->8->16->12->1->3->10->9->11->18->2->7->14."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph18",
        "image": "data/hamilton/graph_visual/hard/graph18.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 10, and the edges are: (10,5) (6,10) (0,6) (5,8) (5,2) (6,5) (1,3) (4,8) (4,0) (8,3) (1,8) (4,9) (2,3) (4,3) (3,0) (1,7) (10,3) (2,8) (9,5)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->6->10->5->9->4->8->2->3->1->7."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph505",
        "image": "data/hamilton/graph_visual/hard/graph505.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (18,0) (8,9) (17,14) (5,4) (0,12) (6,14) (18,8) (14,15) (18,3) (17,16) (14,3) (7,4) (17,9) (15,11) (13,14) (10,1) (18,4) (12,14) (13,11) (18,2) (0,9) (10,0) (5,14) (4,8) (15,16) (7,2) (0,6) (12,5) (18,11) (9,11) (16,5) (7,18) (17,11) (13,5) (7,5) (13,15) (1,12) (13,18) (2,10) (8,13) (6,8) (15,5) (15,12) (6,16) (8,17) (14,1) (17,18) (13,16) (9,5) (4,0) (14,11) (2,6) (2,14) (2,15) (17,15) (0,7) (12,18) (4,15) (10,16) (10,6) (11,7) (12,6)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->18->8->9->17->16->15->11->13->5->4->7->2->10->1->12->6->14->3."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph305",
        "image": "data/hamilton/graph_visual/hard/graph305.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (5,15) (10,11) (15,3) (1,15) (11,8) (12,9) (5,13) (11,5) (1,5) (3,10) (3,2) (11,4) (3,11) (7,8) (3,9) (7,1) (9,5) (4,1) (14,7) (13,4) (15,7) (9,7) (4,12) (11,13) (15,12) (9,6) (13,6) (11,9) (9,14) (6,5) (12,10) (8,14) (13,10) (9,10) (13,7) (12,2) (13,2) (4,7) (1,6) (1,0) (4,3) (13,14) (0,4) (2,7) (6,10) (11,6) (2,9) (15,2) (6,7) (11,7) (12,7) (0,13)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->1->15->5->13->4->11->10->3->2->12->9->6->7->8->14."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph21",
        "image": "data/hamilton/graph_visual/hard/graph21.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 10, and the edges are: (5,4) (0,10) (9,8) (3,5) (3,7) (1,4) (2,7) (1,0) (5,7) (4,9) (1,3) (9,0) (2,9) (0,6) (9,6) (4,8) (6,2) (2,1) (4,7) (8,5) (0,8) (8,7) (3,10) (10,9) (10,4)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->10->3->5->4->1->2->7->8->9->6."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph190",
        "image": "data/hamilton/graph_visual/hard/graph190.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 13, and the edges are: (1,6) (9,3) (9,7) (11,7) (3,7) (11,12) (5,4) (5,2) (9,8) (12,1) (0,12) (3,4) (9,6) (2,8) (2,4) (10,7) (9,4) (13,6) (5,6) (8,0) (1,4) (4,6) (0,10) (7,1) (4,13) (3,12) (13,2) (13,1) (8,10) (6,12) (11,4)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->12->11->7->9->3->4->5->6->1->13->2->8->10."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph358",
        "image": "data/hamilton/graph_visual/hard/graph358.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (1,3) (8,10) (15,3) (11,2) (12,6) (10,2) (15,12) (12,8) (4,9) (7,8) (15,9) (13,14) (9,8) (0,6) (4,13) (13,3) (0,15) (6,3) (15,7) (8,0) (8,3) (0,1) (2,9) (0,7) (0,10) (8,13) (2,0) (14,7) (11,0) (1,7) (3,4) (13,2) (6,8) (9,6) (4,15) (14,4) (10,12) (5,7) (10,7) (8,11) (7,13) (7,3) (3,5) (1,11) (12,2) (8,4) (8,2) (4,1) (5,2) (9,7) (4,0) (1,10) (5,12) (11,9) (0,5) (1,9) (2,3) (2,15) (13,10) (14,15) (3,11) (6,2) (15,1) (14,1) (12,0) (2,7) (1,5) (13,9) (15,6) (8,14) (13,1) (12,14) (12,4) (3,10) (3,14) (5,9) (1,2)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->6->12->15->3->1->7->8->10->13->14->4->9->11->2->5."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph508",
        "image": "data/hamilton/graph_visual/hard/graph508.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (9,18) (0,13) (5,9) (4,9) (2,17) (3,12) (7,6) (4,18) (1,18) (11,16) (13,8) (10,12) (9,12) (10,9) (18,5) (14,13) (9,14) (8,15) (14,3) (2,10) (1,10) (6,4) (16,6) (16,3) (17,18) (4,10) (3,6) (7,16) (5,13) (11,8) (12,18) (8,2) (16,9) (13,4) (7,5) (6,0) (13,3) (15,2) (9,3) (8,16) (16,2) (18,10) (5,2) (17,4) (12,14) (7,1) (14,5) (10,5) (13,17) (1,11) (13,16) (16,17) (18,11) (12,6) (14,8) (17,11) (1,14) (17,9) (15,12) (9,7) (12,7) (12,5) (1,9) (0,3) (8,12) (15,3) (7,8) (9,0)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->13->8->15->2->17->18->9->5->7->6->4->10->12->3->14->1->11->16."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph355",
        "image": "data/hamilton/graph_visual/hard/graph355.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (6,5) (14,9) (4,0) (6,11) (4,14) (1,12) (15,5) (14,10) (1,7) (12,5) (6,4) (4,15) (9,7) (10,2) (8,7) (3,10) (8,14) (11,13) (7,15) (12,0) (2,1) (8,1) (6,7) (5,10) (9,8) (7,14) (2,13) (13,12) (10,0) (3,12) (15,10) (8,12) (15,8) (11,1) (15,12) (4,10) (6,13) (7,13) (3,7) (9,2) (2,5) (3,15) (14,3) (0,8) (4,7) (0,2) (10,1) (10,8) (15,9) (0,1) (9,4) (14,13) (0,13) (14,6) (12,2) (0,5) (2,7) (15,11) (7,10) (3,13) (0,6) (11,2) (0,15) (9,13) (14,2) (13,10) (11,10) (4,8) (2,15) (12,4)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->4->14->9->7->1->12->5->6->11->13->2->10->3->15->8."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph434",
        "image": "data/hamilton/graph_visual/hard/graph434.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 17, and the edges are: (1,16) (15,7) (8,16) (15,9) (2,7) (13,7) (13,17) (5,9) (3,17) (3,11) (0,12) (13,12) (12,8) (13,4) (13,5) (8,3) (8,7) (3,4) (4,10) (9,1) (16,13) (2,1) (3,13) (2,4) (10,15) (9,17) (6,12) (6,13) (15,6) (8,10) (9,14) (8,14) (8,2) (4,12) (11,4) (12,10) (8,15) (5,2) (12,9) (1,3) (10,13) (14,10) (14,5) (12,1) (16,9) (15,4) (4,0) (5,12) (16,11) (8,1) (9,2) (0,15) (7,11) (2,10) (9,6) (0,7) (7,6) (0,1) (0,14) (6,4) (1,11) (1,14) (17,6) (3,10)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->12->13->7->15->9->5->2->1->16->8->14->10->4->11->3->17->6."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph51",
        "image": "data/hamilton/graph_visual/hard/graph51.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 10, and the edges are: (4,10) (1,0) (6,2) (3,10) (10,6) (10,9) (9,7) (8,5) (5,4) (8,6) (0,9) (6,9) (4,2) (5,9) (3,0) (5,2) (8,4) (6,4) (2,0) (9,8) (0,6) (6,3) (2,8) (6,1) (3,1) (7,8) (9,4) (0,5) (0,8) (3,4) (7,10) (1,4) (0,7) (2,7) (2,9)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->1->6->2->4->5->8->9->7->10->3."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph331",
        "image": "data/hamilton/graph_visual/hard/graph331.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (1,10) (3,12) (3,13) (12,2) (14,8) (0,8) (11,14) (9,2) (15,8) (15,2) (1,15) (7,9) (13,15) (4,9) (14,6) (2,10) (10,14) (10,7) (8,10) (12,6) (7,4) (4,2) (13,9) (4,11) (3,10) (11,6) (13,2) (11,2) (4,5) (8,3) (5,14) (2,7) (12,1) (1,9) (9,15) (1,6) (11,5) (6,8) (9,0) (2,3) (5,12) (10,0) (14,7) (9,14) (14,2) (11,3) (12,0) (12,15) (12,4) (8,12) (1,14) (15,7) (4,1) (2,8) (10,4) (14,12) (5,8) (4,13) (15,0) (6,7) (1,2) (8,4) (15,10) (5,0) (1,13) (4,3) (9,10) (13,8) (3,15) (13,14) (5,1) (5,13)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->8->14->11->4->9->2->12->3->13->15->10->7->6->1->5."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph357",
        "image": "data/hamilton/graph_visual/hard/graph357.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (1,8) (5,0) (15,3) (10,4) (6,4) (0,1) (14,5) (14,4) (4,5) (14,13) (0,2) (11,6) (12,13) (14,3) (7,13) (10,5) (12,14) (12,8) (10,15) (10,8) (1,6) (11,1) (3,0) (8,13) (3,2) (9,2) (5,9) (5,3) (14,8) (5,2) (15,4) (4,12) (3,13) (9,8) (7,11) (4,13) (10,6) (9,15) (1,4) (7,10) (3,10) (5,1) (1,10) (9,13) (1,7) (2,4) (5,7) (0,14) (11,2) (13,11) (12,10) (15,11) (15,7) (8,7) (9,4) (3,12) (7,6) (8,4) (4,3) (0,8) (8,15) (0,10) (2,1) (15,0) (13,10) (15,6) (6,14) (2,15) (13,6) (10,11) (8,5) (12,15) (14,7) (0,11) (11,5) (4,0) (7,4)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->5->14->4->10->15->3->2->9->8->1->6->11->7->13->12."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph450",
        "image": "data/hamilton/graph_visual/hard/graph450.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 17, and the edges are: (13,3) (6,12) (4,9) (6,14) (10,13) (6,2) (9,3) (10,12) (8,14) (9,11) (16,3) (17,16) (1,16) (15,14) (1,15) (4,0) (15,5) (6,10) (2,17) (13,6) (17,4) (6,3) (1,3) (3,15) (7,9) (14,7) (0,12) (1,17) (11,6) (14,5) (3,8) (17,12) (7,11) (1,7) (0,1) (16,9) (4,8) (16,6) (12,16) (8,0) (13,5) (2,0) (0,14) (13,7) (14,12) (16,7) (5,2) (13,9) (0,9) (14,13) (1,8) (10,5) (10,1) (16,0) (9,12) (11,17) (9,6) (10,17) (2,4) (6,1) (4,10) (0,5) (7,10) (11,13) (5,6) (8,5) (1,13) (15,2) (7,17) (7,15) (15,4) (3,14) (4,5) (10,0) (3,11) (13,8) (15,8) (12,1) (4,6) (17,9) (12,15) (16,10) (8,2) (2,3) (11,14) (5,11) (0,6) (10,3)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->4->9->3->13->10->12->6->14->8->1->16->17->2->5->15->7->11."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph86",
        "image": "data/hamilton/graph_visual/hard/graph86.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 11, and the edges are: (7,6) (7,8) (0,3) (2,9) (1,4) (5,3) (5,4) (1,10) (0,7) (9,3) (2,7) (2,3) (0,2) (6,0) (6,10) (8,5) (4,2) (1,0) (0,8) (5,11) (1,6) (10,11) (11,0) (7,1) (3,7) (5,6) (4,10) (1,5)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->9->2->7->6->1->4->10->11->5->8."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph409",
        "image": "data/hamilton/graph_visual/hard/graph409.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 16, and the edges are: (0,16) (2,11) (14,16) (7,2) (6,12) (13,15) (9,15) (2,14) (15,2) (2,1) (4,5) (14,9) (11,3) (11,13) (15,14) (11,5) (1,15) (6,3) (5,16) (6,5) (6,15) (9,4) (8,0) (16,15) (1,6) (8,3) (10,1) (12,0) (15,5) (12,3) (15,12) (10,7) (10,13) (15,0) (13,5) (10,0) (7,0) (8,2) (16,7) (1,5) (13,9) (6,4) (13,7) (10,14) (8,11) (3,7) (1,14) (0,3) (11,16) (3,15) (16,3) (11,15) (2,16) (2,13) (8,9) (12,13) (11,4) (6,10) (4,2) (1,9) (4,10) (1,8) (7,12) (4,15) (1,12) (15,8) (14,5) (8,12) (4,13) (7,5) (13,3) (16,4) (2,10) (2,5)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->16->14->2->11->3->6->12->15->13->10->1->8->9->4->5->7."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph140",
        "image": "data/hamilton/graph_visual/hard/graph140.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 12, and the edges are: (7,8) (12,10) (0,1) (0,2) (10,5) (8,1) (2,12) (5,11) (11,8) (10,7) (9,12) (9,0) (6,2) (0,10) (3,7) (9,4) (6,4) (2,1) (0,12) (3,5) (9,6) (8,0) (2,3) (1,5) (4,3) (12,3) (6,0) (2,10) (9,3) (0,11) (2,4) (12,5) (3,6) (11,7) (8,5) (1,3)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->1->8->7->10->12->2->6->4->9->3->5->11."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph157",
        "image": "data/hamilton/graph_visual/hard/graph157.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 12, and the edges are: (2,4) (9,12) (10,9) (11,3) (9,2) (7,9) (7,0) (7,2) (11,9) (4,7) (6,5) (2,6) (7,8) (10,12) (11,0) (6,3) (10,4) (2,8) (4,1) (0,4) (6,0) (8,10) (3,9) (10,1) (9,8) (3,5) (1,6) (7,1) (2,10) (12,4) (8,12) (11,12) (8,1) (5,8) (9,0) (3,8) (7,11) (0,12) (7,3) (6,8) (5,7) (11,10) (3,12) (2,3) (0,5) (1,9) (6,4) (1,12) (2,5) (6,11)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->7->9->12->10->4->2->6->1->8->5->3->11."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph270",
        "image": "data/hamilton/graph_visual/hard/graph270.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 14, and the edges are: (11,7) (8,0) (2,8) (5,0) (4,3) (6,3) (10,8) (14,13) (11,2) (0,11) (7,5) (13,7) (5,3) (11,3) (0,2) (5,14) (13,8) (2,9) (5,4) (0,3) (2,4) (6,8) (10,14) (8,7) (12,4) (9,7) (3,13) (0,13) (13,5) (6,10) (9,5) (6,14) (12,13) (5,11) (4,11) (2,3) (12,6) (6,0) (7,4) (7,14) (6,13) (0,10) (10,3) (8,14) (7,1) (9,6) (12,10) (12,1) (0,7) (12,0) (9,11) (10,5) (1,2) (3,14) (10,4) (2,10) (1,8) (13,9) (10,11) (5,12) (1,5) (5,6) (9,12) (4,6)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->8->2->11->7->5->3->4->10->14->13->6->9->12->1."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph371",
        "image": "data/hamilton/graph_visual/hard/graph371.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 16, and the edges are: (11,13) (15,11) (7,1) (3,11) (1,2) (3,10) (6,11) (6,2) (10,8) (0,4) (9,8) (7,13) (15,1) (13,2) (4,3) (12,13) (4,5) (5,2) (13,5) (8,14) (15,3) (9,10) (14,13) (5,3) (4,12) (6,3) (10,2) (8,11) (14,10) (8,13) (7,14) (16,3) (4,7) (11,0) (2,11) (13,0) (7,15) (1,8) (3,14) (16,4) (6,0) (16,14) (1,3) (12,16) (16,13) (5,14) (8,15) (9,15) (12,2) (5,10) (12,10) (4,8) (1,12) (12,6) (16,6) (10,11) (1,11)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->4->3->11->13->7->1->2->6->12->16->14->8->15->9->10->5."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph455",
        "image": "data/hamilton/graph_visual/hard/graph455.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 17, and the edges are: (15,14) (1,13) (1,17) (4,13) (14,9) (11,12) (11,13) (7,17) (4,17) (8,16) (17,16) (0,1) (0,16) (15,11) (10,1) (17,3) (11,8) (9,11) (6,2) (11,17) (12,17) (15,2) (4,2) (15,10) (16,13) (1,3) (7,8) (2,8) (2,17) (0,15) (11,14) (17,9) (4,1) (7,6) (10,9) (4,11) (9,3) (0,17) (3,12) (2,1) (6,3) (16,11) (11,7) (1,5) (16,15) (1,9) (13,8) (5,14) (10,7) (6,12) (9,16) (12,8) (16,10) (2,9) (11,10) (3,4) (15,13) (13,5) (5,12) (2,0) (10,14) (7,2) (5,6) (12,16) (12,13) (10,4) (14,3) (4,16) (9,8) (11,3) (15,1) (10,6) (5,7) (11,6) (7,9) (6,17) (4,7) (10,13) (6,16) (2,14) (17,8) (1,7) (12,0) (3,8) (4,6) (2,3) (15,7) (4,8) (0,5) (7,12) (13,7) (9,15) (2,13) (9,5) (12,2) (0,13) (5,17) (4,14) (2,5) (4,0) (13,9) (14,0) (10,0) (12,14) (5,11)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->1->13->4->17->7->8->16->11->12->3->9->14->15->2->5->6->10."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph67",
        "image": "data/hamilton/graph_visual/hard/graph67.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 11, and the edges are: (1,6) (4,8) (9,10) (10,3) (1,9) (2,0) (11,10) (4,11) (2,1) (3,6) (8,5) (3,9) (7,9) (4,6) (9,8) (6,7) (10,7) (3,5) (1,4) (9,2) (1,10) (1,0)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->1->6->3->9->7->10->11->4->8->5."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph374",
        "image": "data/hamilton/graph_visual/hard/graph374.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 16, and the edges are: (8,2) (7,5) (14,1) (11,15) (4,7) (0,14) (0,5) (4,11) (6,4) (11,13) (12,1) (14,16) (5,11) (3,7) (8,15) (0,1) (0,6) (1,10) (16,4) (5,10) (7,9) (4,0) (5,4) (2,0) (2,13) (0,9) (7,2) (1,6) (3,14) (3,11) (10,4) (6,16) (12,16) (15,4) (10,8) (16,8) (6,10) (12,9) (13,3) (1,15) (13,10) (15,14) (3,0) (8,12) (8,11) (3,15) (3,5) (15,6) (9,1) (8,14) (16,10) (10,9)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->14->1->12->16->4->7->5->11->3->13->2->8->15->6->10->9."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph406",
        "image": "data/hamilton/graph_visual/hard/graph406.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 16, and the edges are: (15,1) (7,13) (1,13) (11,5) (7,15) (1,7) (9,0) (13,10) (4,1) (5,1) (2,1) (10,6) (10,1) (0,4) (6,7) (13,3) (2,15) (16,7) (8,14) (2,14) (2,3) (5,13) (13,11) (9,10) (14,1) (14,5) (6,15) (13,12) (6,14) (15,5) (6,13) (6,8) (16,9) (12,2) (16,5) (1,9) (8,2) (10,12) (3,1) (13,16) (5,6) (7,14) (11,2) (9,12) (8,4) (12,3) (8,11) (15,10) (0,6) (5,7) (15,4) (5,9) (13,4) (0,8) (11,15) (5,3) (7,8) (3,4) (13,14) (8,5) (8,10) (14,10) (2,13) (3,9) (4,6) (15,12) (2,9) (15,16) (6,9) (15,0) (11,3) (4,9) (2,0) (7,2) (11,16) (1,16)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->9->10->13->7->15->1->4->8->14->2->12->3->11->16->5->6."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph218",
        "image": "data/hamilton/graph_visual/hard/graph218.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 13, and the edges are: (10,11) (13,6) (3,6) (11,3) (9,5) (11,2) (8,7) (6,10) (1,2) (1,13) (13,8) (1,0) (5,3) (8,12) (9,3) (6,5) (2,5) (1,8) (4,5) (8,10) (2,13) (7,4) (1,11) (5,7) (9,12) (12,13) (2,3) (0,3) (4,1) (4,10) (4,2) (11,7) (4,0) (0,13) (12,2) (13,4) (1,6) (1,10) (3,12) (10,3) (12,7) (2,6) (1,7) (11,12) (3,8) (13,10) (8,5) (9,1) (12,10) (1,5) (0,10) (11,0) (9,0) (6,7) (11,8)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->1->2->11->10->6->13->8->7->4->5->9->3->12."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph119",
        "image": "data/hamilton/graph_visual/hard/graph119.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 11, and the edges are: (4,6) (8,2) (5,4) (6,5) (11,0) (0,6) (8,0) (9,6) (8,5) (9,8) (11,4) (1,6) (1,9) (9,3) (8,10) (4,10) (7,4) (7,9) (5,7) (8,6) (1,10) (3,6) (8,7) (11,8) (11,7) (2,5) (4,9) (3,8) (0,4) (0,3) (9,11) (7,1) (0,2) (11,5) (10,2) (5,0) (4,3)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->11->4->6->5->8->2->10->1->7->9->3."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph425",
        "image": "data/hamilton/graph_visual/hard/graph425.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 17, and the edges are: (6,17) (13,5) (12,5) (13,0) (4,10) (17,3) (0,15) (15,8) (9,2) (5,1) (8,17) (12,8) (6,3) (15,2) (11,15) (13,10) (0,1) (4,0) (11,2) (10,17) (14,6) (4,12) (14,5) (13,8) (3,7) (2,5) (10,5) (16,12) (11,4) (7,15) (3,5) (12,7) (6,2) (15,14) (14,1) (6,13) (3,16) (14,13) (6,7) (9,4) (11,9) (12,1) (5,8) (3,10) (8,11) (9,6) (6,10) (13,17) (3,11) (15,13) (7,16) (10,12) (4,1) (12,9) (13,16) (14,12) (16,0) (9,10) (2,0) (6,8) (17,2) (16,14) (17,11)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->13->5->12->8->15->2->9->4->10->17->11->3->6->7->16->14->1."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph436",
        "image": "data/hamilton/graph_visual/hard/graph436.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 17, and the edges are: (14,8) (6,14) (10,3) (7,11) (12,15) (11,15) (4,11) (6,9) (13,3) (16,11) (13,10) (2,1) (17,9) (17,7) (0,14) (2,17) (15,14) (14,5) (13,8) (12,6) (10,16) (7,4) (13,11) (0,11) (16,1) (2,4) (0,3) (6,2) (3,11) (14,13) (17,6) (14,11) (13,4) (12,10) (10,15) (10,1) (14,12) (3,9) (8,17) (16,6) (4,8) (15,3) (17,0) (14,3) (2,9) (17,14) (16,12) (16,9) (3,1) (2,12) (7,14) (12,4) (10,17) (2,0) (1,7) (17,5) (9,7) (0,13) (3,7) (13,1) (7,16) (2,5) (1,14) (4,14) (3,5) (12,17) (0,10)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->14->8->13->3->10->16->11->15->12->6->9->17->5->2->1->7->4."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph329",
        "image": "data/hamilton/graph_visual/hard/graph329.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (13,10) (9,6) (15,4) (7,10) (13,0) (2,15) (8,11) (13,4) (0,14) (2,4) (14,2) (12,15) (14,3) (5,9) (0,2) (8,15) (4,1) (1,2) (11,1) (8,13) (12,5) (5,13) (13,9) (0,10) (2,7) (6,2) (15,1) (10,9) (8,7) (12,13) (3,13) (12,2) (4,9) (11,9) (8,10) (14,15) (15,10) (13,6) (3,8) (14,9) (12,4) (11,5) (0,1) (6,3) (11,14) (6,10)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->13->10->7->2->15->8->11->1->4->12->5->9->6->3->14."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph489",
        "image": "data/hamilton/graph_visual/hard/graph489.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (14,15) (13,16) (13,15) (14,0) (4,8) (10,17) (5,9) (15,6) (6,17) (16,15) (9,7) (7,11) (18,15) (4,17) (5,0) (2,17) (16,11) (8,1) (16,4) (13,0) (10,1) (5,7) (1,4) (3,8) (0,8) (0,15) (5,12) (11,10) (11,14) (13,3) (5,18) (4,0) (7,16) (4,12) (3,6) (12,2) (7,17) (12,6) (11,9) (10,14) (12,7) (13,11) (8,11) (7,2) (11,17) (15,10) (6,18) (11,6) (6,13) (3,15) (18,17) (1,11) (17,9) (14,13) (10,13) (10,18) (15,7) (16,8) (1,16) (11,12) (3,9) (0,17) (13,5) (14,12) (9,6) (0,10) (4,10) (4,13) (8,18) (6,14) (8,5) (7,6) (12,15)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->14->15->13->16->11->7->9->5->12->4->1->8->3->6->18->10->17->2."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph480",
        "image": "data/hamilton/graph_visual/hard/graph480.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (13,15) (17,13) (5,3) (13,9) (2,1) (2,15) (17,5) (3,9) (15,4) (16,10) (10,8) (5,14) (14,1) (15,3) (4,11) (15,16) (9,12) (11,3) (0,7) (8,14) (11,16) (7,1) (0,8) (10,12) (6,12) (17,12) (15,18) (8,9) (10,11) (5,4) (4,2) (2,9) (1,18) (14,16) (7,6) (4,17) (1,9) (0,13) (12,1) (15,7) (6,15) (12,2) (4,6) (8,16) (15,8) (3,1) (0,2) (6,17) (12,5) (3,8) (3,17) (12,16) (8,12) (5,1) (9,14) (17,1) (2,8) (16,0) (12,7) (2,16) (5,16) (6,11) (18,10) (2,10)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->7->1->2->15->13->17->5->3->9->12->6->4->11->16->14->8->10->18."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph150",
        "image": "data/hamilton/graph_visual/hard/graph150.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 12, and the edges are: (12,6) (11,0) (1,12) (12,4) (7,5) (5,0) (7,12) (2,12) (5,4) (9,3) (1,8) (2,6) (5,3) (1,9) (11,7) (11,6) (8,7) (5,9) (11,12) (10,12) (3,2) (7,6) (4,2) (3,11) (5,12) (2,1) (11,5) (11,9) (3,4) (5,2) (5,6) (1,3) (11,4) (1,11) (1,6) (4,0) (1,4) (8,12) (12,9) (4,9) (4,6) (8,11)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->11->7->5->4->9->3->2->6->1->8->12->10."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph344",
        "image": "data/hamilton/graph_visual/hard/graph344.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (12,0) (7,0) (15,11) (0,15) (5,0) (13,2) (0,8) (15,12) (10,14) (12,3) (0,4) (9,15) (6,1) (15,5) (3,15) (2,15) (15,14) (13,10) (8,10) (14,3) (8,4) (8,6) (6,2) (14,9) (11,1) (9,1) (1,5) (7,9) (15,7) (9,6) (12,9) (8,11) (3,4) (0,1) (4,14) (1,12) (6,4) (3,8) (0,9) (11,5) (11,3) (6,10) (0,2) (4,11) (3,1) (1,13) (12,13) (0,3) (3,7) (8,9) (1,4) (11,14) (5,6) (7,11) (2,14) (13,3) (13,7) (15,1) (7,8) (9,2) (1,8) (13,8) (5,9) (15,13) (14,0) (5,14)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->12->15->11->1->6->8->10->14->4->3->7->13->2->9->5."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph346",
        "image": "data/hamilton/graph_visual/hard/graph346.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (10,15) (15,6) (10,4) (10,9) (3,14) (3,9) (0,3) (15,14) (4,9) (12,11) (12,4) (5,13) (7,2) (2,15) (13,7) (14,11) (9,5) (4,1) (15,4) (2,13) (10,1) (4,0) (8,3) (1,6) (5,4) (13,11) (15,3) (5,12) (7,15) (4,8) (3,6) (11,3) (7,3) (9,11) (0,2) (2,9) (13,3) (0,5) (13,14) (5,7) (2,10) (13,0) (12,0) (1,9) (10,6) (4,3) (14,8) (9,8) (6,5) (8,11) (1,2) (4,2) (2,12) (4,14) (9,15) (3,5) (12,15) (8,5) (13,10) (8,2) (0,11) (1,5) (1,15) (6,13) (1,11) (13,15) (0,10) (2,3) (9,7) (10,12) (6,2) (12,3) (10,11) (6,14) (6,8) (14,2)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->14->15->10->4->9->5->13->7->2->1->6->8->11->12."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph174",
        "image": "data/hamilton/graph_visual/hard/graph174.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 12, and the edges are: (3,6) (2,0) (8,0) (4,0) (2,11) (1,12) (8,5) (12,6) (9,0) (9,6) (1,3) (0,11) (4,11) (7,2) (6,4) (12,11) (9,5) (2,12) (1,11) (0,5) (7,0) (6,0) (9,2) (8,10) (5,4) (3,11) (8,12) (10,12) (7,4) (10,2) (2,1) (9,11) (10,5) (3,4) (4,10) (2,3) (6,8) (10,7) (10,9)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->11->4->6->3->1->12->8->5->9->10->7."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph335",
        "image": "data/hamilton/graph_visual/hard/graph335.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (11,9) (13,6) (8,1) (6,14) (0,9) (13,1) (11,10) (10,4) (13,10) (2,8) (7,5) (4,7) (2,10) (10,14) (8,15) (14,12) (3,0) (11,3) (10,6) (8,7) (15,5) (7,6) (14,8) (7,14) (1,7) (2,11) (6,2) (7,9) (0,1) (12,10) (5,6) (0,15) (7,0) (6,12) (11,6) (7,13) (5,2) (8,3) (5,3) (4,8) (1,3) (4,9) (8,11) (13,0) (9,10) (5,0) (9,5) (13,8) (13,5) (12,7) (8,5) (3,7) (4,2) (3,12) (0,2) (9,13) (12,2) (5,4) (8,6) (14,2) (14,13) (12,1) (15,11) (1,15) (3,14) (2,15) (11,1) (1,5) (13,2) (0,4) (12,4) (14,15) (6,3) (5,12)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->9->11->10->4->7->5->15->8->1->13->6->14->2->12->3."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph78",
        "image": "data/hamilton/graph_visual/hard/graph78.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 11, and the edges are: (8,6) (3,1) (1,10) (7,3) (2,0) (10,2) (6,10) (3,10) (5,10) (10,8) (9,6) (2,3) (1,2) (11,6) (10,0) (3,8) (4,9) (4,3) (0,1) (7,5) (8,11) (5,11) (7,10) (3,0) (7,4) (7,6) (8,5)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->10->1->3->7->5->11->8->6->9->4."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph138",
        "image": "data/hamilton/graph_visual/hard/graph138.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 12, and the edges are: (3,11) (7,11) (6,3) (2,9) (8,3) (2,1) (2,3) (2,12) (3,12) (0,1) (5,6) (8,10) (7,5) (10,0) (10,5) (2,4) (2,8) (6,1) (6,4) (5,11) (0,8) (6,9) (10,2) (4,12) (9,8) (11,8) (11,0) (2,0) (5,12) (9,10) (5,4) (5,2) (3,4) (10,7) (10,6) (6,0) (8,12) (2,7) (4,1) (9,12) (4,9)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->1->2->9->6->3->11->7->5->10->8->12->4."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph183",
        "image": "data/hamilton/graph_visual/hard/graph183.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 13, and the edges are: (10,0) (9,2) (11,0) (3,9) (10,4) (8,9) (11,13) (2,12) (11,6) (8,7) (13,1) (11,7) (6,2) (7,9) (4,2) (10,7) (2,10) (0,1) (6,5) (7,13) (13,3) (5,1) (3,7) (7,2) (3,10) (6,4)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->10->3->9->8->7->11->13->1->5->6->4->2->12."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph55",
        "image": "data/hamilton/graph_visual/hard/graph55.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 10, and the edges are: (4,3) (3,8) (4,6) (2,4) (5,3) (3,10) (0,6) (4,0) (0,5) (1,8) (7,5) (2,1) (4,5) (6,5) (6,1) (3,9) (5,10) (2,10) (8,4) (8,2) (6,9) (6,2) (1,7) (5,2) (2,0) (5,9) (1,3) (8,7) (9,4) (0,7) (10,1) (4,7) (8,5) (0,9)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->6->4->3->8->2->10->1->7->5->9."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph3",
        "image": "data/hamilton/graph_visual/hard/graph3.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 10, and the edges are: (8,5) (7,2) (6,4) (4,5) (3,6) (6,5) (4,0) (7,4) (1,5) (9,7) (0,9) (10,9) (0,2) (5,3) (6,7) (10,1) (4,8) (3,1) (1,7) (10,4) (6,1) (0,1) (2,6) (8,7)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->4->6->2->7->9->10->1->3->5->8."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph441",
        "image": "data/hamilton/graph_visual/hard/graph441.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 17, and the edges are: (1,12) (2,4) (4,16) (6,0) (8,17) (4,11) (2,10) (0,3) (14,12) (17,16) (10,9) (6,2) (5,15) (0,12) (12,13) (17,15) (1,0) (14,10) (1,13) (4,3) (5,10) (6,4) (9,15) (3,13) (4,12) (16,1) (17,9) (7,16) (7,14) (16,6) (16,3) (0,2) (1,4) (0,15) (13,6) (2,3) (8,2) (5,8) (15,6) (6,7) (16,2) (4,5) (12,7) (3,1) (7,17) (9,7) (2,15) (5,13) (15,7) (4,0) (1,2) (6,8) (9,8) (17,10) (4,13) (11,14) (13,15) (9,4) (7,1) (10,4) (14,1) (17,3) (5,16) (14,13) (5,0) (2,11)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->6->2->4->16->17->8->5->15->13->3->1->12->7->9->10->14->11."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph324",
        "image": "data/hamilton/graph_visual/hard/graph324.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (11,3) (8,9) (5,4) (4,14) (5,15) (0,1) (1,14) (0,2) (11,1) (12,11) (9,10) (2,14) (2,8) (3,10) (10,8) (10,7) (11,6) (15,9) (8,13) (4,6) (15,0) (15,8) (6,8) (7,12) (7,14) (12,3) (7,13) (3,14) (4,12) (1,3) (4,11) (2,7) (13,14) (14,0) (1,13) (6,14) (11,15) (6,12) (7,9) (2,13) (1,8) (7,1) (8,11) (15,13) (15,1) (12,0)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->1->14->4->5->15->9->8->2->13->7->10->3->11->12->6."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph217",
        "image": "data/hamilton/graph_visual/hard/graph217.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 13, and the edges are: (10,1) (4,3) (7,3) (9,12) (0,12) (3,9) (9,6) (11,1) (10,12) (6,12) (0,7) (8,6) (4,9) (5,8) (4,12) (1,12) (7,12) (11,0) (6,7) (2,1) (5,4) (8,3) (2,0) (13,3) (2,12) (1,4) (8,13) (2,6) (9,7) (4,6) (7,11) (8,2) (1,8) (11,8) (12,5) (1,3) (7,8) (10,0) (0,6) (3,0) (3,6) (10,7) (3,5) (13,11) (11,4) (11,10) (1,9) (10,3) (7,1) (13,4)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->12->9->3->4->5->8->6->2->1->10->7->11->13."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph6",
        "image": "data/hamilton/graph_visual/hard/graph6.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 10, and the edges are: (3,5) (8,5) (6,0) (7,0) (3,9) (10,4) (6,1) (9,4) (10,1) (9,8) (6,4) (0,2) (5,2) (3,7) (3,4) (2,3) (8,2) (6,10) (9,1) (6,9)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->6->1->10->4->9->8->5->2->3->7."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph504",
        "image": "data/hamilton/graph_visual/hard/graph504.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (10,14) (18,9) (13,4) (15,6) (13,1) (12,11) (0,15) (1,17) (7,17) (3,1) (7,18) (7,1) (9,0) (18,13) (4,7) (4,18) (4,16) (16,6) (5,10) (9,10) (5,4) (14,8) (11,15) (17,5) (0,8) (17,0) (14,7) (10,2) (6,9) (16,2) (6,18) (0,5) (18,11) (6,13) (13,2) (4,15) (12,10) (15,17) (13,14) (16,10) (6,5) (14,18) (2,8) (14,6) (2,18) (12,2) (10,15) (3,9) (3,10) (3,5) (12,16) (9,8) (17,13) (15,1) (16,8) (0,10) (4,12) (7,8) (17,8) (18,0) (9,1) (17,14) (6,3)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->15->6->16->4->13->1->17->7->18->9->3->5->10->14->8->2->12->11."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph178",
        "image": "data/hamilton/graph_visual/hard/graph178.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 12, and the edges are: (8,9) (7,8) (12,10) (4,6) (11,0) (2,3) (0,9) (6,5) (0,8) (0,12) (10,8) (5,11) (1,2) (10,1) (1,4) (5,9) (11,4) (2,7) (7,6) (1,3) (3,11) (11,8) (12,5) (2,10) (9,1) (5,1) (7,9) (11,7) (6,1) (8,1) (8,12) (10,4) (5,3) (7,4) (2,4) (4,8) (9,10) (4,3) (9,3) (0,4) (11,2) (10,0) (3,6) (7,0) (2,9) (12,1) (11,6) (8,2) (5,0) (8,6) (4,9)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->11->5->6->4->1->2->3->9->7->8->10->12."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph215",
        "image": "data/hamilton/graph_visual/hard/graph215.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 13, and the edges are: (8,10) (13,5) (2,9) (9,11) (11,13) (7,3) (13,3) (5,12) (2,6) (3,5) (7,2) (2,3) (10,11) (7,12) (3,9) (13,0) (8,9) (9,5) (3,4) (0,2) (8,3) (4,6) (0,4) (8,13) (10,9) (4,2) (5,6) (11,8) (2,8) (1,7) (11,0) (1,3) (1,8) (9,6) (3,0) (2,1) (12,11) (1,6) (0,9) (7,10) (12,1) (13,7) (7,11) (10,0) (11,5) (11,1) (6,12) (10,13) (5,1)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->13->5->12->7->3->2->9->11->10->8->1->6->4."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph149",
        "image": "data/hamilton/graph_visual/hard/graph149.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 12, and the edges are: (9,3) (7,3) (10,4) (0,9) (9,1) (11,10) (9,4) (5,4) (12,7) (5,1) (9,2) (1,4) (10,7) (0,6) (5,9) (2,8) (12,3) (8,7) (8,4) (7,11) (7,2) (12,1) (4,7) (10,0) (4,12) (10,1) (0,2) (5,2) (3,11) (9,11) (8,1) (4,0) (5,6) (5,12) (11,5)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->9->3->7->11->10->4->12->1->8->2->5->6."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph580",
        "image": "data/hamilton/graph_visual/hard/graph580.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 19, and the edges are: (10,0) (19,13) (6,12) (16,0) (19,3) (7,0) (10,8) (3,8) (18,7) (13,16) (15,11) (2,5) (17,4) (19,18) (19,12) (11,7) (5,7) (9,14) (2,17) (2,12) (14,4) (15,18) (10,5) (18,5) (0,17) (2,8) (13,15) (14,6) (5,4) (11,17) (0,19) (10,15) (4,7) (9,11) (18,0) (14,8) (14,16) (12,14) (3,17) (14,3) (2,4) (13,8) (18,8) (1,13) (17,15) (15,0) (7,19) (18,13) (13,6) (16,6) (0,6) (13,5) (1,2) (15,14) (12,9) (14,0) (5,17) (15,19) (1,10) (5,11) (6,8) (16,4) (4,1) (13,3) (11,3) (19,16) (8,17) (1,14) (8,9) (12,10) (9,2) (3,16) (3,9) (19,5) (15,9) (3,4) (12,0) (1,18) (17,16) (14,7) (10,13) (18,9) (6,4) (6,7) (13,14) (6,9) (17,12) (19,6) (5,12) (19,11) (11,12) (18,11) (15,2) (17,10) (0,2) (9,16) (6,3) (1,7) (9,19) (3,15) (3,10) (8,7) (13,2) (1,19) (17,13) (4,18) (10,7) (14,17) (17,7) (5,0)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->10->8->3->19->13->16->14->9->11->15->18->7->5->2->17->12->6->4->1."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph222",
        "image": "data/hamilton/graph_visual/hard/graph222.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 13, and the edges are: (6,9) (3,6) (6,12) (3,11) (3,7) (2,1) (5,8) (9,1) (0,3) (9,0) (4,7) (5,1) (2,5) (7,10) (5,3) (1,4) (9,8) (13,6) (13,8) (9,7) (7,5) (4,10) (0,11) (9,13) (3,2) (3,9) (12,0) (5,0) (13,12) (10,8) (1,13) (13,5) (3,4) (6,8) (0,2) (0,10) (13,2) (11,12) (12,4) (0,4) (11,7) (11,2) (11,5) (0,7) (3,1) (5,4) (7,12) (7,6) (6,10) (1,0) (9,2) (6,4) (11,10) (10,3) (13,10)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->6->9->1->2->5->8->13->12->11->7->4->10."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph407",
        "image": "data/hamilton/graph_visual/hard/graph407.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 16, and the edges are: (12,0) (6,15) (9,11) (8,14) (9,0) (1,10) (11,0) (11,4) (1,7) (0,15) (3,9) (3,2) (12,15) (15,3) (5,9) (15,8) (10,12) (16,11) (13,4) (15,9) (15,5) (1,3) (10,16) (11,8) (7,2) (14,6) (15,11) (7,0) (7,8) (16,7) (2,8) (12,1) (6,3) (14,9) (7,11) (0,5) (13,1) (14,5) (2,10) (3,5) (14,1) (12,3) (3,14) (0,14) (8,4) (16,3) (7,15) (4,5) (13,0) (5,11) (7,5) (12,4) (6,16) (12,14) (5,16) (9,6) (9,10) (14,15) (2,14) (14,13) (0,3) (10,5) (6,12) (9,2) (13,9) (9,12) (6,11) (13,6) (4,9) (4,6) (16,12) (11,13) (0,16) (6,10) (8,1) (12,11) (7,12) (3,7) (7,9) (4,15)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->12->15->6->14->8->11->9->3->2->7->1->10->16->5->4->13."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph260",
        "image": "data/hamilton/graph_visual/hard/graph260.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 14, and the edges are: (8,9) (1,3) (5,2) (0,10) (10,2) (14,11) (2,7) (5,0) (8,3) (9,1) (4,7) (11,6) (9,10) (2,6) (10,14) (2,3) (12,14) (6,13) (1,14) (11,10) (4,9) (9,12) (1,11) (13,1) (12,3) (5,8) (11,5) (10,3) (7,3) (14,6) (0,7) (8,13) (8,4) (9,0)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->10->2->5->8->9->1->13->6->11->14->12->3->7->4."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph526",
        "image": "data/hamilton/graph_visual/hard/graph526.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (2,6) (6,7) (14,9) (4,6) (13,17) (6,9) (16,9) (17,2) (5,16) (2,16) (7,14) (8,18) (17,4) (18,10) (18,7) (10,2) (18,16) (6,1) (10,6) (9,15) (11,9) (15,18) (3,11) (8,5) (16,11) (12,5) (1,16) (9,1) (17,5) (4,18) (7,17) (14,10) (0,4) (1,11) (10,15) (10,3) (3,13) (9,3) (8,17) (13,16) (14,6) (5,11) (5,14) (18,5) (5,4) (6,11) (2,8) (13,11) (3,17) (7,12) (6,3) (3,15) (9,5) (10,9) (0,1) (7,10) (8,16) (0,5) (4,10) (16,4) (0,6) (10,8) (13,7) (16,15) (15,13) (8,7) (3,2) (14,0) (13,4) (17,11) (0,16) (16,12) (7,16) (18,14) (10,11) (5,15) (1,15) (12,11) (2,5) (0,3) (13,14) (4,8) (13,12) (11,14) (4,12) (10,5) (11,4) (3,4) (1,18) (11,7) (10,13) (18,2) (12,14) (15,4) (14,15) (18,11) (1,4) (9,0) (7,9)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->4->6->2->17->13->3->11->9->14->7->18->8->5->12->16->1->15->10."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph259",
        "image": "data/hamilton/graph_visual/hard/graph259.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 14, and the edges are: (9,10) (8,13) (2,12) (12,1) (5,1) (13,12) (14,7) (0,14) (12,5) (6,1) (10,0) (9,4) (8,2) (5,2) (14,10) (13,9) (0,13) (5,4) (7,3) (12,3) (0,3) (11,13) (4,11) (5,6) (3,14) (13,2) (3,13) (10,3) (10,6) (9,12) (5,3) (12,0) (11,6) (13,6) (9,2) (7,1) (7,5) (14,4) (4,13) (10,11)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->14->7->3->12->2->8->13->9->10->6->1->5->4->11."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph435",
        "image": "data/hamilton/graph_visual/hard/graph435.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 17, and the edges are: (13,11) (16,7) (2,7) (14,9) (9,10) (13,12) (17,0) (12,6) (0,7) (17,8) (0,15) (6,17) (4,1) (12,4) (2,12) (2,11) (9,5) (3,2) (11,16) (12,17) (10,6) (3,16) (3,11) (16,15) (5,3) (1,16) (15,12) (5,17) (2,10) (11,14) (4,0) (6,2) (12,0) (8,14) (2,4) (12,16) (3,17) (3,15) (10,8) (1,0) (9,15) (15,2) (13,6) (17,13) (15,6) (0,2) (1,7) (0,16) (3,7) (6,14) (9,7) (0,10) (14,3) (4,3) (9,0) (3,9) (14,2) (16,9) (10,13) (13,8) (14,0) (2,13) (6,16) (4,15) (16,4)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->17->8->14->9->10->6->12->13->11->2->7->16->1->4->15->3->5."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph226",
        "image": "data/hamilton/graph_visual/hard/graph226.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 13, and the edges are: (13,7) (7,9) (8,2) (11,2) (8,9) (13,2) (8,13) (13,5) (12,1) (12,9) (5,2) (2,0) (5,10) (0,1) (0,7) (8,3) (5,9) (6,3) (6,2) (13,6) (7,5) (8,12) (6,7) (9,10) (11,13) (6,1) (12,5) (12,7) (0,8) (13,12) (3,1) (1,13) (0,5) (9,0) (3,5) (3,7) (0,13) (6,12) (12,3) (5,8) (11,5) (1,9) (11,7) (4,13) (6,4) (7,10) (4,9) (8,1)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->8->9->7->12->1->3->6->4->13->11->5->10."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph354",
        "image": "data/hamilton/graph_visual/hard/graph354.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (10,12) (10,8) (6,9) (8,14) (3,1) (10,14) (5,9) (8,6) (11,3) (13,0) (9,4) (8,0) (0,14) (15,10) (3,7) (1,13) (8,4) (1,11) (3,10) (13,9) (2,15) (9,11) (9,12) (14,11) (0,5) (13,5) (6,0) (5,11) (3,15) (6,11) (0,9) (9,1) (6,7) (10,11) (14,9) (2,11) (4,14) (4,2) (12,13) (15,1) (12,8) (2,12) (10,1) (2,8) (2,5) (4,5) (4,3) (0,4) (14,5) (6,5) (13,4) (10,7) (11,13) (5,3) (11,7) (1,5) (13,7) (15,0) (1,7) (8,1) (14,7) (0,10) (1,0) (12,3) (3,0) (10,5) (2,6)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->13->1->3->11->9->6->8->4->5->14->7->10->12->2->15."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph593",
        "image": "data/hamilton/graph_visual/hard/graph593.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 19, and the edges are: (14,19) (17,16) (18,15) (15,4) (8,5) (15,8) (4,3) (15,16) (16,1) (3,16) (0,5) (10,8) (4,19) (7,6) (18,17) (14,8) (10,7) (5,16) (6,18) (4,7) (0,2) (15,11) (9,1) (13,0) (17,0) (10,15) (10,13) (7,2) (0,19) (3,10) (14,4) (11,2) (6,1) (5,2) (11,19) (9,14) (19,10) (8,0) (2,6) (14,6) (1,12) (2,10) (1,11) (13,3) (18,19) (11,7) (16,14) (10,18) (16,19) (3,1) (2,4) (11,0) (8,13) (8,19) (13,19) (15,0) (12,8) (3,14) (0,12) (19,9) (11,13) (7,14) (16,12) (11,18) (15,14) (16,11) (9,0) (17,13) (11,9) (4,8) (8,2) (3,9) (12,17) (13,7) (12,18) (3,12) (11,3) (0,4) (16,2) (19,3) (14,10) (14,18) (4,17) (16,13) (1,13) (11,8) (3,8) (4,10) (15,9) (15,5) (18,13) (16,0) (5,14) (4,9) (12,14) (6,12) (6,0) (8,9) (6,3) (7,8) (13,6) (13,12) (15,13) (17,5) (16,9) (19,12) (10,9) (7,17) (5,18) (14,2) (16,8) (17,15) (3,0) (17,3) (11,5) (15,3) (8,18) (0,7)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->5->8->15->18->17->16->1->9->14->19->4->3->10->7->6->2->11->13->12."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph104",
        "image": "data/hamilton/graph_visual/hard/graph104.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 11, and the edges are: (11,8) (0,3) (8,3) (7,9) (7,1) (5,3) (3,6) (10,6) (1,2) (5,7) (1,5) (3,1) (11,0) (8,1) (0,1) (0,2) (5,6) (4,2) (1,9) (2,5) (10,4) (11,9) (9,10) (0,7) (9,3) (2,9) (0,6) (2,3) (5,11) (3,4) (7,8) (6,4) (5,0) (9,6) (6,7) (3,7) (10,11) (4,11) (2,11) (1,11) (6,11) (10,0) (5,10)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->8->11->9->7->1->2->4->10->6->5."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph380",
        "image": "data/hamilton/graph_visual/hard/graph380.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 16, and the edges are: (13,9) (4,14) (10,0) (2,1) (13,2) (12,3) (8,4) (9,2) (0,9) (5,3) (8,14) (8,7) (15,13) (16,0) (9,4) (4,6) (1,11) (10,12) (4,13) (3,10) (10,16) (2,6) (12,5) (10,7) (1,4) (4,16) (15,8) (6,3) (13,10) (3,11) (16,7) (10,15) (13,7) (7,9) (8,12) (8,2) (5,14) (16,11) (6,12) (12,4) (2,10) (10,1) (7,5) (8,5) (1,3) (12,13) (16,3) (14,2) (16,9) (9,10) (11,12) (10,11) (14,15)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->10->12->3->5->14->4->8->15->13->9->7->16->11->1->2->6."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph263",
        "image": "data/hamilton/graph_visual/hard/graph263.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 14, and the edges are: (7,0) (2,1) (5,2) (2,10) (11,8) (12,1) (13,12) (3,10) (14,1) (14,3) (11,13) (2,3) (5,0) (11,6) (4,5) (7,8) (9,1) (9,8) (9,6) (8,0) (10,7) (6,12) (12,9) (10,14) (12,11) (11,9) (14,13) (10,9) (4,2) (6,7) (4,1) (14,8) (8,1) (9,7) (1,11) (4,13) (3,13) (3,7) (13,10) (13,5) (4,0) (12,5) (10,0) (14,4) (11,5) (9,4)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->7->8->11->13->12->1->2->5->4->14->3->10->9->6."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph584",
        "image": "data/hamilton/graph_visual/hard/graph584.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 19, and the edges are: (15,2) (10,9) (18,12) (16,14) (2,6) (17,6) (16,4) (6,10) (5,3) (2,13) (18,14) (9,0) (4,2) (1,14) (16,9) (10,0) (19,17) (2,1) (10,8) (16,17) (0,16) (9,14) (3,13) (2,18) (18,5) (10,15) (16,11) (2,10) (2,12) (7,8) (1,19) (3,6) (14,10) (17,9) (10,12) (4,10) (2,19) (4,13) (11,18) (3,17) (6,18) (15,16) (4,12) (7,0) (5,9) (15,8) (9,18) (18,0) (17,0) (7,4) (4,0) (15,11) (3,9) (6,5) (0,15) (5,11) (9,4) (5,7) (8,16) (4,1) (7,6) (18,13) (0,14) (8,0) (9,7) (6,19) (4,6) (1,12) (0,2) (16,10) (9,1) (10,18) (12,8) (18,17) (8,11) (4,18) (12,15) (12,11) (1,0) (5,19) (17,12) (14,11) (3,16) (3,14) (19,15) (10,17) (13,19) (8,4) (12,13) (14,5) (3,10) (15,5) (3,12) (4,5) (11,1) (16,18) (18,3) (12,19) (4,3) (8,19) (12,9) (7,11) (6,13) (0,12) (13,5) (11,4) (15,18) (3,7) (4,15) (5,12) (19,10)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->9->10->6->2->15->16->14->18->12->4->13->3->5->7->8->11->1->19->17."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph485",
        "image": "data/hamilton/graph_visual/hard/graph485.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (11,7) (16,2) (5,12) (9,10) (10,12) (7,15) (13,3) (2,6) (14,16) (9,12) (4,10) (3,0) (10,16) (15,9) (10,3) (12,2) (11,10) (8,13) (6,1) (13,15) (8,14) (18,10) (7,17) (1,3) (1,18) (1,7) (8,4) (0,18) (8,0) (11,2) (17,6) (4,9) (16,17) (2,7) (6,14) (16,1) (9,18) (18,14) (17,1) (6,0) (16,3) (14,3) (12,6) (18,6) (1,13) (1,2) (4,5) (17,14) (14,2) (5,10) (7,5) (15,4) (16,13) (17,5) (14,15) (3,17) (1,15) (18,5) (3,12) (12,0) (12,16) (17,4) (16,9)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->13->8->14->16->2->6->1->18->10->9->12->5->17->4->15->7->11."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph123",
        "image": "data/hamilton/graph_visual/hard/graph123.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 12, and the edges are: (10,9) (0,8) (5,12) (3,10) (12,11) (6,1) (9,12) (10,0) (4,8) (2,0) (5,7) (10,5) (7,1) (6,11) (3,2) (11,8) (11,1) (5,1) (11,5) (12,8) (7,0) (3,11) (8,10) (6,4) (12,7) (3,9) (6,12) (9,5)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->8->4->6->1->7->5->10->9->12->11->3->2."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph518",
        "image": "data/hamilton/graph_visual/hard/graph518.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (3,14) (12,17) (12,16) (2,6) (14,11) (11,13) (2,7) (3,12) (12,13) (2,15) (15,4) (10,15) (10,14) (0,10) (15,16) (3,18) (15,13) (9,18) (8,3) (17,10) (9,4) (6,17) (3,4) (4,0) (1,17) (8,17) (7,9) (4,5) (15,11) (13,6) (3,10) (9,1) (5,18) (14,2) (12,15) (11,0) (16,1) (0,14) (5,13) (4,12) (18,10) (2,13) (5,17) (8,18) (4,13) (0,1) (2,17) (15,7) (1,13) (1,6) (14,15) (7,3) (1,5) (9,0) (16,13) (11,5) (14,17) (5,6) (0,2) (18,6) (1,10) (4,7) (18,16) (10,7) (12,9) (3,1) (18,0) (7,16) (12,1) (14,5) (15,3) (9,8) (16,17) (15,6) (7,17) (16,9) (10,16) (13,10) (8,7) (13,9) (7,1) (7,11) (4,6) (4,8) (7,0) (5,2) (18,2) (6,14) (15,5) (9,6) (5,10) (1,18) (12,0) (6,7) (8,11) (11,1) (14,18) (16,6) (11,12) (8,14) (17,18) (7,12) (4,14) (9,15)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->10->15->2->6->17->12->16->1->9->18->3->14->11->13->5->4->7->8."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph9",
        "image": "data/hamilton/graph_visual/hard/graph9.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 10, and the edges are: (6,0) (4,10) (10,3) (1,5) (6,3) (3,4) (10,7) (5,4) (1,2) (1,7) (3,7) (3,9) (0,9) (8,0) (3,8) (5,10) (6,7) (7,8) (4,6) (4,7) (1,3) (10,1) (9,4) (5,6) (7,0) (4,0)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->6->7->8->3->9->4->10->5->1->2."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph10",
        "image": "data/hamilton/graph_visual/hard/graph10.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 10, and the edges are: (0,8) (3,7) (4,8) (7,10) (9,8) (1,2) (2,10) (6,8) (10,4) (7,2) (4,0) (5,3) (0,5) (4,9) (9,0) (5,7) (9,10) (3,10) (2,6) (0,3) (5,6) (1,0)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->8->4->9->10->7->3->5->6->2->1."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph509",
        "image": "data/hamilton/graph_visual/hard/graph509.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (6,12) (14,18) (7,8) (18,17) (14,4) (10,1) (18,11) (11,14) (12,4) (15,10) (7,10) (11,2) (9,16) (7,3) (12,7) (6,2) (2,10) (12,1) (9,6) (17,12) (8,11) (3,4) (14,5) (8,17) (17,16) (5,8) (5,11) (6,3) (0,4) (11,0) (14,12) (12,9) (18,6) (8,6) (15,16) (8,4) (8,3) (7,16) (9,13) (16,3) (9,1) (13,6) (18,12) (11,4) (14,3) (3,18) (16,6) (0,17) (9,7) (12,5) (15,0) (3,17) (1,11) (9,10) (3,12) (10,17) (15,6) (3,15) (7,13) (10,8) (0,9) (15,2) (5,13) (6,0) (7,4) (5,17) (6,1) (13,8) (0,14) (14,8) (14,13) (18,4) (16,8) (2,17)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->4->14->18->17->12->6->2->11->8->7->3->16->15->10->1->9->13->5."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph75",
        "image": "data/hamilton/graph_visual/hard/graph75.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 11, and the edges are: (2,0) (4,0) (0,9) (7,2) (10,6) (4,5) (9,1) (3,2) (7,4) (1,0) (9,3) (8,11) (0,6) (11,2) (10,7) (3,6) (8,1) (8,5) (7,3) (0,11) (10,8) (1,4) (10,0) (5,0) (2,8) (1,6)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->7->10->6->3->9->1->4->5->8->11."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph486",
        "image": "data/hamilton/graph_visual/hard/graph486.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (6,10) (7,4) (5,16) (6,12) (5,8) (0,7) (9,1) (9,6) (13,15) (3,6) (17,10) (9,7) (6,14) (14,12) (5,15) (9,8) (16,18) (9,5) (7,6) (9,14) (0,2) (18,11) (5,12) (11,6) (12,4) (17,3) (5,18) (1,4) (11,1) (1,13) (3,15) (8,15) (4,0) (13,4) (0,14) (10,7) (4,18) (14,18) (16,0) (11,5) (6,2) (2,10) (9,3) (1,10) (1,6) (17,14) (0,10) (16,10) (3,16) (7,14) (0,1) (12,7) (1,14) (1,7) (16,15) (1,2) (5,13) (10,12) (3,5) (13,18) (2,9) (18,15) (18,17) (2,15) (0,3) (15,0)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->7->4->12->6->10->17->3->16->5->8->9->14->18->11->1->13->15->2."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph188",
        "image": "data/hamilton/graph_visual/hard/graph188.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 13, and the edges are: (6,4) (0,11) (13,2) (5,13) (2,6) (11,10) (4,10) (2,7) (7,1) (13,7) (10,7) (5,11) (10,6) (6,13) (12,3) (11,13) (11,2) (2,9) (12,9) (7,5) (3,13) (4,12) (12,7) (5,6) (12,8) (4,3) (7,9) (8,10) (9,4) (10,3) (8,3) (4,8) (1,9) (11,6) (1,5) (6,0)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->11->10->4->6->2->13->5->7->1->9->12->3->8."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph474",
        "image": "data/hamilton/graph_visual/hard/graph474.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 17, and the edges are: (7,16) (2,7) (0,1) (2,17) (13,12) (13,17) (14,5) (11,6) (10,1) (10,7) (12,17) (0,5) (7,0) (4,7) (13,5) (10,0) (12,2) (13,2) (14,15) (6,5) (0,6) (17,9) (0,17) (8,10) (3,16) (6,17) (4,12) (7,9) (14,16) (14,3) (12,11) (11,5) (3,5) (11,17) (2,0) (13,0) (15,16) (5,4) (3,11) (10,15) (9,16) (11,14) (4,6) (1,17) (9,4) (12,5) (14,8) (5,9) (4,17) (15,1) (13,10) (4,3) (15,9) (16,6) (6,12) (4,0) (7,5) (16,1) (1,12) (3,8) (1,2) (13,6) (15,12) (4,11) (16,10) (15,6) (9,0) (3,0) (15,4) (2,8) (16,0) (2,3) (14,9) (6,3) (2,9) (15,11) (10,14) (3,12) (1,3) (10,12) (15,8) (2,6) (4,10) (8,7) (1,4) (8,16) (9,8) (13,1) (8,17) (5,10) (15,17) (9,6) (9,3) (11,10)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->1->10->7->16->3->14->5->13->12->17->2->8->15->9->4->6->11."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph444",
        "image": "data/hamilton/graph_visual/hard/graph444.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 17, and the edges are: (15,14) (16,0) (14,5) (14,1) (5,10) (6,4) (17,0) (9,16) (1,8) (7,11) (3,1) (11,9) (7,17) (0,8) (5,13) (4,12) (9,14) (13,17) (5,4) (10,11) (16,4) (7,12) (16,5) (10,9) (7,0) (4,13) (8,2) (12,0) (7,14) (11,16) (9,3) (8,15) (16,3) (17,11) (0,2) (4,10) (7,15) (17,10) (15,12) (10,6) (4,1) (12,1) (7,3) (2,1) (14,12) (6,16) (12,17) (10,16) (12,8) (17,16) (1,16) (17,4) (5,2) (8,11) (6,1) (13,12) (4,15) (9,1) (6,12) (3,6) (6,15) (8,13) (4,3) (3,8) (11,13) (1,7) (5,17) (4,11) (11,14)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->16->9->11->7->17->13->5->14->15->8->2->1->3->6->10->4->12."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph254",
        "image": "data/hamilton/graph_visual/hard/graph254.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 14, and the edges are: (8,5) (6,13) (14,3) (14,7) (12,7) (8,4) (14,2) (4,9) (5,7) (10,1) (3,2) (0,5) (4,11) (9,12) (9,14) (10,4) (6,10) (1,12) (13,10) (7,8) (4,3) (5,3) (9,1) (11,8) (14,0) (10,8) (0,7) (5,11) (14,10) (3,8) (11,13) (6,14) (5,12) (2,8) (4,2) (13,12) (9,2) (5,9) (1,11) (13,9) (12,11) (0,4) (2,12)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->5->8->4->9->1->10->6->13->11->12->7->14->3->2."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph438",
        "image": "data/hamilton/graph_visual/hard/graph438.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 17, and the edges are: (0,6) (16,9) (8,3) (0,7) (5,15) (3,7) (4,14) (11,6) (5,3) (9,17) (4,17) (3,12) (3,16) (4,8) (9,2) (13,6) (7,14) (2,11) (15,7) (3,15) (11,4) (16,14) (15,12) (15,17) (0,3) (12,0) (17,16) (1,4) (14,3) (4,15) (4,7) (5,0) (16,2) (5,16) (10,11) (3,11) (4,2) (9,0) (10,12) (8,17) (7,13) (17,14) (14,0) (0,17) (15,2) (10,2) (9,10) (17,5) (13,17) (4,10) (5,6) (12,13) (2,17) (5,8) (1,14) (7,8) (11,0) (16,10) (11,9) (1,0) (2,14) (1,15) (11,14)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->6->11->2->9->16->3->8->4->17->5->15->1->14->7->13->12->10."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph235",
        "image": "data/hamilton/graph_visual/hard/graph235.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 13, and the edges are: (5,8) (1,6) (7,10) (9,2) (6,11) (3,11) (4,1) (8,10) (0,1) (13,12) (1,12) (0,2) (9,0) (9,5) (11,10) (11,5) (0,8) (5,2) (1,9) (3,9) (12,9) (4,10) (8,3) (5,1) (11,1) (3,10) (6,2) (13,2) (0,7) (2,7) (4,12) (3,0) (9,4) (5,13) (5,0) (5,6) (11,8) (5,7) (0,12) (2,3) (8,9) (11,13) (4,5) (13,1) (6,3) (3,13) (7,11) (8,2) (9,6) (6,13)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->1->6->11->3->9->2->13->12->4->10->7->5->8."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph467",
        "image": "data/hamilton/graph_visual/hard/graph467.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 17, and the edges are: (1,8) (14,17) (6,3) (10,14) (9,5) (6,5) (10,11) (17,2) (7,17) (14,13) (9,15) (0,6) (0,17) (0,3) (17,6) (13,2) (7,12) (15,13) (11,15) (5,12) (15,6) (13,17) (16,11) (13,7) (1,16) (4,7) (9,14) (15,16) (2,8) (11,14) (13,4) (15,5) (7,11) (16,8) (13,6) (7,5) (8,15) (4,3) (4,2) (10,2) (0,14) (1,5) (7,9) (10,17) (16,14) (14,1) (6,11) (14,6) (16,7) (8,14) (2,12) (2,14) (4,8) (13,9) (13,5) (5,8) (5,14) (11,0) (13,3) (1,3) (14,3) (11,9) (3,11) (9,10) (11,5) (7,0) (5,4) (10,8) (6,9) (12,3) (17,15) (8,3) (2,16) (3,9) (9,0) (15,0) (10,12) (4,6) (1,11) (8,11) (9,17) (0,12) (2,9) (14,15) (3,17) (11,17) (11,4) (9,4) (12,16) (13,8) (0,5) (2,11) (2,5)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->6->3->4->7->17->14->10->11->15->9->5->12->2->13->8->1->16."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph512",
        "image": "data/hamilton/graph_visual/hard/graph512.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (17,12) (5,18) (16,14) (7,9) (1,10) (5,8) (2,11) (0,6) (16,6) (16,9) (2,7) (0,12) (1,4) (18,10) (9,18) (17,8) (18,13) (3,18) (2,8) (14,17) (8,3) (17,7) (10,4) (12,7) (16,15) (14,2) (1,12) (15,14) (3,7) (5,16) (1,3) (12,5) (13,12) (6,13) (11,4) (6,11) (0,9) (15,4) (7,1) (3,2) (10,11) (9,17) (16,13) (13,5) (2,17) (5,4) (2,5) (18,12) (1,16) (13,3) (0,3) (8,0) (1,15) (6,5) (0,16) (5,7) (11,18) (5,0) (15,12) (18,15) (16,18) (11,16) (1,17) (14,7) (10,7) (16,17) (11,3) (10,16) (8,14) (3,9) (7,8) (7,13) (18,2) (14,3) (7,18) (8,11) (6,10) (12,14) (2,13) (2,0) (14,9) (8,9) (5,14) (17,15) (10,2) (11,7) (8,16) (2,4) (10,17) (3,12) (15,11) (10,9) (18,0) (3,17) (6,9) (9,13) (7,15) (8,12) (17,0) (13,8) (11,12) (0,1) (17,13) (0,10) (0,7) (4,16) (1,8)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->6->16->14->17->12->7->9->18->5->8->2->11->10->4->15->1->3->13."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph302",
        "image": "data/hamilton/graph_visual/hard/graph302.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (12,9) (2,13) (4,0) (3,12) (12,7) (15,10) (10,7) (12,15) (1,9) (15,5) (7,5) (11,4) (1,7) (8,9) (2,15) (1,11) (8,7) (13,6) (8,4) (3,0) (8,5) (12,1) (5,13) (10,0) (14,15) (5,12) (2,5) (11,10) (6,12) (9,14) (3,8) (3,14) (13,1)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->4->11->1->9->12->3->14->15->10->7->8->5->2->13->6."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph396",
        "image": "data/hamilton/graph_visual/hard/graph396.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 16, and the edges are: (14,7) (0,6) (13,8) (15,6) (6,9) (16,2) (4,3) (10,6) (10,11) (12,9) (3,1) (9,4) (14,0) (12,11) (8,16) (7,1) (5,15) (5,3) (4,2) (16,4) (7,13) (1,12) (13,12) (3,2) (11,4) (12,4) (2,7) (15,0) (14,8) (8,6) (10,12) (11,8) (4,1) (11,5) (16,3) (14,10) (6,5) (1,15) (6,14) (7,10) (12,14) (0,8) (6,2) (13,11) (13,9) (4,7) (0,9) (12,5) (1,10) (14,16) (7,11) (4,13) (16,9) (0,1) (12,7) (2,11) (5,9) (12,3) (3,9) (9,2) (12,0) (2,13) (1,8) (15,4) (0,7) (15,10) (12,15) (5,8) (16,5) (12,8) (16,0) (5,10) (13,10) (11,1) (13,15) (3,0) (14,4) (10,9) (1,9) (10,3) (9,7) (3,13) (8,15) (16,12)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->6->15->5->3->4->9->12->11->10->14->7->1->8->13->2->16."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph154",
        "image": "data/hamilton/graph_visual/hard/graph154.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 12, and the edges are: (7,5) (6,11) (12,4) (7,10) (2,7) (2,1) (7,1) (7,11) (5,10) (8,4) (11,9) (3,9) (8,11) (0,3) (3,11) (2,3) (0,12) (2,6) (10,9) (0,7) (10,3) (11,10) (5,12) (1,8) (3,6) (4,1) (10,8) (0,10) (1,5) (6,1) (4,0) (10,1) (8,9) (0,11) (2,4) (6,10) (6,12) (3,7) (9,7) (2,10) (11,2) (6,5) (10,12) (3,5) (4,10) (6,4) (9,2) (7,4) (5,2) (1,9) (7,12) (8,0) (5,9) (8,6)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->9->11->6->2->7->5->10->8->1->4->12."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph139",
        "image": "data/hamilton/graph_visual/hard/graph139.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 12, and the edges are: (12,4) (3,5) (0,2) (7,9) (10,4) (3,9) (12,8) (7,6) (1,11) (10,8) (4,7) (12,1) (2,3) (2,9) (11,8) (7,3) (11,12) (11,9) (8,6) (0,7) (4,8) (5,2) (1,7) (3,12) (5,12) (6,4) (9,1) (1,3)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->3->5->12->4->10->8->11->1->9->7->6."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph261",
        "image": "data/hamilton/graph_visual/hard/graph261.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 14, and the edges are: (3,1) (14,9) (2,0) (7,3) (14,8) (2,5) (9,2) (6,0) (2,11) (3,0) (6,14) (12,1) (4,1) (13,14) (9,8) (8,5) (13,4) (13,8) (2,14) (13,11) (3,10) (0,5) (0,11) (4,10) (3,2) (5,13) (1,5) (6,7) (13,0) (10,12) (4,5) (7,12) (11,7) (13,12) (9,6) (2,8) (6,8) (12,11) (14,10) (14,4) (8,10)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->5->8->14->9->6->7->3->1->12->10->4->13->11."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph338",
        "image": "data/hamilton/graph_visual/hard/graph338.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (3,8) (8,5) (3,5) (6,7) (12,11) (2,7) (10,2) (2,4) (11,2) (14,7) (5,2) (6,1) (15,8) (2,3) (0,15) (15,13) (14,9) (4,8) (0,1) (11,1) (9,11) (0,8) (15,14) (10,12) (10,6) (10,7) (3,0) (0,4) (14,0) (4,3) (14,12) (10,9) (12,8) (8,1) (2,0) (14,5) (11,4) (4,13) (3,1) (15,12) (9,15) (9,3) (3,13) (15,6) (1,10) (0,10) (5,1) (6,13) (13,10) (3,14) (12,6) (10,8) (7,13) (7,4) (9,0) (2,6) (5,7) (9,7) (4,15) (5,12) (13,12) (15,7) (13,2) (10,4) (2,1)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->15->8->3->5->2->7->6->1->11->12->14->9->10->13->4."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph404",
        "image": "data/hamilton/graph_visual/hard/graph404.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 16, and the edges are: (7,11) (11,10) (8,5) (1,8) (7,13) (2,5) (9,11) (11,16) (9,5) (13,14) (7,2) (13,1) (13,3) (11,14) (9,14) (12,14) (0,13) (0,14) (15,1) (9,8) (15,0) (13,6) (8,6) (14,5) (16,3) (11,12) (12,16) (9,16) (1,6) (4,1) (7,14) (11,15) (0,16) (12,4) (4,9) (8,7) (15,13) (2,14) (2,6) (4,8) (10,5) (2,3) (1,5) (11,6) (10,0) (3,5) (9,12) (8,14) (16,7) (8,15) (10,1) (5,15) (3,7) (6,14) (8,13) (4,3) (15,10) (5,6) (5,11) (1,16) (11,1) (10,4) (2,16) (3,10) (2,9) (1,3) (6,15) (1,2) (7,15) (10,2) (1,7) (2,4) (12,2) (9,13) (11,8) (0,3) (1,9) (16,5) (12,6) (1,14) (14,15) (13,10) (8,16) (11,3) (3,14) (10,14) (13,4)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->13->7->11->10->5->8->1->15->6->2->14->9->16->3->4->12."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph367",
        "image": "data/hamilton/graph_visual/hard/graph367.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 16, and the edges are: (9,10) (3,9) (14,1) (1,5) (3,5) (9,13) (5,0) (11,1) (8,4) (2,8) (8,10) (9,14) (12,4) (8,12) (14,13) (6,15) (9,2) (13,4) (15,13) (4,14) (3,6) (12,7) (14,7) (15,5) (0,15) (3,16) (7,10) (9,7) (15,3) (3,8) (11,2) (15,4) (6,7) (1,6) (7,16) (10,16) (13,5) (0,1) (10,4) (9,8) (11,6) (14,16) (13,3) (6,0) (16,1) (15,9) (8,1) (10,11) (12,16) (10,6) (11,15) (8,7) (9,16) (0,9) (10,0) (5,11) (10,13) (6,2) (11,13) (1,7)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->5->1->14->9->10->8->4->12->7->6->2->11->15->13->3->16."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph92",
        "image": "data/hamilton/graph_visual/hard/graph92.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 11, and the edges are: (3,1) (1,5) (3,4) (0,8) (1,11) (10,9) (11,3) (10,0) (10,8) (8,2) (4,11) (1,10) (3,2) (1,7) (2,9) (3,10) (8,7) (1,9) (6,4) (8,11) (8,5) (8,4) (0,4) (11,6) (7,10) (4,5) (2,4) (7,11) (8,1) (6,8) (10,2) (5,6) (6,7) (3,5) (6,2) (9,0) (4,7) (11,0) (10,6) (9,4) (2,5) (6,9)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->8->10->9->2->3->1->5->4->11->6->7."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph572",
        "image": "data/hamilton/graph_visual/hard/graph572.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 19, and the edges are: (0,16) (19,11) (15,19) (8,19) (9,12) (10,9) (8,12) (12,0) (7,0) (10,14) (5,1) (1,8) (19,18) (1,11) (15,9) (11,9) (11,4) (1,3) (8,0) (16,7) (1,4) (8,4) (19,17) (7,15) (6,2) (10,5) (1,14) (3,18) (2,15) (3,5) (7,14) (11,6) (16,3) (6,5) (4,19) (15,18) (11,17) (15,6) (10,11) (7,3) (19,13) (2,9) (4,9) (11,12) (19,9) (3,17) (6,3) (12,3) (3,14) (18,4) (12,15) (2,11) (5,2) (10,12) (0,9) (14,18) (17,4) (12,13) (15,4) (12,14) (8,15) (10,7) (6,13) (12,17) (14,16) (1,9) (10,13) (19,2) (1,19) (14,4) (13,11) (17,8) (1,2) (13,0) (15,1) (6,9) (16,4) (12,7) (3,0) (0,6) (14,5) (16,5) (17,18) (7,19) (7,6) (14,8) (3,2) (16,12) (15,5) (18,7) (13,16) (7,1) (16,1) (6,18) (9,16) (19,3) (8,3) (2,8) (11,0) (19,0) (9,18) (15,17) (10,4) (9,17) (8,13) (19,6) (0,2) (6,12) (14,6) (8,16) (5,12) (10,2) (16,10) (5,18) (15,13) (10,18) (13,2) (17,16) (0,1) (2,18) (7,9)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->16->7->15->19->11->1->5->10->9->12->8->4->18->14->6->13->2->3->17."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph245",
        "image": "data/hamilton/graph_visual/hard/graph245.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 14, and the edges are: (1,4) (11,6) (2,11) (4,14) (0,9) (14,6) (0,2) (0,6) (13,1) (1,0) (12,14) (0,3) (12,0) (3,2) (9,8) (6,2) (8,11) (13,8) (12,6) (3,14) (7,10) (12,5) (1,5) (2,1) (4,5) (3,6) (1,14) (13,10) (1,9) (8,2) (2,12) (12,8) (0,14) (6,1) (2,13) (14,2) (7,2) (10,9) (8,0) (11,7) (6,13) (11,3) (11,4) (13,7) (9,12) (5,8) (9,13) (13,11) (5,10) (11,1) (6,4)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->9->8->11->6->14->4->1->13->7->10->5->12->2->3."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph68",
        "image": "data/hamilton/graph_visual/hard/graph68.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 11, and the edges are: (11,9) (8,5) (10,11) (0,6) (6,10) (10,0) (1,9) (7,10) (9,8) (10,4) (3,8) (1,5) (10,9) (5,7) (11,3) (5,9) (9,4) (7,6) (6,4) (1,2) (3,2) (7,9) (0,1) (7,1) (8,11) (3,4)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->6->10->11->9->8->5->7->1->2->3->4."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph363",
        "image": "data/hamilton/graph_visual/hard/graph363.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 16, and the edges are: (12,6) (1,16) (14,7) (1,6) (15,4) (1,13) (4,16) (15,11) (13,14) (2,4) (16,12) (16,13) (11,4) (4,7) (8,7) (5,11) (14,16) (6,15) (10,7) (15,3) (15,9) (16,9) (5,16) (10,1) (13,12) (2,8) (5,9) (7,15) (3,12) (14,8) (1,12) (5,6) (11,10) (9,14) (2,15) (12,2) (0,8) (1,5) (9,10) (10,3) (3,6) (1,7) (6,4) (13,10) (7,12) (13,3) (15,0) (16,2) (15,16) (12,11) (11,14) (9,1) (3,11) (11,0) (8,12) (4,8) (14,12) (6,9)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->8->7->14->13->1->16->4->15->11->5->9->10->3->6->12->2."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph303",
        "image": "data/hamilton/graph_visual/hard/graph303.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (11,2) (7,0) (0,4) (0,2) (2,3) (5,3) (8,0) (13,4) (7,6) (10,11) (13,9) (2,14) (0,5) (4,9) (1,4) (11,3) (14,7) (11,5) (6,15) (12,3) (15,2) (7,4) (15,7) (4,3) (0,12) (0,1) (3,10) (12,14) (8,15) (5,7) (4,2) (1,11) (8,9) (9,3) (5,12) (3,6) (8,11) (15,12) (3,13) (9,15) (11,4) (14,10) (10,9) (11,6) (0,10) (6,9) (10,4) (9,7)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->7->6->15->2->11->1->4->13->3->5->12->14->10->9->8."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph530",
        "image": "data/hamilton/graph_visual/hard/graph530.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (8,7) (3,4) (2,17) (15,16) (3,9) (5,17) (8,15) (12,5) (11,17) (16,9) (10,4) (15,0) (1,14) (4,9) (18,6) (17,4) (5,4) (12,0) (12,2) (10,2) (3,1) (18,4) (9,7) (14,2) (3,11) (18,15) (6,7) (7,12) (8,2) (2,16) (5,7) (15,11) (15,14) (13,4) (10,16) (18,0) (13,6) (8,0) (4,16) (11,18) (3,15) (6,12) (14,12) (3,6) (0,9) (17,12) (14,10) (4,12) (9,8) (5,8) (7,11) (1,13) (6,4) (17,10) (10,5) (3,0) (15,1) (12,11) (11,1) (15,17) (7,15) (1,2) (18,13) (18,16) (1,18) (2,7) (8,3) (6,5) (5,2) (3,7) (3,16) (10,11) (7,14) (14,6) (2,18) (11,4) (10,8) (5,14) (5,13) (5,15) (4,1) (0,16) (6,11) (5,0) (16,17) (8,1) (2,6) (7,1) (1,16) (0,1) (5,9) (12,8) (1,9) (18,14) (11,9) (13,3) (18,9) (10,13) (13,17) (8,11)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->15->16->9->3->4->10->2->17->5->12->7->8->1->14->6->13->18->11."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph582",
        "image": "data/hamilton/graph_visual/hard/graph582.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 19, and the edges are: (6,15) (8,2) (5,2) (11,18) (7,9) (12,17) (2,16) (10,15) (9,11) (7,1) (10,2) (0,19) (19,5) (2,7) (17,14) (9,14) (11,6) (2,3) (5,15) (18,3) (18,2) (12,0) (14,4) (18,14) (7,5) (7,14) (6,14) (18,19) (7,19) (3,4) (10,9) (0,17) (18,16) (2,17) (11,5) (13,6) (9,18) (13,16) (16,15) (15,3) (9,12) (3,8) (3,6) (10,12) (6,16) (6,17) (10,19) (10,17) (3,5) (19,9) (4,17) (11,1) (5,10) (13,9) (4,9) (19,16) (0,16) (19,4) (16,17) (16,8) (9,8) (19,15) (4,12) (13,0) (3,13) (17,5) (18,5) (1,8) (0,6) (13,7) (10,11) (12,7) (12,14) (18,1) (13,5) (18,4) (11,19) (19,12) (2,13) (8,12) (10,8) (2,9) (7,4) (8,5) (7,16) (11,8) (12,1) (15,0) (11,14) (7,0) (6,19) (17,8) (11,16) (16,4) (1,17) (17,15) (16,1) (8,6) (2,4) (1,3) (11,13) (6,2) (14,16)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->19->5->2->8->3->18->11->9->7->1->12->17->14->4->16->13->6->15->10."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph567",
        "image": "data/hamilton/graph_visual/hard/graph567.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 19, and the edges are: (2,17) (4,5) (7,3) (6,1) (8,1) (6,0) (2,10) (12,11) (2,3) (16,9) (15,7) (14,9) (13,16) (13,17) (10,15) (6,13) (4,14) (6,4) (18,12) (3,13) (10,11) (1,13) (9,6) (18,19) (5,7) (19,4) (6,10) (12,15) (14,17) (18,2) (1,16) (18,4) (1,7) (4,7) (10,19) (17,8) (17,7) (10,0) (6,18) (3,10) (15,2) (6,2) (3,4) (8,10) (8,2) (13,5) (1,12) (18,16) (0,18) (9,4) (5,6) (4,12) (0,12) (6,19) (10,18) (16,7) (14,3) (11,5) (17,3) (19,2) (16,19) (11,9) (17,0) (0,4) (18,9) (18,15) (13,19) (1,3) (19,7) (4,17) (2,13) (13,7) (19,12) (3,9) (5,9) (14,13) (0,16) (4,13) (10,4) (14,2) (10,7) (14,5) (5,16) (11,3)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->6->1->8->17->2->10->15->7->3->13->16->9->14->4->5->11->12->18->19."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph225",
        "image": "data/hamilton/graph_visual/hard/graph225.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 13, and the edges are: (7,6) (10,9) (12,8) (3,0) (11,10) (8,0) (0,9) (7,9) (7,2) (8,7) (3,1) (13,11) (6,5) (2,12) (12,9) (2,0) (0,13) (6,11) (11,3) (9,11) (4,6) (4,10) (2,13) (1,4) (4,11) (2,4) (13,7) (4,0) (10,12) (3,7) (11,8) (3,9) (3,5) (11,7) (5,11) (4,12) (11,0) (0,5) (9,1) (10,5) (13,9) (13,10) (7,0) (1,13) (8,13) (0,1) (3,10) (12,11) (1,6) (13,3) (12,0) (8,9) (4,8)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->1->4->6->7->9->10->12->2->13->8->11->5."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph488",
        "image": "data/hamilton/graph_visual/hard/graph488.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (10,4) (16,6) (6,0) (5,13) (17,2) (9,10) (3,6) (14,3) (15,8) (5,0) (12,13) (0,9) (11,10) (3,2) (13,17) (10,12) (12,5) (9,14) (6,17) (5,16) (3,9) (18,12) (16,17) (7,5) (14,17) (11,9) (13,16) (17,9) (8,2) (10,5) (18,7) (4,2) (15,3) (13,14) (17,1) (13,15) (3,13) (6,14) (7,8) (17,10) (11,0) (0,14) (15,6) (6,10) (12,9) (18,16) (9,16) (18,17) (4,0) (6,11) (18,6) (4,17) (11,14) (18,3) (17,8) (13,8) (4,5) (8,0) (4,9) (1,10) (7,10) (2,10) (2,12) (15,10) (9,8) (8,18) (12,0) (0,10) (7,0) (3,16) (15,4) (5,8) (15,7) (5,17) (9,18) (4,18)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->6->16->5->13->12->10->4->2->3->14->11->9->8->15->7->18->17->1."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph146",
        "image": "data/hamilton/graph_visual/hard/graph146.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 12, and the edges are: (10,12) (7,4) (6,1) (5,3) (8,6) (7,8) (5,9) (10,8) (4,2) (0,3) (12,11) (0,5) (2,9) (0,2) (3,10) (11,1) (2,10) (1,10) (10,9) (7,2) (1,0) (3,7) (4,1) (8,0) (7,9) (0,4) (12,0) (12,7) (5,8) (3,1)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->5->9->2->4->7->8->6->1->11->12->10."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph93",
        "image": "data/hamilton/graph_visual/hard/graph93.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 11, and the edges are: (2,6) (11,0) (10,4) (11,10) (7,5) (11,9) (5,0) (3,1) (11,8) (3,8) (4,9) (10,5) (3,4) (6,9) (4,2) (7,6) (11,7) (8,6) (5,4) (7,9) (0,2) (5,1) (8,4) (9,1) (0,6) (8,0) (2,5) (0,10) (2,11) (11,4) (0,4) (1,7) (6,10) (8,1) (8,9) (2,7) (8,10) (7,8) (10,3) (4,7) (7,0) (5,8) (5,6) (1,10)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->11->10->4->9->6->2->5->7->1->3->8."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph80",
        "image": "data/hamilton/graph_visual/hard/graph80.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 11, and the edges are: (8,9) (0,11) (1,4) (0,7) (9,1) (9,3) (1,11) (5,4) (1,10) (10,2) (0,4) (7,3) (11,2) (5,9) (4,3) (7,1) (4,8) (5,11) (2,8) (4,6) (2,5) (9,6) (6,2) (9,4)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->11->5->4->3->7->1->10->2->8->9->6."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph196",
        "image": "data/hamilton/graph_visual/hard/graph196.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 13, and the edges are: (6,2) (9,0) (1,0) (10,5) (5,13) (12,9) (3,10) (12,13) (8,3) (6,11) (8,2) (9,13) (4,2) (1,5) (13,4) (13,10) (5,0) (13,0) (11,5) (8,0) (12,2) (6,5) (2,3) (4,12) (6,13) (9,2) (9,10) (1,4) (9,5) (7,10) (11,10) (10,6) (2,11) (10,12) (12,7) (7,6) (4,3) (1,3) (4,8) (1,13) (5,8) (2,13) (11,13) (6,4) (0,10) (10,1) (1,8)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->9->12->13->5->10->3->8->1->4->2->11->6->7."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph60",
        "image": "data/hamilton/graph_visual/hard/graph60.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 11, and the edges are: (7,0) (1,2) (4,3) (5,1) (6,9) (6,1) (2,8) (9,11) (10,3) (8,4) (7,11) (4,11) (4,2) (5,0) (1,4) (10,4) (3,7) (4,9) (4,6) (10,11) (5,8) (0,8) (10,7) (11,6) (4,7) (0,6)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->7->11->9->6->1->5->8->2->4->3->10."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph207",
        "image": "data/hamilton/graph_visual/hard/graph207.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 13, and the edges are: (2,12) (3,7) (4,5) (11,12) (0,4) (13,10) (10,0) (7,12) (2,11) (5,7) (9,7) (1,7) (13,11) (0,6) (2,13) (8,4) (1,10) (1,5) (3,2) (12,5) (2,8) (10,9) (13,6) (9,0) (4,11) (10,8) (2,1) (8,9) (9,3) (12,6) (5,6) (11,0) (5,8) (12,10) (6,11)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->4->5->7->3->9->8->2->12->11->6->13->10->1."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph96",
        "image": "data/hamilton/graph_visual/hard/graph96.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 11, and the edges are: (7,10) (3,5) (9,0) (1,0) (11,1) (8,11) (3,10) (11,3) (2,8) (7,3) (5,2) (9,2) (2,6) (4,7) (7,5) (1,10) (9,7) (9,3) (9,6) (5,9) (4,10) (4,3) (11,4) (2,3) (7,2) (1,2) (8,7) (4,0) (3,1) (1,5) (8,1) (11,5) (8,3) (5,10) (9,8) (6,11) (0,2) (11,2) (10,11)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->9->2->8->7->10->1->5->3->4->11->6."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph252",
        "image": "data/hamilton/graph_visual/hard/graph252.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 14, and the edges are: (9,7) (12,14) (2,10) (4,1) (4,13) (5,3) (4,11) (4,8) (8,10) (7,1) (4,3) (2,12) (14,7) (6,11) (8,3) (7,10) (2,1) (10,13) (9,6) (4,10) (5,13) (9,14) (12,3) (1,3) (12,13) (12,6) (5,2) (10,14) (13,1) (5,9) (8,2) (5,1) (9,2) (11,14) (9,13) (12,8) (6,4) (5,0) (0,12) (11,3) (12,5) (10,1) (3,2) (2,11)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->5->3->4->1->7->9->6->11->14->12->2->8->10->13."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph416",
        "image": "data/hamilton/graph_visual/hard/graph416.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 16, and the edges are: (4,13) (11,15) (4,11) (1,0) (5,6) (11,16) (4,7) (7,3) (9,10) (3,2) (13,2) (7,11) (4,2) (16,10) (8,2) (16,12) (12,9) (4,3) (4,6) (6,3) (2,0) (1,10) (6,2) (4,16) (15,2) (6,16) (1,16) (16,7) (12,10) (15,5) (7,0) (5,1) (4,10) (12,4) (11,9) (13,3) (2,1) (5,3) (5,8) (3,10) (8,3) (9,3) (0,5) (13,8) (7,6) (13,1) (9,13) (2,16) (6,13) (9,2) (8,0) (10,15) (14,1) (15,9) (7,1) (3,16) (0,4) (15,14) (14,2) (13,16) (10,5) (0,9) (3,1) (6,9) (1,15) (10,2) (5,9) (12,7) (13,11) (12,15)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->1->10->9->12->16->11->15->5->6->4->7->3->13->8->2->14."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph91",
        "image": "data/hamilton/graph_visual/hard/graph91.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 11, and the edges are: (5,1) (11,4) (6,11) (2,0) (5,11) (2,9) (2,4) (8,3) (6,8) (3,9) (0,3) (8,5) (1,8) (4,8) (1,9) (6,1) (4,10) (11,2) (11,3) (6,2) (1,0) (9,5) (9,8) (9,11) (2,7) (7,5) (4,9) (3,1) (11,8) (0,11) (10,3) (9,7) (0,4) (7,3) (3,4) (2,8) (9,0) (2,1) (7,4) (6,10) (8,7) (8,0) (11,7) (3,5)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->9->3->8->6->1->5->11->7->4->10."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph366",
        "image": "data/hamilton/graph_visual/hard/graph366.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 16, and the edges are: (6,8) (15,0) (1,8) (5,10) (9,4) (7,6) (13,16) (13,6) (0,4) (3,1) (5,6) (13,0) (5,12) (4,15) (14,16) (10,2) (12,9) (15,7) (14,13) (8,13) (8,4) (9,16) (11,6) (2,15) (13,10) (7,14) (8,2) (11,12) (1,2) (5,11) (0,11) (1,6) (10,9) (2,0) (15,3) (0,16) (13,12) (13,2) (6,2) (12,1) (14,2) (9,6) (0,14) (16,8) (2,12) (11,1) (13,9) (16,3) (6,16) (5,0) (9,1) (1,4) (4,14) (5,2) (16,1) (9,15) (9,0) (6,14) (7,5) (11,15)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->15->4->9->12->5->10->2->8->6->7->14->13->16->3->1->11."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph242",
        "image": "data/hamilton/graph_visual/hard/graph242.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 14, and the edges are: (12,5) (6,8) (14,8) (3,8) (0,3) (6,9) (13,8) (11,3) (13,6) (8,5) (3,7) (14,5) (12,9) (14,13) (10,14) (14,1) (5,3) (10,4) (4,7) (2,3) (4,8) (6,10) (2,9) (14,7) (7,12) (3,14) (11,6) (12,3) (12,14) (9,7) (13,4) (2,6) (9,8) (0,11) (0,5) (2,7) (4,9) (8,7) (1,3) (9,1) (5,2) (10,7) (11,13) (13,7) (4,2) (4,14) (5,9) (13,5) (0,4) (10,9) (6,0) (1,8) (8,0) (0,12) (6,7) (4,5)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->8->6->9->1->14->10->4->2->7->12->5->13->11."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph289",
        "image": "data/hamilton/graph_visual/hard/graph289.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 14, and the edges are: (9,8) (1,6) (13,7) (11,4) (10,4) (5,3) (8,7) (1,13) (0,13) (9,14) (5,1) (5,4) (6,13) (14,10) (7,9) (4,0) (14,2) (2,6) (11,8) (5,2) (8,1) (12,3) (2,3) (1,10) (7,5) (8,4) (1,7) (13,4) (9,12) (2,8) (6,0) (2,9) (5,10) (6,5) (12,6) (3,8) (12,13) (1,2) (0,7) (7,6) (6,9) (11,3) (9,10) (13,10) (14,8) (14,11) (10,6) (7,2) (4,7) (0,2) (12,1) (9,1) (13,14) (3,7) (11,10) (4,9) (0,10) (11,0) (11,12) (8,0) (14,4) (4,12)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->13->7->8->9->14->10->4->11->3->5->1->2->6->12."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph136",
        "image": "data/hamilton/graph_visual/hard/graph136.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 12, and the edges are: (11,7) (3,10) (0,12) (11,2) (3,8) (11,5) (0,1) (1,7) (2,4) (6,10) (1,11) (7,10) (12,3) (6,5) (8,10) (6,9) (2,12) (9,0) (9,5) (11,3) (10,4) (5,2) (4,9) (7,9) (1,4) (3,9) (0,2) (4,11) (0,7) (5,7) (3,5) (2,10) (10,0)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->12->3->8->10->6->5->11->2->4->9->7->1."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph66",
        "image": "data/hamilton/graph_visual/hard/graph66.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 11, and the edges are: (0,8) (5,2) (1,9) (11,10) (5,6) (9,10) (1,11) (8,3) (5,4) (10,7) (10,1) (4,9) (11,4) (11,3) (10,2) (2,4) (8,7) (2,9) (3,7) (6,8) (1,6) (8,2) (11,2) (3,9) (9,11) (5,11) (4,0) (0,5)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->8->3->11->1->6->5->2->4->9->10->7."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph478",
        "image": "data/hamilton/graph_visual/hard/graph478.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 17, and the edges are: (7,5) (0,4) (0,14) (5,6) (0,15) (10,4) (2,16) (3,17) (14,4) (11,1) (16,12) (14,5) (8,17) (10,15) (11,14) (3,16) (12,4) (10,5) (2,3) (17,9) (15,11) (12,11) (4,13) (16,15) (9,6) (9,8) (8,2) (11,13) (12,13) (14,16) (9,1) (10,12) (9,12) (5,0) (6,1) (13,6) (1,10) (5,8) (13,10) (11,9) (14,2) (16,1) (11,2) (4,2) (3,1) (6,11) (13,3) (6,8) (5,16) (7,8) (9,16) (9,15) (5,11) (15,1) (3,0) (14,1) (4,3) (16,10) (10,8) (0,10) (12,8) (7,15) (10,7) (10,9) (4,17) (1,13) (2,1) (17,1) (16,0) (2,12) (15,3) (7,1) (5,2) (13,16) (14,6) (12,3) (1,8) (17,13) (3,7) (10,3) (15,12) (9,5) (9,7) (10,11) (7,0) (4,1) (11,8)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->4->10->15->11->1->9->17->3->16->2->8->7->5->14->6->13->12."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph405",
        "image": "data/hamilton/graph_visual/hard/graph405.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 16, and the edges are: (7,5) (4,15) (9,11) (0,1) (14,6) (2,1) (12,15) (2,7) (8,14) (12,2) (9,0) (3,5) (11,10) (15,0) (13,12) (9,16) (13,0) (10,2) (1,4) (4,8) (8,11) (14,10) (4,10) (6,15) (14,15) (9,14) (5,13) (5,15) (6,10) (13,11) (2,5) (8,15) (9,5) (3,8) (8,12) (1,14) (14,3) (10,0) (10,7) (1,12) (15,7) (6,8) (3,6) (5,8) (0,16) (11,4) (5,1) (5,11) (4,13) (4,0) (3,1) (9,10) (16,14) (6,16) (13,3) (3,11) (15,13) (9,6) (6,4) (0,5) (3,4) (6,12) (2,11) (9,15) (8,16) (7,9) (4,9) (14,11) (16,5) (16,7) (15,10) (1,8) (8,2) (12,14) (7,12) (13,6) (3,0) (7,8) (16,4) (2,6) (2,14) (1,11)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->1->2->7->5->3->8->14->6->15->4->16->9->10->11->13->12."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph47",
        "image": "data/hamilton/graph_visual/hard/graph47.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 10, and the edges are: (6,9) (5,0) (0,3) (2,7) (3,10) (5,1) (10,9) (1,4) (7,3) (3,2) (9,3) (4,7) (0,1) (10,6) (5,7) (10,2) (8,3) (8,1) (1,7) (4,5) (8,4) (0,4) (2,8) (6,8) (0,6) (8,10) (4,6) (1,6) (0,10)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->5->1->4->7->2->3->10->9->6->8."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph38",
        "image": "data/hamilton/graph_visual/hard/graph38.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 10, and the edges are: (0,1) (7,8) (0,9) (5,7) (10,9) (6,5) (4,8) (1,5) (9,3) (6,1) (3,4) (9,7) (8,9) (2,3) (9,1) (4,6) (8,1) (9,2) (4,0) (7,6) (10,2) (6,0) (10,1) (3,6) (5,4) (2,8) (8,6) (2,7)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->1->5->7->8->4->6->3->9->10->2."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph483",
        "image": "data/hamilton/graph_visual/hard/graph483.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (11,15) (6,10) (1,11) (15,2) (17,16) (15,0) (0,1) (13,10) (17,14) (16,1) (6,15) (7,6) (7,11) (5,15) (4,16) (1,5) (0,3) (1,12) (15,4) (9,17) (13,17) (18,15) (12,8) (17,8) (16,10) (1,9) (3,7) (3,10) (3,8) (16,13) (6,0) (15,16) (12,3) (11,3) (5,3) (2,7) (6,9) (18,16) (8,6) (18,13) (15,9) (0,10) (7,14) (5,7) (16,3) (18,7) (0,7) (8,15) (0,9) (7,9) (11,5) (3,2) (17,6) (17,0) (13,0) (11,2) (6,1) (12,2) (12,15) (18,10) (4,10) (10,14) (8,0) (17,1) (2,1) (18,0) (18,17) (10,2) (15,14)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->15->11->1->16->4->10->13->18->7->14->17->9->6->8->12->2->3->5."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph522",
        "image": "data/hamilton/graph_visual/hard/graph522.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (16,6) (2,16) (9,17) (0,2) (10,8) (15,18) (17,10) (3,11) (6,1) (0,3) (8,13) (4,16) (15,10) (16,15) (18,11) (11,2) (16,17) (14,16) (17,7) (17,14) (16,11) (7,0) (10,0) (13,12) (8,1) (13,10) (14,0) (5,10) (9,2) (8,6) (6,11) (6,17) (3,7) (8,17) (1,4) (0,9) (8,0) (7,13) (9,12) (13,17) (2,15) (12,18) (12,10) (3,17) (9,1) (16,1) (7,16) (10,4) (10,11) (13,4) (12,15) (3,9) (12,8) (6,2) (7,9) (8,16) (16,12) (4,11) (18,0) (2,3) (7,11) (14,12) (12,0) (0,17) (1,17) (16,9) (14,9) (18,9) (3,13) (1,14) (7,14) (18,3) (15,13) (6,18) (2,4) (4,7) (15,9) (10,2) (4,3) (4,17) (13,11) (7,1) (9,11) (7,10) (6,12) (9,8) (13,5) (1,11) (0,1) (17,5) (8,3) (9,10) (4,0) (17,12) (14,10) (14,2) (8,4) (0,16) (16,10) (3,16) (14,13) (13,16) (4,14) (16,18) (0,13) (9,6) (0,5)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->16->6->1->8->10->17->9->12->15->18->11->3->7->14->4->13->5."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph446",
        "image": "data/hamilton/graph_visual/hard/graph446.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 17, and the edges are: (2,12) (14,2) (2,9) (2,3) (8,0) (10,6) (14,6) (7,3) (2,7) (14,7) (1,10) (4,10) (13,10) (1,8) (1,11) (11,15) (16,14) (17,11) (5,14) (12,0) (4,14) (3,8) (3,17) (13,14) (12,17) (15,7) (9,3) (4,16) (11,5) (2,6) (5,6) (16,1) (3,5) (14,9) (10,2) (15,17) (0,10) (8,14) (16,6) (7,0) (11,7) (12,5) (7,8) (12,15) (13,6) (16,3) (8,10) (5,7) (5,17) (11,16) (2,5) (7,1) (3,12) (17,0) (10,7) (4,15) (13,16) (12,14) (0,9) (9,1) (7,6)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->8->1->10->6->14->2->9->3->7->11->17->5->12->15->4->16->13."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph214",
        "image": "data/hamilton/graph_visual/hard/graph214.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 13, and the edges are: (3,8) (10,7) (9,8) (6,1) (7,2) (10,5) (4,11) (11,12) (8,5) (0,11) (6,10) (11,7) (13,8) (11,9) (8,11) (12,0) (8,2) (12,4) (2,3) (0,6) (4,2) (4,8) (0,2) (9,1) (13,6) (7,0) (13,11) (4,3) (10,2) (2,5) (4,6) (8,7) (5,9) (9,4) (3,9) (10,13) (3,7) (6,2) (1,0) (8,12) (11,5) (13,1) (12,9) (12,7) (8,1) (10,12) (12,6) (0,10) (3,12) (9,0) (9,7) (3,10) (12,5) (6,5) (7,13) (10,8) (13,2) (11,1) (1,7) (6,9)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->11->4->12->8->3->2->7->10->5->9->1->6->13."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph184",
        "image": "data/hamilton/graph_visual/hard/graph184.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 13, and the edges are: (10,0) (5,3) (8,13) (8,0) (6,2) (3,10) (3,9) (9,5) (12,4) (0,13) (1,6) (3,8) (6,12) (5,0) (9,6) (0,12) (1,2) (1,5) (2,13) (3,13) (0,11) (1,8) (6,3) (13,7) (6,5) (4,11) (6,13) (13,11) (6,11) (10,11) (7,10) (11,2) (6,0) (10,5) (8,2) (10,4) (4,6) (9,12) (11,8) (1,0) (11,9) (12,1) (12,7) (3,0) (9,13) (4,7) (10,9)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->10->3->5->9->6->2->1->8->13->7->12->4->11."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph99",
        "image": "data/hamilton/graph_visual/hard/graph99.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 11, and the edges are: (7,2) (9,7) (1,9) (7,6) (10,1) (0,1) (2,3) (6,1) (4,7) (2,6) (3,10) (10,4) (7,10) (11,9) (5,3) (8,7) (2,0) (2,4) (9,5) (2,1) (5,8) (1,5) (11,3) (11,2) (11,6) (11,10) (1,8) (11,1) (11,8) (0,10) (10,2) (4,8) (6,5) (0,4)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->1->9->7->2->3->10->4->8->5->6->11."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph48",
        "image": "data/hamilton/graph_visual/hard/graph48.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 10, and the edges are: (6,8) (1,3) (3,8) (2,3) (10,1) (7,3) (7,9) (9,1) (6,9) (6,2) (5,7) (7,6) (8,5) (2,1) (9,10) (9,5) (0,5) (8,1) (4,6) (0,10) (2,9) (0,6) (5,1) (3,0) (3,10) (10,6) (7,0) (10,7) (1,7) (5,6) (5,4) (3,4) (9,8) (4,0) (7,2) (4,2) (4,8) (10,5) (4,7) (5,2)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->5->7->3->1->10->9->6->8->4->2."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph288",
        "image": "data/hamilton/graph_visual/hard/graph288.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 14, and the edges are: (11,13) (9,2) (9,7) (10,0) (14,12) (3,14) (13,10) (11,7) (11,12) (8,2) (5,10) (4,13) (3,9) (8,6) (3,4) (5,12) (4,0) (11,10) (5,3) (2,3) (9,0) (14,6) (9,8) (3,12) (9,13) (14,9) (1,8) (2,6) (10,8) (1,9) (10,1) (2,14) (6,4) (10,4) (9,10) (6,12) (10,14) (10,7) (8,11) (8,7) (1,4) (10,3) (8,12) (0,13) (2,0) (0,8) (7,0) (1,14) (0,6) (7,1) (13,3) (13,12) (14,4) (2,10) (5,7) (11,4) (9,6) (5,8) (7,6) (13,2) (3,6) (14,7) (0,5) (1,12) (12,7)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->10->13->11->7->9->2->8->6->14->12->5->3->4->1."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph327",
        "image": "data/hamilton/graph_visual/hard/graph327.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (14,7) (6,1) (8,10) (1,15) (4,2) (9,12) (14,10) (2,9) (5,8) (10,6) (10,0) (14,0) (3,1) (3,8) (12,10) (2,0) (3,13) (3,5) (6,8) (0,6) (8,13) (2,3) (8,14) (4,14) (13,1) (1,8) (11,9) (6,7) (3,6) (12,8) (13,10) (1,0) (6,5) (8,9) (14,9) (0,13) (9,4) (1,11) (0,9) (5,13) (12,3) (10,11) (2,12) (4,10) (5,11) (8,11) (3,15) (0,7) (7,4) (2,5) (3,0) (5,9) (6,2) (10,5)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->10->8->5->6->7->14->4->2->12->9->11->1->15->3->13."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph411",
        "image": "data/hamilton/graph_visual/hard/graph411.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 16, and the edges are: (9,2) (3,7) (16,14) (1,5) (0,3) (16,11) (10,1) (9,6) (16,4) (15,13) (11,12) (2,13) (2,16) (12,13) (14,0) (6,5) (10,0) (13,1) (0,2) (6,12) (13,3) (11,13) (8,11) (0,6) (2,7) (5,14) (1,9) (8,14) (3,4) (10,4) (12,5) (3,9) (10,15) (9,4) (7,5) (1,8) (10,8) (15,4) (16,7) (0,16) (14,10) (13,0) (15,3) (12,1) (0,5) (14,13) (3,6) (8,6) (3,8) (1,3) (10,3) (2,6) (12,0) (4,12) (10,5) (8,15) (7,13) (4,8) (0,11) (13,5) (5,15) (4,2) (0,15) (14,3) (16,5) (8,12) (3,2) (16,12) (15,16) (2,14) (12,3) (4,0) (4,6) (8,7) (7,12) (4,14) (12,2) (11,4) (14,15) (1,15) (1,11) (14,11) (12,14) (2,11) (9,0) (1,0) (15,6)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->7->2->9->6->5->1->10->4->16->14->8->11->12->13->15."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph589",
        "image": "data/hamilton/graph_visual/hard/graph589.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 19, and the edges are: (15,5) (14,9) (14,13) (4,5) (16,1) (11,6) (8,1) (3,14) (4,6) (6,18) (9,17) (8,2) (12,6) (3,12) (18,16) (9,10) (13,3) (13,16) (6,7) (5,17) (17,4) (2,7) (2,11) (2,13) (8,9) (1,9) (2,10) (9,4) (5,11) (8,12) (0,10) (11,3) (10,4) (9,5) (2,19) (1,11) (2,12) (13,0) (3,15) (0,18) (5,10) (18,2) (14,2) (9,6) (13,15) (10,19) (19,1) (9,16) (17,19) (2,3) (16,12) (0,4) (2,15) (1,13) (0,15) (17,12) (16,17) (15,1) (15,18) (5,14) (5,8) (15,10) (15,9) (0,9) (5,7) (13,4) (19,16) (4,8) (15,12) (16,2) (16,14) (1,7) (4,3) (19,9) (9,13) (14,11) (7,4) (13,12) (13,18) (13,10) (14,15) (7,12) (15,17) (10,6) (9,12) (12,18) (7,10) (4,15) (3,5) (2,5) (1,2) (13,11) (19,5) (6,0) (11,0) (7,3) (3,0) (6,3) (18,3) (8,13) (6,16) (16,7) (19,14) (11,18) (1,14) (3,19) (8,16) (18,1) (16,5) (7,17) (0,17) (7,19) (16,11) (17,2) (8,0) (6,14) (13,6) (7,18) (15,19) (14,7) (2,0) (3,9) (10,16) (6,2) (1,6) (1,5) (1,3) (12,11) (12,4)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->10->9->14->13->3->12->6->11->2->8->1->16->18->15->5->4->17->19->7."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph594",
        "image": "data/hamilton/graph_visual/hard/graph594.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 19, and the edges are: (2,16) (3,10) (17,2) (6,0) (16,9) (5,0) (10,5) (1,5) (2,14) (12,15) (18,13) (3,11) (2,18) (8,13) (7,16) (17,0) (11,6) (7,18) (13,15) (12,13) (17,6) (0,12) (0,16) (19,2) (12,1) (17,8) (3,6) (16,6) (8,11) (1,10) (9,14) (19,14) (3,13) (5,7) (4,17) (11,18) (3,19) (12,16) (9,0) (12,5) (5,3) (17,14) (1,15) (4,13) (13,14) (4,18) (11,15) (18,14) (13,5) (17,10) (7,12) (19,12) (5,11) (10,9) (8,18) (16,11) (11,4) (0,2) (1,18) (15,16) (5,15) (10,18) (1,3) (11,7) (9,3) (7,15) (19,4) (1,0) (0,14) (10,8) (2,9) (7,8) (5,17) (4,0) (17,3) (10,15) (3,15) (18,12) (9,11) (9,1) (1,6) (12,9) (4,15) (5,16) (10,19) (14,3) (16,8) (19,16) (6,14) (17,12) (13,16) (16,3) (17,16) (9,4) (9,6) (9,18) (14,7) (12,3) (13,1) (7,13) (6,12) (8,14) (13,2) (11,19) (1,14) (11,10) (15,9) (16,14) (14,15) (19,1) (15,0) (14,12) (14,11)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->6->11->3->10->5->1->12->15->13->18->2->16->9->14->19->4->17->8->7."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph200",
        "image": "data/hamilton/graph_visual/hard/graph200.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 13, and the edges are: (5,10) (7,2) (8,9) (1,7) (6,0) (1,2) (11,8) (9,5) (2,11) (3,5) (0,5) (11,12) (7,0) (10,13) (4,12) (1,13) (11,9) (10,12) (1,10) (0,13) (3,8) (7,12) (9,7) (5,7) (1,6) (9,10) (11,0) (3,2) (4,5) (6,13) (6,10) (3,7) (1,9)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->6->1->7->2->11->12->4->5->3->8->9->10->13."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph515",
        "image": "data/hamilton/graph_visual/hard/graph515.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (11,4) (4,5) (7,17) (8,2) (13,18) (8,0) (14,17) (5,6) (6,13) (14,11) (2,0) (1,0) (15,4) (15,5) (16,4) (3,4) (8,12) (18,5) (13,5) (6,12) (16,11) (11,1) (12,1) (18,9) (8,18) (5,16) (3,0) (16,9) (8,7) (3,13) (14,3) (10,17) (11,10) (15,11) (17,6) (9,10) (0,16) (9,7) (12,7) (12,18) (1,2) (0,10) (11,2) (9,2) (3,2) (8,13) (16,2) (15,9) (13,2) (17,0) (7,1) (16,18) (6,10) (15,16) (7,5) (13,1) (13,14) (1,14) (1,16) (17,13) (1,18) (1,5) (4,7) (5,0) (16,10) (6,8) (12,11) (6,4) (6,16) (11,17) (9,4) (10,5) (14,12) (14,9) (1,10) (14,15) (2,7) (0,9) (16,13) (8,15) (13,11) (12,15) (10,14) (15,10) (12,13) (12,4) (5,17) (6,1) (0,14) (8,10) (3,16) (9,6) (9,11) (1,15) (7,14) (3,7) (0,7) (3,15) (14,5) (17,15) (7,6) (2,5) (3,18) (18,6) (18,2) (11,0) (17,1) (12,10) (5,9) (2,4) (4,13)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->8->2->1->11->4->5->6->13->18->9->16->15->14->17->10->12->7->3."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph296",
        "image": "data/hamilton/graph_visual/hard/graph296.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 14, and the edges are: (1,6) (14,11) (14,5) (3,11) (2,6) (4,12) (14,8) (5,3) (9,8) (2,5) (0,13) (9,3) (12,8) (4,9) (14,6) (2,10) (1,13) (2,3) (10,3) (12,7) (14,1) (1,11) (12,2) (11,0) (7,4) (7,2) (11,9) (2,11) (11,13) (8,10) (10,13) (6,8) (7,10) (10,11) (12,5) (2,4) (5,11) (13,4) (9,2) (2,0) (10,0) (9,6) (7,9) (8,13) (14,2) (2,8) (5,10) (5,1) (8,7) (9,1) (13,9) (12,13) (5,9) (8,1) (1,12) (7,13) (7,3) (4,1) (0,4) (10,1) (6,7) (5,8) (7,5) (3,0)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->13->1->6->2->5->14->11->3->9->8->12->4->7->10."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph586",
        "image": "data/hamilton/graph_visual/hard/graph586.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 19, and the edges are: (19,15) (11,18) (14,0) (2,3) (7,19) (18,3) (19,2) (18,1) (1,6) (5,12) (16,12) (19,5) (14,8) (2,6) (17,15) (12,11) (4,11) (0,8) (1,5) (4,2) (1,7) (16,5) (19,9) (7,9) (12,0) (10,8) (17,8) (9,0) (13,7) (4,15) (4,14) (18,0) (19,8) (18,15) (12,2) (16,15) (3,11) (11,6) (16,18) (11,17) (6,15) (17,18) (1,4) (15,12) (10,11) (18,5) (17,9) (14,15) (3,19) (13,12) (8,12) (8,3) (4,17) (7,14) (10,15) (4,3) (6,3) (18,6) (17,14) (11,15) (9,10) (10,5) (19,1) (10,18) (12,10) (19,14) (2,10) (11,5) (13,18) (7,6) (16,6) (0,5) (13,6) (14,18) (17,19) (3,0) (0,6) (8,5) (13,15) (16,10) (17,7) (2,16) (5,17) (1,14) (4,0) (10,17) (9,16) (7,11) (10,14) (19,13) (5,7) (12,19) (0,15) (0,16) (7,4) (14,5) (0,2) (14,11) (1,2) (12,18) (5,15) (6,8) (16,13) (9,11) (9,2) (16,8) (6,19) (1,17) (11,13) (10,19) (11,16) (16,14) (7,16)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->14->8->10->11->18->3->2->19->15->17->9->7->13->12->5->16->6->1->4."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph249",
        "image": "data/hamilton/graph_visual/hard/graph249.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 14, and the edges are: (7,10) (2,5) (12,9) (0,8) (10,0) (12,11) (13,10) (1,12) (14,1) (3,1) (13,1) (4,5) (0,7) (8,6) (6,0) (0,4) (5,7) (12,3) (5,3) (1,0) (9,8) (8,5) (2,4) (0,13) (14,10) (7,3) (9,7) (11,0) (10,8) (12,7) (10,12) (9,4) (7,6) (8,1) (5,14) (5,13) (6,4) (8,11) (2,14) (7,8) (9,3) (11,9)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->8->6->7->10->13->1->12->11->9->4->2->14->5->3."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph454",
        "image": "data/hamilton/graph_visual/hard/graph454.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 17, and the edges are: (6,0) (4,16) (7,9) (16,7) (1,16) (10,0) (8,14) (5,14) (0,15) (17,9) (17,7) (15,9) (13,1) (17,14) (3,8) (13,8) (5,0) (1,0) (17,0) (5,4) (17,10) (7,5) (16,9) (14,3) (9,3) (0,14) (13,7) (4,3) (13,5) (11,14) (17,3) (12,15) (7,3) (16,5) (10,11) (2,9) (6,8) (6,15) (11,4) (15,13) (15,7) (5,15) (9,4) (16,12) (13,16) (6,16) (2,13) (16,10) (6,13) (0,12) (15,16) (6,4) (0,2) (10,3) (4,8) (0,11) (15,14) (16,0) (1,6) (10,1) (5,6) (16,17) (13,17) (17,11) (17,5) (12,4) (14,9) (12,10) (17,4) (11,13) (1,7) (11,6) (9,12) (12,14) (13,4) (11,3) (1,11) (12,6) (5,3) (4,0)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->6->8->14->5->4->16->7->9->17->3->11->1->10->12->15->13->2."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph124",
        "image": "data/hamilton/graph_visual/hard/graph124.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 12, and the edges are: (12,10) (8,7) (11,0) (6,3) (7,9) (8,12) (7,2) (4,0) (12,1) (9,11) (1,0) (5,1) (5,10) (7,11) (9,10) (8,9) (0,5) (4,9) (1,10) (6,12) (9,12) (5,2) (10,11) (2,4) (8,3) (6,2) (12,7) (6,8)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->11->9->7->8->3->6->12->10->1->5->2->4."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph135",
        "image": "data/hamilton/graph_visual/hard/graph135.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 12, and the edges are: (7,3) (11,2) (6,1) (11,9) (0,6) (7,9) (1,10) (0,2) (5,6) (6,2) (3,11) (1,8) (12,8) (2,9) (2,5) (8,2) (10,12) (4,0) (3,0) (11,6) (12,2) (12,1) (7,1) (2,4) (9,6) (0,5) (6,4) (8,4) (7,5) (11,10) (10,9) (6,3) (9,1)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->6->1->10->12->8->4->2->9->11->3->7->5."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph469",
        "image": "data/hamilton/graph_visual/hard/graph469.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 17, and the edges are: (1,0) (15,9) (3,16) (9,6) (4,5) (17,6) (13,11) (13,7) (3,7) (7,1) (14,5) (7,5) (4,17) (12,10) (10,7) (14,9) (10,3) (4,10) (4,3) (16,7) (5,15) (2,1) (0,15) (3,1) (9,16) (0,17) (2,8) (15,10) (11,15) (10,13) (17,12) (17,2) (14,3) (14,16) (6,4) (5,9) (17,5) (15,4) (1,12) (4,1) (13,3) (6,10) (2,7) (5,6) (12,0) (8,0) (14,1) (7,11) (10,14) (7,4) (2,5) (14,4) (17,10) (7,14) (1,11) (1,13) (13,17) (12,6) (13,6) (4,12) (8,15) (2,0) (17,7) (17,11) (2,12) (3,12) (12,16) (3,11) (15,2) (1,8) (12,5) (17,15) (16,5) (11,6) (9,12) (16,8) (10,8) (9,8) (12,13) (2,9) (2,4) (11,0) (16,10) (3,5) (3,9) (0,7) (16,11) (3,8) (17,14) (15,3) (11,12) (0,5) (2,14) (10,2) (4,0) (0,16)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->1->7->13->11->15->9->6->17->4->5->14->3->16->12->10->8->2."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph384",
        "image": "data/hamilton/graph_visual/hard/graph384.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 16, and the edges are: (11,4) (11,0) (3,1) (2,0) (0,1) (9,8) (14,3) (12,9) (2,6) (2,3) (13,14) (16,0) (16,1) (10,4) (0,8) (7,8) (2,11) (9,16) (16,11) (14,0) (4,9) (4,2) (9,7) (13,2) (11,5) (6,0) (16,6) (8,4) (3,6) (5,6) (15,11) (11,13) (3,7) (14,1) (16,14) (8,6) (13,8) (0,12) (0,13) (14,8) (15,7) (5,9) (3,5) (5,0) (1,8) (9,3) (16,7) (0,3) (15,9) (13,12) (15,16) (5,4) (10,0) (12,1) (10,14) (7,0) (3,12) (1,5) (15,0) (5,12) (14,2)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->11->4->10->14->3->1->16->9->12->5->6->2->13->8->7->15."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph262",
        "image": "data/hamilton/graph_visual/hard/graph262.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 14, and the edges are: (2,13) (4,11) (13,7) (9,7) (8,6) (11,0) (9,14) (14,0) (11,8) (7,5) (6,9) (9,8) (4,5) (9,10) (3,13) (10,1) (4,9) (7,1) (12,14) (5,6) (5,2) (3,8) (1,14) (4,6) (12,1) (10,3) (2,6) (14,6) (0,3) (14,7) (7,6) (12,9) (3,2) (10,14) (7,8) (8,12)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->11->4->5->7->13->2->6->8->9->14->12->1->10->3."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph11",
        "image": "data/hamilton/graph_visual/hard/graph11.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 10, and the edges are: (0,4) (9,5) (10,1) (8,5) (7,8) (5,3) (1,9) (1,2) (10,7) (6,7) (5,4) (7,0) (9,4) (1,8) (0,6) (3,8) (5,7) (6,5) (4,6) (1,0) (6,2) (8,6)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->4->9->5->3->8->7->10->1->2->6."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph7",
        "image": "data/hamilton/graph_visual/hard/graph7.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 10, and the edges are: (2,7) (4,8) (8,2) (10,7) (8,9) (7,3) (2,3) (9,7) (8,5) (1,4) (3,6) (1,5) (0,10) (10,8) (10,9) (6,8) (2,5) (0,3) (1,3) (1,8) (5,7)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->10->7->9->8->4->1->5->2->3->6."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph348",
        "image": "data/hamilton/graph_visual/hard/graph348.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (3,2) (2,4) (9,15) (9,2) (14,5) (2,12) (11,3) (4,1) (3,0) (11,8) (7,10) (15,2) (6,7) (13,2) (9,14) (10,1) (13,0) (12,4) (4,9) (11,2) (12,6) (13,9) (13,11) (0,8) (10,14) (6,5) (3,4) (3,14) (2,7) (4,11) (8,5) (2,0) (5,15) (3,9) (8,4) (14,13) (8,3) (1,12) (5,11) (11,12) (1,8) (6,10) (7,3) (10,11) (0,5) (6,8) (13,4) (4,6) (0,11) (5,13) (8,9) (7,12) (6,9) (14,4) (1,11) (15,10) (14,8) (5,12) (4,10) (9,12) (12,14) (3,5)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->2->4->1->10->7->6->12->11->8->5->14->13->9->15."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph565",
        "image": "data/hamilton/graph_visual/hard/graph565.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 19, and the edges are: (9,5) (7,11) (17,15) (5,10) (6,16) (14,0) (4,12) (7,0) (2,11) (4,2) (19,5) (13,14) (19,4) (17,8) (8,13) (13,3) (1,13) (2,10) (10,3) (16,0) (1,8) (19,13) (19,15) (0,6) (19,10) (2,18) (7,6) (9,8) (16,17) (8,11) (2,12) (5,17) (7,16) (16,5) (3,0) (4,16) (7,10) (10,0) (2,19) (16,15) (16,1) (1,18) (0,15) (10,11) (12,9) (15,6) (16,12) (15,12) (2,5) (2,8) (7,19) (1,5) (11,14) (7,1) (8,5) (0,5) (1,12) (6,4) (2,16) (8,15) (0,17) (5,13) (13,6) (9,11) (10,4) (1,19) (12,19) (14,2) (10,6) (7,4) (14,16) (18,14) (4,11) (7,14) (13,16) (8,3) (16,8) (13,17) (19,14) (0,13) (11,5) (3,11)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->14->13->8->17->15->19->5->9->12->4->2->11->3->10->7->6->16->1->18."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph337",
        "image": "data/hamilton/graph_visual/hard/graph337.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (13,14) (7,6) (15,1) (2,1) (11,9) (13,1) (2,8) (12,8) (13,15) (15,4) (2,9) (14,8) (15,14) (11,6) (5,12) (5,11) (1,9) (8,9) (10,13) (0,15) (8,13) (1,3) (2,11) (3,10) (14,6) (2,15) (11,0) (9,0) (12,2) (15,11) (1,12) (6,10) (0,4) (2,7) (10,12) (0,10) (13,0) (2,6) (7,3) (5,14) (6,3) (14,11) (11,7) (4,6) (8,4) (8,10) (2,5) (8,11) (13,7) (13,6) (4,5) (0,2) (6,0) (1,6) (1,7) (9,7) (15,8) (7,5) (5,3) (3,12) (7,15) (7,4) (7,8) (5,0) (1,5) (10,2) (11,3) (12,15) (5,10) (1,0) (8,0) (5,8) (6,15) (9,3) (1,8) (10,7) (13,2) (9,12) (6,5) (11,13)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->15->1->2->8->12->5->11->9->7->3->10->13->14->6->4."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph451",
        "image": "data/hamilton/graph_visual/hard/graph451.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 17, and the edges are: (6,1) (1,2) (7,11) (17,6) (12,13) (9,4) (1,13) (10,2) (3,13) (15,9) (0,8) (11,16) (9,11) (0,6) (5,6) (14,5) (17,13) (1,15) (8,1) (3,7) (8,9) (5,11) (16,10) (6,2) (0,12) (14,1) (8,6) (8,17) (11,14) (7,12) (2,13) (9,0) (10,3) (9,13) (8,12) (9,16) (4,11) (3,6) (14,2) (11,0) (16,17) (7,5) (3,0) (4,5) (10,1) (15,11) (5,16) (5,10) (15,14) (5,13) (6,16) (8,14) (14,6) (11,2) (17,15) (13,4) (4,3) (11,3) (14,3) (17,1) (14,9) (7,17) (12,15) (10,9) (16,1) (7,16) (10,14) (6,13) (4,8) (17,10) (5,12) (0,15) (13,10) (7,10) (14,17) (13,0) (15,13) (12,9) (11,8) (12,2) (7,14) (2,15) (8,16) (15,8) (1,12) (7,8) (5,17) (0,7) (6,10) (13,16)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->8->1->6->17->13->12->7->11->16->10->2->15->9->4->5->14->3."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph573",
        "image": "data/hamilton/graph_visual/hard/graph573.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 19, and the edges are: (1,14) (4,16) (2,13) (6,7) (13,15) (14,8) (6,1) (12,1) (8,12) (14,12) (15,1) (5,18) (16,12) (15,8) (7,8) (12,18) (11,2) (16,5) (1,10) (14,15) (17,18) (7,11) (7,14) (3,19) (16,18) (19,13) (9,4) (11,0) (0,9) (13,14) (19,9) (0,4) (6,18) (11,13) (18,3) (7,0) (17,13) (12,6) (18,10) (17,8) (6,0) (16,7) (16,10) (14,9) (3,10) (11,4) (3,5) (9,10) (10,15) (8,1) (13,1) (12,9) (3,4) (15,2) (10,6) (2,5) (8,2) (5,15) (11,8) (2,18) (18,7) (1,16) (5,4) (4,12) (18,1) (13,0) (1,11) (7,10) (7,4) (16,9) (13,8) (16,6) (19,17) (4,19) (8,6) (17,16) (7,12) (9,6) (14,16) (3,9) (19,1) (10,14) (13,10) (8,10) (1,17) (13,18) (4,14) (12,5) (3,17) (16,3) (6,5) (14,19) (19,2) (5,0) (6,19) (18,4) (1,9) (9,5) (6,11) (12,10) (17,4) (0,1) (19,15) (15,16) (18,0) (2,9) (0,10) (13,9) (3,0) (17,14) (15,6) (4,2) (1,5) (3,2) (8,0) (11,5)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->11->2->13->15->1->14->8->12->16->4->9->19->3->10->7->6->5->18->17."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph328",
        "image": "data/hamilton/graph_visual/hard/graph328.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (8,5) (13,6) (14,12) (1,8) (0,1) (9,14) (13,14) (15,6) (15,2) (6,5) (15,0) (1,13) (3,1) (0,2) (11,4) (13,7) (14,2) (13,2) (12,15) (13,15) (3,11) (4,10) (15,9) (7,4) (9,5) (11,8) (15,7) (2,11) (11,6) (8,7) (1,7) (15,10) (7,12) (10,3) (5,4) (10,12) (15,4) (0,13) (13,12) (5,7) (11,15) (12,9) (6,4) (14,15) (14,7)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->1->8->5->6->13->14->12->7->4->10->3->11->2->15->9."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph556",
        "image": "data/hamilton/graph_visual/hard/graph556.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 19, and the edges are: (5,4) (9,12) (11,12) (19,17) (15,9) (11,14) (5,13) (3,10) (15,2) (9,13) (9,2) (1,18) (8,19) (5,19) (13,17) (18,15) (14,9) (18,8) (4,0) (14,3) (8,6) (2,12) (15,4) (19,4) (19,9) (2,19) (13,2) (6,15) (5,16) (1,4) (5,12) (14,18) (11,2) (3,13) (18,9) (19,6) (9,4) (1,3) (7,10) (8,14) (19,3) (10,4) (7,16) (2,16) (1,16) (17,2) (7,5) (8,1) (9,8) (7,2) (17,5) (11,6) (18,2) (19,18) (7,3) (7,18) (13,12) (10,19) (0,9) (0,3) (4,14) (17,6) (11,8) (11,1) (15,11) (11,18) (6,18) (19,12) (11,4) (11,17) (0,8) (13,4) (1,10) (13,1) (13,16) (0,2)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->4->5->13->9->12->11->14->3->10->7->16->2->15->18->1->8->19->17->6."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph63",
        "image": "data/hamilton/graph_visual/hard/graph63.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 11, and the edges are: (3,6) (0,9) (3,11) (4,6) (2,7) (8,6) (7,0) (4,2) (2,5) (8,2) (9,6) (3,2) (10,11) (3,1) (7,10) (1,5) (5,0) (0,1) (2,0) (2,11) (4,3) (5,6) (5,4) (4,11) (0,6) (6,7) (10,2)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->9->6->3->1->5->4->11->10->7->2->8."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph427",
        "image": "data/hamilton/graph_visual/hard/graph427.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 17, and the edges are: (2,6) (3,2) (10,0) (13,14) (6,7) (5,9) (13,4) (5,6) (11,0) (1,9) (11,16) (14,3) (10,11) (6,11) (13,9) (12,11) (8,13) (3,7) (2,1) (0,1) (16,10) (5,0) (9,14) (13,16) (5,10) (7,4) (9,11) (0,15) (14,10) (2,14) (16,9) (14,15) (9,4) (12,17) (12,7) (14,12) (2,4) (3,5) (9,17) (16,15) (2,12) (6,3) (10,3) (16,17) (2,9) (7,9) (8,5) (1,10) (8,7) (1,5) (10,7) (8,9) (7,15) (16,1) (8,2) (0,12) (6,14) (16,12) (14,17) (0,2) (10,15) (6,12)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->10->11->16->13->14->3->6->12->17->9->1->5->8->2->4->7->15."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph503",
        "image": "data/hamilton/graph_visual/hard/graph503.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (15,2) (2,12) (5,1) (9,13) (11,15) (17,18) (2,8) (3,5) (15,8) (5,15) (1,9) (11,7) (16,3) (17,14) (9,17) (7,17) (5,7) (15,17) (0,18) (6,1) (7,2) (16,10) (4,8) (6,17) (18,14) (1,7) (7,16) (17,11) (10,18) (14,11) (14,3) (9,12) (12,8) (7,3) (16,18) (5,8) (18,1) (11,13) (9,5) (4,11) (8,1) (2,13) (6,18) (15,16) (17,2) (14,10) (4,2) (12,1) (6,0) (6,7) (6,12) (5,11) (14,8) (3,9) (11,12) (16,6) (6,14) (2,5) (0,16) (18,15) (3,1) (15,4) (3,10) (5,6)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->18->17->14->11->15->2->13->9->1->5->3->10->16->7->6->12->8->4."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph490",
        "image": "data/hamilton/graph_visual/hard/graph490.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (17,16) (13,17) (5,10) (12,8) (17,15) (17,18) (15,1) (6,2) (13,18) (3,16) (14,9) (14,4) (3,4) (15,0) (5,12) (4,7) (4,18) (12,11) (12,0) (3,10) (8,6) (2,18) (7,16) (14,7) (11,10) (7,18) (12,9) (7,15) (3,12) (3,17) (1,5) (6,3) (14,10) (3,5) (10,15) (3,8) (4,2) (14,16) (8,15) (15,18) (6,10) (18,14) (6,13) (5,4) (5,2) (8,2) (2,16) (4,17) (17,6) (0,4) (15,12) (8,1) (11,15) (8,17) (17,10) (14,17) (14,1) (18,8) (5,13) (5,11) (11,18) (17,5) (0,3) (18,12) (11,16) (12,16) (8,4) (10,1) (1,11) (9,0) (2,17) (1,18) (13,3) (7,3)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->15->17->16->3->4->14->9->12->8->2->6->10->11->1->5->13->18->7."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph107",
        "image": "data/hamilton/graph_visual/hard/graph107.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 11, and the edges are: (4,0) (9,8) (2,0) (8,1) (2,7) (0,5) (3,9) (9,4) (6,9) (8,3) (6,10) (7,8) (0,1) (11,0) (3,6) (4,6) (9,1) (7,1) (6,1) (3,0) (4,2) (2,9) (10,5) (1,10) (9,0) (5,2) (11,2) (11,10) (7,4) (1,2) (0,6) (7,9) (10,3) (6,8) (4,5) (2,6) (9,10) (11,5) (8,5) (1,4) (5,3)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->4->9->8->1->7->2->5->11->10->6->3."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph216",
        "image": "data/hamilton/graph_visual/hard/graph216.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 13, and the edges are: (8,7) (8,3) (0,9) (9,1) (2,3) (3,1) (3,13) (1,6) (13,11) (6,3) (12,6) (11,9) (9,6) (6,5) (7,11) (10,5) (10,0) (1,4) (7,13) (0,5) (1,13) (9,3) (4,6) (8,4) (11,0) (2,1) (12,13) (12,0) (11,1) (5,4) (13,2) (7,2) (5,11) (3,7) (2,8) (7,1) (9,2) (2,6) (7,5) (13,5) (10,13) (4,13) (8,0) (7,6) (8,13) (3,5) (1,0) (12,11) (5,8) (10,12) (10,7) (2,5) (3,12) (10,6) (12,1) (9,10) (2,11) (12,8) (0,3) (11,3)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->9->1->3->8->7->11->13->12->10->5->4->6->2."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph547",
        "image": "data/hamilton/graph_visual/hard/graph547.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 19, and the edges are: (15,1) (8,13) (6,5) (7,12) (13,19) (9,11) (3,2) (6,19) (3,4) (16,6) (11,3) (12,3) (6,2) (15,3) (13,16) (10,17) (13,17) (17,15) (14,12) (3,13) (5,15) (14,19) (18,15) (7,1) (6,0) (4,2) (5,12) (13,14) (1,17) (17,12) (5,16) (12,4) (18,4) (14,3) (15,11) (0,4) (17,6) (12,2) (15,16) (5,18) (5,4) (5,3) (19,5) (6,13) (14,10) (1,14) (8,9) (2,16) (17,19) (7,6) (6,4) (1,11) (2,19) (14,0) (9,7) (9,3) (8,1) (1,4) (16,12) (2,5) (15,9) (3,16) (19,3) (18,14) (11,6) (3,0) (3,8) (7,4) (11,12) (19,8) (4,14) (17,3) (0,19) (9,0) (17,14) (16,0) (15,14) (5,1)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->6->5->15->1->7->12->3->4->18->14->10->17->13->16->2->19->8->9->11."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph431",
        "image": "data/hamilton/graph_visual/hard/graph431.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 17, and the edges are: (11,2) (17,8) (12,8) (3,15) (2,13) (11,15) (1,13) (9,12) (2,5) (15,6) (4,8) (16,13) (14,17) (15,17) (6,16) (1,15) (8,13) (5,14) (14,12) (16,15) (5,17) (14,2) (3,6) (14,16) (13,12) (1,3) (6,2) (7,12) (1,2) (10,12) (3,5) (3,10) (4,1) (9,8) (8,14) (8,5) (9,15) (17,16) (3,11) (17,2) (0,17) (17,4) (10,14) (8,1) (5,11) (17,1) (1,9) (2,3) (4,7) (13,9) (10,13) (11,13) (16,3) (7,6) (17,10) (12,3) (1,12) (3,7) (2,15) (9,11) (3,4) (5,13) (2,10)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->17->8->12->9->15->3->6->16->13->10->14->5->11->2->1->4->7."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph177",
        "image": "data/hamilton/graph_visual/hard/graph177.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 12, and the edges are: (6,8) (1,4) (5,12) (6,1) (7,5) (12,0) (6,3) (1,12) (4,12) (10,11) (6,7) (7,9) (7,0) (5,8) (3,0) (9,10) (6,11) (5,10) (0,11) (9,8) (7,11) (6,0) (9,1) (12,6) (2,11) (8,10) (5,4) (9,5) (4,3) (10,7) (7,8) (2,0) (9,0) (9,4) (10,2) (11,12) (7,3) (5,11) (0,1) (4,2) (1,2) (0,4) (6,4) (8,12) (8,0) (12,3) (8,3)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->12->5->7->6->8->9->10->11->2->1->4->3."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph43",
        "image": "data/hamilton/graph_visual/hard/graph43.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 10, and the edges are: (0,8) (10,1) (10,0) (3,7) (1,3) (10,7) (0,9) (6,7) (9,7) (6,4) (2,1) (6,8) (10,6) (10,2) (8,3) (10,3) (8,5) (3,9) (7,4) (8,1) (3,5) (1,7) (1,0) (9,2) (3,4) (10,4) (2,8) (10,9) (2,4) (6,2) (0,7) (10,5) (5,7) (2,0) (3,2) (10,8) (6,5) (5,2) (5,4)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->8->6->7->3->1->10->4->5->2->9."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph414",
        "image": "data/hamilton/graph_visual/hard/graph414.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 16, and the edges are: (5,11) (2,9) (13,8) (3,12) (4,1) (5,15) (9,10) (13,3) (14,11) (9,6) (9,4) (2,12) (14,8) (6,15) (16,0) (3,0) (5,2) (5,13) (10,6) (8,10) (13,14) (11,12) (15,2) (7,5) (13,1) (4,7) (12,6) (3,9) (8,11) (4,15) (11,7) (5,9) (14,9) (8,4) (10,14) (3,16) (4,11) (0,10) (15,3) (14,5) (5,3) (4,5) (0,8) (12,0) (10,13) (15,8) (6,13) (14,7) (7,0) (8,2) (3,14) (11,16) (16,15) (7,13) (2,16) (3,10) (11,6) (5,16) (11,2) (10,12) (10,15) (14,12) (2,10) (4,12) (16,8) (7,6) (7,16) (1,5) (10,7) (14,6) (14,1) (13,12) (14,0) (14,15)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->16->3->12->2->9->10->6->15->5->11->14->8->13->1->4->7."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph251",
        "image": "data/hamilton/graph_visual/hard/graph251.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 14, and the edges are: (9,1) (8,4) (2,14) (6,10) (2,12) (14,8) (3,12) (3,0) (3,5) (3,7) (6,4) (11,14) (5,14) (2,1) (3,1) (0,5) (3,9) (7,2) (1,6) (6,12) (12,4) (6,13) (7,1) (0,14) (2,6) (10,2) (1,0) (13,4) (10,14) (0,7) (10,5) (13,3) (11,13) (9,13) (13,12) (9,11) (10,7) (7,13) (3,10) (3,8) (10,9) (4,0)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->12->2->14->8->4->6->1->9->11->13->7->10->5."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph294",
        "image": "data/hamilton/graph_visual/hard/graph294.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 14, and the edges are: (5,14) (5,2) (13,2) (1,10) (7,13) (9,6) (6,10) (1,7) (1,13) (9,12) (12,13) (13,5) (8,11) (10,12) (8,14) (12,5) (4,1) (0,4) (2,14) (12,4) (12,14) (9,13) (6,14) (4,9) (9,1) (10,8) (10,13) (8,1) (14,11) (14,1) (0,2) (8,13) (7,8) (0,9) (5,10) (6,5) (10,9) (7,10) (11,9) (6,2) (2,4) (6,4) (0,6) (13,0) (11,2) (6,3) (8,6) (0,5) (3,4) (14,7) (2,9) (7,12) (11,6) (9,8) (0,1) (2,12) (2,1) (11,7) (0,12) (11,4) (10,4) (9,3) (11,12) (8,3) (14,10) (8,4) (3,10)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->4->1->10->6->9->12->13->2->5->14->11->7->8->3."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph243",
        "image": "data/hamilton/graph_visual/hard/graph243.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 14, and the edges are: (9,8) (3,0) (11,12) (10,2) (5,4) (7,4) (1,10) (14,10) (7,6) (8,14) (5,6) (5,12) (11,5) (1,14) (0,7) (14,4) (2,13) (14,12) (10,5) (6,2) (4,0) (2,4) (14,11) (11,13) (6,13) (9,14) (13,0) (1,9) (13,8) (8,10) (7,13) (7,11) (5,3) (13,1) (5,13) (11,3) (0,2) (14,0) (12,3) (13,9)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->5->4->7->6->2->10->1->14->8->9->13->11->12."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph108",
        "image": "data/hamilton/graph_visual/hard/graph108.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 11, and the edges are: (5,4) (9,0) (9,1) (4,3) (0,10) (3,10) (6,0) (7,0) (5,1) (5,10) (7,1) (6,9) (0,2) (11,0) (3,0) (9,7) (1,11) (2,3) (5,8) (8,11) (10,9) (2,1) (6,7) (5,11) (4,7) (11,3) (5,2) (6,8) (5,3) (1,6) (4,6) (11,6) (0,1) (11,9) (1,3) (10,11) (4,10) (1,8) (2,6) (8,10)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->9->1->5->4->7->6->8->11->10->3->2."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph551",
        "image": "data/hamilton/graph_visual/hard/graph551.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 19, and the edges are: (13,3) (9,1) (5,14) (5,2) (3,2) (15,12) (8,10) (3,18) (16,6) (10,1) (16,9) (5,15) (8,15) (10,16) (15,4) (1,8) (9,11) (4,8) (14,11) (12,11) (9,15) (19,1) (11,5) (6,17) (2,4) (11,15) (8,6) (18,11) (13,0) (15,7) (14,3) (8,2) (12,19) (19,11) (0,17) (13,2) (0,5) (6,18) (11,3) (8,17) (17,16) (11,10) (5,7) (19,0) (18,14) (14,1) (8,5) (11,17) (2,17) (13,15) (14,4) (13,1) (17,7) (12,3) (9,0) (3,16) (11,4) (3,1) (7,16) (9,5) (2,18) (16,2) (11,7) (0,15) (18,0) (2,7) (10,12) (6,4) (7,0) (5,18) (10,7) (9,14) (14,16) (7,14) (8,11) (7,6) (14,6) (8,0)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->13->3->2->5->14->11->9->1->19->12->15->4->8->10->16->17->7->6->18."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph116",
        "image": "data/hamilton/graph_visual/hard/graph116.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 11, and the edges are: (7,2) (0,7) (3,4) (6,8) (3,7) (8,9) (7,9) (5,4) (7,1) (2,4) (8,10) (9,1) (2,0) (10,3) (6,4) (4,11) (11,9) (2,10) (0,4) (1,6) (8,4) (2,3) (0,5) (3,6) (1,10) (10,5) (3,11) (5,8) (6,10) (4,10) (1,2) (7,11) (3,0) (4,9)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->7->2->4->3->10->5->8->6->1->9->11."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph352",
        "image": "data/hamilton/graph_visual/hard/graph352.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (9,4) (5,10) (9,5) (2,9) (12,11) (12,4) (11,3) (13,7) (4,13) (11,10) (7,11) (3,8) (10,0) (7,9) (12,10) (3,9) (13,2) (15,14) (4,3) (6,0) (6,1) (15,0) (12,7) (11,5) (3,1) (2,14) (15,8) (0,3) (8,4) (11,8) (0,5) (8,2) (7,10) (12,15) (4,5) (11,9) (14,5) (3,6) (11,1) (5,3) (11,4) (12,1) (2,12) (15,2) (4,14) (0,13) (14,10) (10,15) (4,15) (15,6) (4,10) (4,2) (0,11) (6,14) (10,13) (15,9) (4,7) (8,7) (8,6) (5,12) (0,8) (2,3) (2,7) (0,12) (1,10) (13,9) (2,11) (11,14) (7,0) (3,14)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->10->5->9->4->12->11->3->8->7->13->2->14->15->6->1."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph498",
        "image": "data/hamilton/graph_visual/hard/graph498.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (1,14) (16,3) (8,0) (18,15) (13,3) (1,11) (12,4) (15,2) (10,5) (17,8) (11,6) (7,11) (14,0) (2,8) (4,5) (13,5) (13,14) (12,7) (11,13) (12,17) (8,1) (2,0) (12,8) (6,10) (3,11) (4,14) (6,9) (11,10) (13,6) (17,1) (7,2) (14,5) (18,9) (18,17) (0,4) (16,7) (4,10) (4,2) (2,13) (12,5) (15,5) (4,7) (15,1) (6,16) (9,13) (6,14) (10,9) (4,9) (9,11) (2,11) (18,5) (8,9) (8,11) (18,4) (2,12) (6,2) (7,10) (17,5) (4,16) (7,9) (6,4) (14,10) (8,4) (0,9) (14,3)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->8->17->12->4->5->10->6->11->1->14->13->3->16->7->2->15->18->9."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph65",
        "image": "data/hamilton/graph_visual/hard/graph65.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 11, and the edges are: (1,3) (10,3) (6,9) (9,4) (1,9) (1,2) (10,6) (10,5) (11,0) (11,10) (8,7) (3,6) (5,6) (6,0) (1,10) (1,7) (8,9) (3,11) (5,0) (5,11) (11,2) (0,4) (9,5) (11,4) (7,9)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->6->3->10->5->11->2->1->7->8->9->4."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph223",
        "image": "data/hamilton/graph_visual/hard/graph223.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 13, and the edges are: (4,1) (2,12) (8,12) (0,8) (0,4) (8,10) (4,8) (12,10) (6,10) (9,2) (7,6) (3,6) (13,11) (9,4) (5,11) (5,8) (7,11) (11,12) (5,10) (8,11) (4,3) (13,4) (7,8) (11,3) (2,1) (1,11) (7,1) (12,6) (9,13) (9,1) (5,9) (6,9) (0,3) (1,10) (0,5) (10,0) (7,5) (4,5) (2,7) (3,7) (5,2) (10,3) (0,13) (13,6) (10,7) (11,10) (5,6) (6,1) (11,0) (11,6) (9,7) (8,2) (5,13) (12,4) (8,1)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->8->12->2->9->4->1->11->13->6->10->5->7->3."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph57",
        "image": "data/hamilton/graph_visual/hard/graph57.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 10, and the edges are: (7,6) (2,9) (2,4) (10,0) (5,9) (3,5) (4,6) (9,8) (0,6) (8,5) (10,8) (0,5) (7,2) (10,2) (10,1) (6,3) (4,3) (3,0) (0,2) (8,0) (10,7) (1,6) (8,7) (5,1) (9,0) (9,10) (7,0) (9,3) (9,6) (1,8) (5,7) (1,4) (6,10) (1,7) (6,2)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->10->8->9->2->4->6->7->1->5->3."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph315",
        "image": "data/hamilton/graph_visual/hard/graph315.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (4,14) (12,2) (1,0) (11,15) (9,14) (13,2) (6,1) (2,8) (2,10) (11,2) (3,7) (5,0) (1,10) (4,9) (9,11) (0,2) (5,2) (0,7) (4,10) (14,2) (6,10) (14,8) (4,12) (10,0) (5,15) (15,7) (6,13) (3,5) (13,4) (13,15) (15,14) (3,10) (1,11) (7,6) (15,2) (4,7) (4,5) (9,0) (3,14) (6,12) (0,3) (8,0) (12,9) (7,10) (12,3) (7,11) (0,14) (2,4) (14,11) (10,5) (9,6) (6,3) (12,15) (4,1) (14,6)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->1->6->10->2->12->4->13->15->5->3->7->11->9->14->8."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph423",
        "image": "data/hamilton/graph_visual/hard/graph423.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 17, and the edges are: (9,8) (15,2) (17,1) (9,12) (16,7) (11,6) (17,10) (12,0) (0,5) (7,13) (13,6) (11,17) (3,16) (17,16) (11,3) (15,17) (4,2) (0,13) (14,5) (17,3) (3,12) (13,8) (6,9) (13,15) (7,2) (1,0) (8,17) (8,5) (13,1) (16,4) (1,6) (8,10) (9,1) (2,11) (17,0) (6,15) (10,0) (6,12) (6,14) (1,16) (5,13) (10,14) (5,4) (12,10) (7,14) (2,5) (13,11) (2,9) (13,3) (4,3) (15,10) (2,6) (17,12) (1,7) (10,6) (5,17) (8,7) (15,5) (11,5) (10,2) (9,17) (12,4) (15,3) (0,14) (1,15) (16,8) (13,10) (5,12)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->12->9->8->13->7->16->3->11->6->1->17->10->14->5->4->2->15."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph321",
        "image": "data/hamilton/graph_visual/hard/graph321.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (5,4) (2,11) (5,12) (1,5) (15,7) (9,0) (7,1) (8,11) (1,13) (3,6) (7,9) (4,8) (14,7) (1,2) (6,12) (4,12) (0,7) (15,13) (12,13) (3,11) (12,8) (0,13) (8,13) (5,14) (2,4) (14,1) (2,5) (4,0) (8,6) (9,15) (10,5) (15,6) (0,15) (13,10) (10,3) (13,3) (8,3) (15,11) (12,11) (7,2) (2,9) (12,9) (6,1) (6,10) (15,2) (0,5) (7,11) (3,2) (14,3) (3,1) (3,7) (14,2) (11,9) (13,11) (5,7) (12,1)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->9->7->15->13->1->5->4->8->11->2->14->3->10->6->12."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph424",
        "image": "data/hamilton/graph_visual/hard/graph424.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 17, and the edges are: (12,6) (11,12) (7,6) (16,3) (2,9) (13,9) (16,1) (17,6) (13,11) (11,5) (1,3) (0,7) (16,11) (2,12) (15,2) (7,3) (10,1) (14,2) (6,0) (5,17) (15,10) (2,4) (5,4) (0,9) (17,7) (5,8) (5,0) (11,2) (0,8) (15,7) (0,17) (8,11) (5,1) (7,1) (16,9) (4,16) (0,11) (5,16) (15,5) (3,12) (13,6) (7,9) (5,7) (1,13) (1,12) (5,10) (11,9) (3,17) (10,2) (0,14) (0,12) (0,15) (8,14) (17,2) (10,11) (1,14) (16,7) (14,7) (8,4) (0,10) (15,6) (12,14) (14,15) (4,7) (16,8) (1,11) (9,12) (9,10) (11,15) (6,2) (3,11)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->7->6->12->11->13->9->2->15->10->1->16->3->17->5->4->8->14."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph230",
        "image": "data/hamilton/graph_visual/hard/graph230.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 13, and the edges are: (12,0) (9,4) (3,5) (6,0) (1,6) (6,5) (5,10) (7,0) (6,12) (12,5) (5,2) (9,6) (3,8) (7,12) (9,13) (0,5) (2,10) (7,6) (1,3) (11,8) (11,5) (11,3) (3,10) (10,4) (2,11) (7,5) (12,2) (6,3) (2,9) (2,0) (12,4) (11,4) (0,8) (7,1) (1,10) (1,9) (8,4) (13,11) (11,0) (8,5) (5,1) (10,11) (6,11) (7,11) (1,2) (2,4) (13,6) (7,8) (2,3) (12,3) (8,12)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->12->6->1->3->5->10->2->11->13->9->4->8->7."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph391",
        "image": "data/hamilton/graph_visual/hard/graph391.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 16, and the edges are: (16,2) (10,8) (4,7) (5,1) (1,10) (2,14) (9,12) (4,9) (9,16) (3,16) (14,6) (12,3) (10,13) (16,10) (12,6) (2,13) (11,8) (9,8) (14,8) (14,1) (10,4) (16,7) (3,6) (9,13) (15,13) (5,4) (12,7) (4,8) (9,1) (5,7) (6,7) (12,14) (11,13) (2,8) (6,15) (2,4) (0,2) (5,3) (1,15) (10,11) (12,16) (5,6) (0,6) (12,0) (15,8) (16,13) (1,16) (3,11) (0,11) (10,6) (2,9) (16,0) (13,4) (1,11) (10,7) (12,5) (0,9) (12,15) (0,1) (13,7) (6,2) (5,8) (4,11) (13,6) (15,10) (15,5) (10,14) (12,8) (14,7) (1,3) (12,11) (9,15) (0,8) (7,3) (6,16) (9,14) (7,8) (10,2) (8,3) (0,10)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->16->9->12->3->6->14->8->10->1->5->7->4->11->13->15."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph285",
        "image": "data/hamilton/graph_visual/hard/graph285.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 14, and the edges are: (12,4) (10,14) (0,12) (13,12) (5,8) (12,8) (8,0) (10,3) (6,1) (3,13) (8,6) (11,7) (0,11) (1,8) (6,14) (14,5) (13,2) (0,4) (10,11) (9,12) (13,0) (13,10) (11,3) (3,1) (4,1) (2,6) (5,11) (1,14) (13,8) (9,10) (9,8) (1,13) (10,1) (5,1) (7,0) (0,9) (12,2) (2,5) (4,14) (9,4) (8,4) (2,1) (3,6) (14,7) (5,7) (4,3) (7,12) (7,8) (2,8) (11,8) (9,7) (5,4) (7,13) (3,14) (13,5) (7,1) (5,10) (0,6) (13,14) (0,14)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->12->4->1->6->8->5->14->10->9->7->11->3->13->2."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph579",
        "image": "data/hamilton/graph_visual/hard/graph579.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 19, and the edges are: (10,7) (13,19) (16,18) (5,17) (11,0) (15,4) (9,16) (11,17) (7,16) (0,6) (8,6) (3,1) (11,14) (18,12) (18,6) (5,13) (16,6) (19,9) (12,5) (5,2) (9,11) (14,18) (8,19) (1,4) (7,1) (16,15) (17,19) (13,2) (18,5) (18,10) (7,9) (8,0) (10,12) (17,18) (14,3) (8,15) (17,8) (2,8) (16,14) (2,1) (5,0) (1,16) (11,4) (5,6) (5,9) (8,3) (1,19) (17,13) (11,2) (10,2) (5,15) (15,2) (8,14) (17,10) (8,4) (2,6) (15,9) (7,2) (2,17) (5,8) (3,11) (15,13) (8,12) (3,17) (5,7) (1,18) (3,2) (11,7) (7,8) (14,17) (13,10) (7,18) (5,19) (9,1) (14,5) (5,10) (15,14) (19,4) (6,14) (4,12) (4,3) (1,14) (0,16) (5,16) (0,19) (3,5) (3,9) (10,9) (2,12) (17,16) (18,15) (13,3) (15,11) (4,14) (3,10) (18,3) (18,8) (12,19) (0,15) (10,14) (13,6) (1,17) (15,6) (6,1) (10,16) (13,18) (5,4) (14,0) (17,0) (13,1) (13,4) (8,10) (18,2) (17,7) (14,13)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->11->17->5->13->19->9->16->18->12->10->7->1->3->14->8->6->2->15->4."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph540",
        "image": "data/hamilton/graph_visual/hard/graph540.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 19, and the edges are: (17,5) (12,13) (5,13) (8,13) (5,19) (6,2) (10,18) (9,4) (2,11) (8,16) (0,9) (9,11) (6,8) (10,0) (19,13) (10,17) (5,7) (3,10) (18,9) (0,3) (18,8) (11,10) (1,17) (0,2) (2,15) (9,10) (3,16) (0,1) (3,15) (14,0) (8,2) (7,4) (10,6) (14,9) (16,10) (18,11) (12,4) (18,2) (3,14) (7,13) (7,14) (4,2) (1,12) (3,6) (14,8) (17,2) (11,1) (0,5) (17,18) (6,5) (10,13) (2,10) (2,5) (3,18) (16,6) (13,17) (11,8) (19,1) (9,5) (12,2) (12,16) (5,8) (1,8) (13,2) (11,16) (19,3) (18,19) (14,11) (3,9) (2,1) (6,19) (0,12) (17,14) (0,4) (1,14)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->9->4->7->5->17->10->18->8->13->12->1->19->3->15->2->6->16->11->14."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph129",
        "image": "data/hamilton/graph_visual/hard/graph129.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 12, and the edges are: (2,11) (6,8) (10,3) (0,1) (11,6) (4,7) (12,1) (2,12) (1,3) (4,2) (10,7) (4,1) (2,5) (12,4) (2,0) (11,5) (3,6) (2,8) (3,5) (2,3) (5,12) (1,6) (2,9) (2,6) (8,4) (3,7) (5,10) (8,7)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->1->12->4->7->10->3->5->11->6->8->2->9."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph364",
        "image": "data/hamilton/graph_visual/hard/graph364.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 16, and the edges are: (10,9) (8,5) (15,6) (10,15) (8,9) (6,8) (14,6) (6,5) (7,9) (2,3) (4,7) (6,1) (16,11) (2,4) (1,10) (1,4) (9,11) (10,12) (7,11) (5,10) (9,5) (4,9) (7,13) (14,16) (2,10) (10,11) (5,14) (2,5) (2,9) (12,3) (15,13) (7,8) (6,9) (2,1) (4,0) (3,7) (2,6) (16,4) (10,13) (12,16) (0,3) (12,5) (13,12) (12,7) (4,15) (7,6) (8,10) (8,15) (5,11) (8,14) (14,15) (12,11) (16,8) (0,13) (15,5)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->4->7->9->10->15->13->12->3->2->5->11->16->14->8->6->1."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph339",
        "image": "data/hamilton/graph_visual/hard/graph339.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (10,0) (10,4) (2,13) (10,6) (0,3) (0,13) (10,2) (15,4) (10,1) (9,15) (2,4) (8,1) (8,13) (10,9) (2,5) (6,11) (14,1) (11,8) (4,1) (7,14) (10,15) (8,2) (13,1) (9,4) (4,6) (5,11) (11,13) (6,8) (1,6) (13,15) (1,5) (12,0) (8,14) (10,14) (11,2) (2,3) (6,13) (8,7) (3,8) (2,6) (13,12) (14,9) (3,14) (15,5) (8,4) (13,10) (7,12) (15,1) (5,10) (7,13) (4,3) (15,7) (11,0) (6,5) (7,4) (1,11) (6,0) (3,9) (3,5) (15,12) (2,15) (10,7) (14,15) (12,11) (3,11) (7,2) (11,7) (3,12) (0,8) (7,1) (12,6) (4,5)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->10->4->15->9->14->1->8->13->2->5->6->11->3->12->7."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph29",
        "image": "data/hamilton/graph_visual/hard/graph29.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 10, and the edges are: (4,6) (1,2) (6,2) (10,6) (1,6) (10,9) (4,8) (9,2) (10,1) (4,5) (1,3) (3,6) (9,0) (1,7) (8,5) (1,4) (5,7) (10,2) (8,2) (3,5) (0,3) (10,4) (7,9)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->9->10->6->4->8->2->1->3->5->7."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph534",
        "image": "data/hamilton/graph_visual/hard/graph534.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (1,16) (12,13) (12,2) (1,11) (11,4) (11,12) (3,18) (6,5) (3,1) (11,5) (2,6) (12,1) (18,1) (10,8) (0,1) (9,1) (16,8) (15,7) (4,0) (5,1) (13,7) (8,3) (7,12) (12,16) (3,16) (5,3) (10,9) (17,15) (0,12) (7,14) (6,8) (11,9) (13,0) (13,15) (4,3) (8,9) (0,18) (13,16) (4,13) (0,14) (10,18) (5,10) (10,1) (17,1) (2,17) (14,10) (3,10) (5,14) (11,16) (8,7) (17,14) (13,14) (4,5) (8,15) (7,6) (18,14) (9,4) (2,11) (12,18) (0,5) (11,10) (13,3) (10,16) (16,4) (13,6) (15,9) (17,11) (12,14) (5,13) (16,9) (14,1) (15,10) (2,0) (11,15) (3,6) (2,13) (4,18) (8,5) (13,18) (2,4) (8,17) (9,0) (17,10) (15,6) (18,11) (15,3) (18,6) (6,16) (18,2) (15,5) (18,7) (12,9) (13,9) (11,8) (18,17) (0,8) (0,3) (2,16) (11,14) (10,0) (16,18)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->1->16->8->10->9->11->4->3->18->14->7->15->17->2->12->13->6->5."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph110",
        "image": "data/hamilton/graph_visual/hard/graph110.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 11, and the edges are: (8,6) (2,1) (10,9) (6,10) (7,5) (0,9) (8,9) (4,6) (3,8) (10,5) (2,6) (0,1) (0,2) (11,5) (10,0) (0,8) (11,1) (9,5) (7,10) (8,10) (3,4) (0,5) (10,3) (11,7) (11,9) (6,9) (10,1) (3,0) (6,0) (11,8) (3,11) (10,4) (2,10) (4,1) (4,2)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->9->10->6->8->3->4->2->1->11->5->7."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph153",
        "image": "data/hamilton/graph_visual/hard/graph153.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 12, and the edges are: (3,1) (0,7) (1,12) (10,12) (12,3) (0,9) (6,7) (7,8) (5,11) (5,12) (2,8) (3,9) (2,10) (3,5) (9,12) (6,9) (5,2) (10,5) (0,10) (9,10) (3,0) (10,1) (3,7) (5,4) (1,0) (7,10) (8,5) (8,0) (8,9) (4,6) (2,11) (8,3) (4,3) (4,0) (7,9) (2,12) (6,10) (6,2) (6,11) (7,2)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->7->6->9->12->1->10->2->8->3->4->5->11."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph482",
        "image": "data/hamilton/graph_visual/hard/graph482.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (14,3) (5,0) (7,17) (18,16) (15,5) (10,12) (1,17) (5,17) (15,2) (18,11) (4,9) (0,15) (7,11) (4,2) (17,12) (6,9) (12,18) (17,4) (12,15) (17,14) (12,4) (17,10) (6,12) (0,2) (0,14) (4,14) (16,14) (7,8) (12,9) (16,2) (8,3) (15,10) (10,8) (8,18) (9,5) (11,6) (6,14) (14,15) (10,4) (18,3) (4,8) (8,11) (12,2) (6,8) (7,1) (2,13) (10,7) (1,14) (6,7) (5,13) (12,5) (14,12) (15,18) (12,8) (18,14) (12,3) (7,18) (3,11) (10,13) (9,1) (15,8) (3,17) (0,17) (7,3) (3,10) (4,7) (2,10) (16,11)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->5->15->2->4->9->6->12->17->7->1->14->3->18->16->11->8->10->13."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph576",
        "image": "data/hamilton/graph_visual/hard/graph576.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 19, and the edges are: (11,5) (9,0) (0,17) (16,4) (4,13) (3,19) (15,3) (17,3) (10,1) (1,13) (18,11) (16,1) (1,17) (11,6) (6,4) (19,4) (10,9) (11,0) (11,12) (12,14) (1,19) (1,2) (17,16) (19,11) (4,8) (19,16) (13,14) (18,5) (4,5) (6,12) (7,15) (7,2) (8,18) (3,5) (0,1) (14,16) (8,13) (4,0) (17,12) (10,13) (18,6) (9,16) (15,5) (0,12) (12,10) (15,14) (7,9) (9,17) (11,9) (3,6) (14,0) (13,16) (0,19) (7,12) (9,12) (0,3) (11,13) (3,7) (14,7) (3,4) (15,13) (10,15) (17,10) (7,4) (14,6) (18,1) (3,11) (10,2) (2,12) (6,10) (10,16) (8,15) (14,4) (4,9) (1,11) (14,10) (15,2) (18,9) (5,9) (5,0) (4,11) (4,18) (17,15) (10,8) (0,8) (8,19) (11,15) (16,18) (2,3) (19,2) (11,10) (3,18) (15,9) (14,5) (2,16) (19,12) (18,14) (7,18) (14,3) (12,5) (12,15) (1,5) (7,8) (8,17) (9,6) (14,9) (9,13) (18,17) (13,0) (8,2) (1,3) (3,12) (6,2) (16,6) (5,13) (18,2) (14,2) (6,19) (6,5) (17,2) (2,4) (4,10)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->9->10->1->13->4->16->17->3->19->11->5->18->8->15->7->2->12->14->6."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph533",
        "image": "data/hamilton/graph_visual/hard/graph533.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (2,3) (3,4) (15,13) (4,18) (6,12) (7,8) (11,7) (6,17) (1,15) (15,12) (13,7) (3,16) (16,14) (11,1) (6,5) (3,15) (9,15) (15,0) (18,0) (9,1) (9,10) (3,10) (16,13) (18,17) (5,0) (0,17) (7,3) (14,4) (3,0) (17,16) (15,6) (5,14) (8,17) (9,17) (10,1) (1,6) (10,12) (13,4) (13,18) (10,4) (15,18) (5,12) (14,9) (14,7) (11,13) (7,17) (15,11) (18,12) (3,12) (9,3) (3,1) (4,6) (6,10) (11,4) (0,1) (11,14) (9,12) (8,0) (2,5) (4,16) (16,10) (15,8) (13,14) (0,16) (12,17) (10,5) (17,2) (6,0) (2,0) (9,0) (16,11) (12,14) (12,16) (1,18) (8,6) (7,4) (13,10) (17,4) (7,0) (5,18) (9,8) (8,16) (11,10) (11,2) (2,14) (1,13) (10,8) (15,17) (5,9) (15,4) (9,11) (11,6) (5,8) (9,13) (17,14) (4,9) (6,16) (9,2) (10,17) (16,9) (13,6)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->15->13->7->8->17->6->12->10->9->1->11->4->3->16->14->2->5->18."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph278",
        "image": "data/hamilton/graph_visual/hard/graph278.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 14, and the edges are: (2,6) (12,0) (6,5) (2,14) (6,14) (7,6) (13,1) (3,13) (12,5) (14,7) (8,9) (11,0) (12,11) (11,10) (0,1) (11,1) (0,4) (8,10) (3,8) (8,4) (8,1) (13,6) (7,10) (14,0) (7,4) (10,0) (12,9) (5,2) (3,4) (13,11) (5,13) (4,12) (4,13) (10,14) (10,2) (8,14) (12,8) (4,5) (0,3) (1,14) (10,1) (14,13) (12,1) (8,7) (11,5) (9,13) (9,2) (10,13) (9,14) (7,2) (0,2) (14,3) (3,12) (11,8) (5,9) (6,4) (0,13) (11,7) (0,8) (14,12) (1,5) (7,3) (9,10) (1,9) (10,12) (5,7) (11,3) (7,13) (9,4) (10,3)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->12->5->6->2->14->7->10->11->1->13->3->8->9->4."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph349",
        "image": "data/hamilton/graph_visual/hard/graph349.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (4,11) (0,12) (5,6) (9,10) (9,12) (13,14) (8,14) (10,2) (13,2) (5,14) (1,15) (12,6) (10,5) (4,13) (2,12) (10,0) (10,4) (9,14) (5,1) (10,6) (10,12) (1,3) (0,7) (12,14) (15,13) (4,6) (5,9) (8,15) (0,3) (12,7) (14,3) (8,4) (14,2) (6,3) (13,11) (9,0) (1,12) (5,3) (9,4) (5,8) (4,12) (1,6) (15,9) (3,10) (8,1) (0,4) (7,8) (8,9) (1,9) (11,8) (10,15) (4,1) (8,3) (11,3) (12,13) (1,10) (11,10) (9,7) (2,4)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->12->9->10->2->13->14->5->6->4->11->3->1->15->8->7."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph318",
        "image": "data/hamilton/graph_visual/hard/graph318.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (14,8) (4,14) (5,1) (0,3) (14,3) (10,9) (10,13) (0,6) (0,11) (13,1) (9,6) (8,2) (11,3) (3,13) (9,15) (10,7) (11,12) (14,1) (7,2) (6,15) (13,12) (8,6) (7,5) (11,1) (15,1) (9,12) (4,3) (2,0) (11,8) (2,6) (8,15) (3,2) (4,2) (9,8) (15,11) (12,15) (2,11) (9,5) (7,13) (14,6) (2,14) (4,13) (4,0) (2,5) (15,14) (10,11) (1,4) (8,4) (11,6) (11,13) (5,14)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->14->8->2->7->10->9->6->15->11->12->13->4->1->5."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph370",
        "image": "data/hamilton/graph_visual/hard/graph370.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 16, and the edges are: (16,5) (13,1) (9,2) (3,7) (15,8) (3,9) (4,1) (15,16) (16,9) (6,8) (15,14) (7,16) (7,1) (16,13) (13,2) (7,9) (3,2) (1,5) (16,8) (1,11) (3,14) (15,13) (12,3) (10,5) (15,1) (11,4) (2,16) (12,1) (11,16) (16,6) (15,10) (8,2) (10,12) (0,5) (15,12) (3,11) (11,10) (0,14) (1,2) (6,3) (4,2) (0,4) (7,11) (4,10) (12,7) (2,10) (0,7) (12,0) (13,10) (14,5) (3,13) (1,10) (4,15)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->5->16->15->14->3->9->7->11->4->1->12->10->13->2->8->6."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph535",
        "image": "data/hamilton/graph_visual/hard/graph535.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (15,6) (7,14) (6,18) (1,12) (1,16) (6,2) (16,10) (1,10) (12,10) (12,15) (4,6) (11,4) (1,14) (1,11) (0,2) (8,12) (7,15) (8,16) (2,8) (2,5) (7,2) (17,4) (11,8) (9,2) (11,9) (0,15) (5,10) (0,9) (14,4) (15,3) (1,6) (12,6) (0,8) (16,3) (14,15) (2,3) (2,11) (10,15) (3,14) (4,3) (10,0) (3,18) (11,10) (11,12) (16,13) (10,13) (3,13) (8,17) (1,4) (10,7) (18,8) (13,7) (1,13) (1,17) (16,4) (16,18) (15,16) (18,17) (6,5) (2,16) (1,0) (14,16) (10,3) (5,9) (2,10) (16,12) (8,9) (10,6) (7,11) (6,0) (3,9) (18,4) (8,3) (8,5) (5,15) (2,13) (18,10) (18,7) (17,0) (16,6) (13,18) (8,6) (9,12) (0,18) (9,7) (2,4) (17,10) (15,4) (4,13) (6,17) (2,17) (14,8) (14,10) (2,1) (2,15) (8,15) (5,12) (9,16)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->6->15->12->1->16->10->5->9->11->4->17->8->18->3->14->7->13."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph81",
        "image": "data/hamilton/graph_visual/hard/graph81.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 11, and the edges are: (10,2) (6,11) (3,7) (4,5) (8,6) (3,8) (1,7) (3,6) (7,2) (5,11) (1,5) (3,9) (7,11) (7,9) (3,10) (1,2) (1,6) (6,0) (7,10) (9,5) (0,3) (9,1) (11,3) (4,1) (4,11) (5,6)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->6->11->5->4->1->2->10->7->9->3->8."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph247",
        "image": "data/hamilton/graph_visual/hard/graph247.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 14, and the edges are: (2,4) (9,1) (0,2) (1,7) (11,1) (0,13) (4,8) (10,6) (6,8) (2,7) (14,13) (14,6) (7,0) (5,9) (11,14) (11,12) (2,1) (13,6) (13,7) (8,5) (2,10) (12,13) (14,2) (2,3) (11,13) (11,10) (1,8) (2,13) (0,6) (3,4) (11,9) (4,12) (10,5) (7,3) (2,9) (0,8) (1,10) (8,12) (12,7) (14,7) (10,12) (6,11) (3,9) (12,6) (1,0) (1,6) (5,3) (11,7)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->4->8->6->10->11->1->9->5->3->7->12->13->14."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph23",
        "image": "data/hamilton/graph_visual/hard/graph23.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 10, and the edges are: (10,6) (7,0) (0,1) (5,6) (10,1) (2,9) (7,9) (5,8) (3,4) (4,6) (7,10) (10,8) (2,4) (7,4) (9,0) (5,9) (8,4) (1,4) (6,1) (10,5) (3,0) (0,6) (1,2)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->7->9->2->1->10->6->5->8->4->3."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph310",
        "image": "data/hamilton/graph_visual/hard/graph310.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (14,11) (6,3) (14,3) (4,2) (12,5) (7,13) (6,12) (2,10) (0,10) (1,4) (4,9) (13,12) (2,9) (2,7) (5,15) (11,15) (4,7) (5,7) (3,11) (0,1) (13,6) (8,4) (1,5) (10,7) (2,11) (14,12) (11,8) (13,0) (15,10) (10,1) (11,6) (9,12) (14,7) (1,13) (0,3) (13,10) (4,6) (10,4) (7,0) (8,13) (0,9) (9,6) (3,13) (14,10) (15,9) (15,3) (15,2) (12,11)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->10->2->4->1->5->12->6->3->14->7->13->8->11->15->9."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph544",
        "image": "data/hamilton/graph_visual/hard/graph544.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 19, and the edges are: (7,4) (10,15) (15,3) (8,19) (18,11) (4,5) (4,0) (1,2) (5,0) (14,6) (1,3) (15,9) (9,8) (12,13) (7,15) (3,8) (18,19) (13,15) (16,4) (15,4) (17,8) (1,0) (16,3) (0,11) (12,8) (5,2) (16,13) (17,14) (2,7) (0,3) (4,8) (13,6) (16,10) (1,12) (11,7) (17,7) (7,5) (1,4) (10,17) (19,3) (16,5) (14,13) (0,9) (18,17) (11,1) (14,8) (18,4) (6,17) (19,7) (1,15) (7,0) (19,1) (15,5) (6,4) (3,18) (4,10) (17,4) (6,0) (15,6) (1,8) (12,4) (10,11) (11,16) (10,8) (15,0) (1,13) (13,19) (3,17) (13,4) (12,19) (14,19) (6,18) (19,0) (0,10) (2,11) (13,0) (15,17) (9,11)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->4->7->15->10->16->3->1->12->13->6->14->17->18->19->8->9->11->2->5."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph457",
        "image": "data/hamilton/graph_visual/hard/graph457.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 17, and the edges are: (8,16) (8,7) (10,17) (15,14) (12,1) (17,6) (3,1) (1,15) (15,12) (14,10) (15,5) (7,9) (2,1) (1,10) (13,15) (3,10) (9,0) (4,10) (15,16) (15,0) (15,8) (8,14) (9,2) (5,1) (14,2) (13,12) (8,11) (4,8) (8,9) (14,11) (4,5) (10,7) (2,5) (2,17) (8,17) (9,3) (17,11) (11,9) (8,6) (8,3) (6,7) (13,5) (10,5) (17,14) (14,12) (4,9) (17,12) (10,11) (10,9) (0,11) (2,7) (0,6) (6,3) (7,4) (0,4) (3,17) (14,1) (0,5) (6,16) (3,4) (0,17) (17,15) (1,4) (12,2) (0,3) (6,4) (6,2) (6,15) (3,12) (3,14) (6,10) (17,5) (1,11) (14,4) (16,14) (7,12) (9,5) (4,17) (17,1) (13,6) (10,0) (14,7) (2,0) (10,15) (13,16) (13,4) (5,8) (9,12) (8,12) (13,3) (1,6) (2,3) (12,11) (1,13) (10,13)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->9->7->8->16->15->14->10->17->6->3->1->2->5->4->13->12->11."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph168",
        "image": "data/hamilton/graph_visual/hard/graph168.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 12, and the edges are: (11,7) (12,3) (4,5) (10,6) (11,0) (1,9) (4,7) (6,3) (4,3) (4,9) (7,6) (5,9) (2,12) (9,10) (11,10) (11,8) (12,11) (4,6) (1,6) (9,12) (8,9) (6,9) (9,7) (1,3) (1,8) (9,11) (2,4) (1,11) (10,3) (3,2) (6,12) (0,12) (1,10) (0,7) (4,12) (2,5) (5,7) (2,7) (0,4) (0,1) (7,8) (4,1) (1,12) (8,0) (4,10) (0,10) (8,2)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->11->7->4->5->9->1->6->10->3->12->2->8."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph319",
        "image": "data/hamilton/graph_visual/hard/graph319.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (11,10) (10,13) (2,9) (10,8) (10,6) (12,14) (0,6) (15,8) (15,4) (13,5) (1,0) (2,0) (13,14) (14,3) (2,3) (12,8) (11,12) (9,11) (7,12) (0,4) (2,5) (11,5) (3,8) (3,12) (6,9) (2,10) (10,1) (1,5) (6,12) (10,14) (4,7) (14,5) (10,5) (8,0) (3,9) (13,7) (5,12) (14,1) (1,7) (15,3) (2,15) (0,3) (6,13) (12,2) (7,5) (9,1) (10,3) (15,13) (13,9)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->6->10->11->12->14->13->5->2->9->3->8->15->4->7->1."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph392",
        "image": "data/hamilton/graph_visual/hard/graph392.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 16, and the edges are: (13,6) (0,4) (15,3) (11,4) (10,1) (16,8) (12,1) (7,14) (8,2) (4,5) (7,3) (0,15) (3,1) (11,12) (15,13) (5,8) (5,2) (15,5) (1,14) (6,9) (2,13) (9,2) (4,2) (1,0) (9,11) (15,4) (7,9) (8,14) (13,7) (10,7) (8,13) (13,5) (6,11) (10,9) (16,3) (6,14) (8,0) (10,4) (0,5) (5,11) (2,0) (16,15) (1,4) (14,4) (5,10) (16,1) (6,1) (16,9) (14,0) (8,1) (13,0) (15,2) (12,5) (5,7) (9,12) (10,16) (13,16) (6,12) (13,4) (8,11) (9,13) (7,15) (5,14) (13,14) (5,1) (8,9) (10,2) (3,0) (9,5) (14,11) (6,4) (7,11) (4,16) (13,1) (6,15) (15,8) (3,12) (16,11) (11,1) (4,7) (7,2) (10,6) (3,13) (2,6) (4,9) (12,15) (3,4) (15,11) (6,16) (10,0) (3,5)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->4->11->12->1->10->7->14->8->16->3->15->13->6->9->2->5."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph412",
        "image": "data/hamilton/graph_visual/hard/graph412.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 16, and the edges are: (6,8) (0,11) (8,15) (11,9) (7,12) (6,3) (8,5) (16,3) (15,5) (2,13) (10,8) (9,6) (12,15) (5,13) (15,7) (9,1) (12,1) (16,9) (12,14) (14,9) (10,3) (16,7) (6,5) (12,6) (15,0) (2,3) (5,0) (4,16) (2,1) (6,14) (1,13) (0,9) (0,1) (1,15) (11,7) (6,10) (1,6) (9,13) (8,4) (8,1) (14,4) (0,4) (15,2) (8,7) (10,7) (12,11) (0,12) (2,12) (0,14) (7,9) (11,14) (1,14) (5,16) (0,2) (9,10) (4,6) (10,4) (13,0) (5,1) (9,4) (16,10) (3,0) (6,16) (11,16) (12,13) (14,7) (7,3) (5,14) (12,3) (13,6) (13,15) (2,9) (15,6) (4,12) (3,5) (11,4) (2,4) (15,4) (0,6) (16,13)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->11->9->6->8->15->5->13->2->3->16->7->12->1->14->4->10."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph236",
        "image": "data/hamilton/graph_visual/hard/graph236.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 13, and the edges are: (7,2) (6,13) (10,12) (10,6) (5,9) (4,7) (10,0) (8,3) (9,4) (10,3) (3,13) (8,1) (12,9) (9,1) (5,1) (12,6) (0,3) (8,7) (1,10) (11,7) (11,9) (11,5) (2,4) (0,1) (6,11) (11,12) (7,1) (2,1) (7,12) (0,5) (5,12) (2,11) (3,2) (12,2) (7,0) (3,11) (11,8) (11,0) (5,4) (11,10) (4,6) (10,7) (10,5) (13,8) (8,10) (6,3) (13,1) (6,7) (0,8) (6,9) (2,5) (5,6) (11,1) (6,8) (0,6) (0,4) (12,8) (3,1) (2,9)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->10->12->9->5->1->8->3->13->6->11->7->2->4."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph244",
        "image": "data/hamilton/graph_visual/hard/graph244.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 14, and the edges are: (12,4) (10,1) (10,7) (0,5) (9,8) (7,14) (7,1) (0,2) (12,7) (6,3) (1,0) (5,7) (11,14) (9,7) (2,6) (14,1) (11,0) (9,2) (7,0) (8,6) (14,3) (7,13) (10,12) (2,1) (6,5) (4,0) (13,14) (6,10) (12,6) (8,13) (12,11) (10,9) (7,6)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->5->7->13->8->9->2->1->10->6->3->14->11->12->4."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph311",
        "image": "data/hamilton/graph_visual/hard/graph311.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (12,14) (14,5) (3,11) (4,9) (3,6) (10,8) (3,9) (4,11) (8,6) (6,15) (7,2) (14,2) (15,2) (3,7) (7,12) (1,14) (2,4) (5,3) (15,3) (1,10) (9,2) (14,4) (15,14) (12,9) (5,15) (5,11) (6,11) (3,10) (0,3) (11,13) (5,9) (11,1) (5,10) (6,1) (4,3) (2,12) (7,13) (8,12) (4,7) (5,1) (14,10) (10,4) (7,9) (13,2) (13,4)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->11->4->9->2->14->5->15->6->1->10->8->12->7->13."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph542",
        "image": "data/hamilton/graph_visual/hard/graph542.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 19, and the edges are: (9,1) (8,1) (17,10) (19,15) (13,12) (0,1) (4,0) (16,9) (2,18) (18,1) (0,14) (6,17) (6,1) (6,9) (15,2) (8,5) (19,3) (14,2) (7,9) (1,13) (17,4) (16,15) (1,19) (19,6) (6,3) (18,15) (14,19) (5,12) (1,15) (2,8) (7,18) (1,12) (11,17) (3,11) (19,0) (18,9) (8,18) (17,18) (2,13) (16,13) (14,4) (4,12) (8,19) (19,18) (10,9) (4,5) (8,15) (12,19) (5,7) (5,2) (12,11) (14,7) (16,19) (8,0) (16,1) (18,13) (2,1) (16,4) (13,19) (1,10) (1,3) (7,13) (13,8) (17,0) (17,7) (11,6) (10,3) (2,19) (17,16) (13,14) (5,19) (9,14) (4,9) (6,12) (11,15) (3,0) (13,5) (7,19) (3,18) (13,0)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->1->9->16->15->19->3->6->11->12->13->2->18->8->5->4->14->7->17->10."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph332",
        "image": "data/hamilton/graph_visual/hard/graph332.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (12,2) (9,4) (13,6) (4,2) (0,6) (0,10) (10,9) (7,15) (12,10) (13,2) (0,14) (7,5) (8,4) (2,9) (10,6) (0,5) (1,11) (4,12) (5,13) (8,10) (10,15) (15,13) (14,1) (11,7) (13,11) (7,4) (3,7) (12,13) (3,0) (7,6) (15,1) (15,11) (9,0) (1,4) (13,9) (15,3) (12,15) (1,6) (3,2) (7,14) (8,12) (10,5) (11,12) (2,5) (10,14) (9,1) (13,3) (0,1) (15,14) (1,3) (10,3) (9,8) (8,11) (9,11) (8,5) (12,7) (8,3) (10,7) (8,2) (13,4) (15,0) (6,14) (4,10) (3,12) (8,0) (15,8) (1,2) (0,2) (10,2)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->6->13->2->12->10->9->4->8->11->1->14->15->3->7->5."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph524",
        "image": "data/hamilton/graph_visual/hard/graph524.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (2,8) (15,14) (10,5) (5,0) (6,9) (10,17) (11,18) (11,1) (1,16) (6,5) (13,2) (4,10) (13,8) (15,3) (2,11) (12,2) (0,11) (2,10) (5,8) (13,18) (13,7) (7,0) (6,11) (18,15) (0,4) (3,9) (1,5) (2,15) (16,10) (3,6) (17,5) (16,14) (13,0) (10,15) (14,11) (12,18) (1,7) (4,15) (15,0) (16,15) (16,0) (3,17) (4,7) (3,7) (10,1) (0,18) (6,7) (12,8) (14,18) (10,12) (8,0) (6,12) (17,4) (0,17) (7,8) (17,14) (16,17) (3,1) (8,11) (3,0) (17,18) (13,14) (1,9) (4,14) (9,7) (11,7) (7,17) (9,5) (12,4) (2,7) (9,4) (18,9) (3,18) (0,12) (8,6) (6,13) (6,4) (2,3) (4,8) (2,5) (5,18) (18,7) (10,9) (12,11) (12,9) (3,10) (14,1) (4,2) (1,0) (3,5) (15,6) (16,12) (8,18) (10,13) (10,14) (13,17) (10,18) (2,1)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->5->10->17->3->15->14->16->1->11->18->13->2->8->12->6->9->7->4."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph33",
        "image": "data/hamilton/graph_visual/hard/graph33.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 10, and the edges are: (10,9) (3,8) (3,1) (4,0) (4,1) (3,7) (6,0) (6,10) (0,9) (5,1) (4,2) (7,5) (9,5) (6,7) (4,3) (7,8) (9,8) (2,9) (10,0) (8,5) (8,4) (6,5) (0,2) (0,1) (6,1) (2,6)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->4->1->3->8->7->5->9->10->6->2."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph312",
        "image": "data/hamilton/graph_visual/hard/graph312.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (1,3) (14,5) (12,1) (4,1) (4,6) (13,6) (14,3) (12,11) (15,0) (15,7) (7,14) (3,13) (9,1) (7,4) (11,1) (9,8) (15,4) (2,15) (6,9) (14,10) (12,15) (2,11) (0,10) (8,2) (12,13) (3,11) (11,6) (7,13) (9,15) (6,1) (9,11) (2,9) (6,14) (1,7) (13,8) (11,13) (3,10) (12,8) (1,10) (5,15) (14,1) (4,10) (15,14) (14,4) (10,8) (5,13) (7,10) (15,8) (2,0)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->15->7->14->5->13->6->4->1->3->10->8->9->2->11->12."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph62",
        "image": "data/hamilton/graph_visual/hard/graph62.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 11, and the edges are: (6,1) (11,2) (11,6) (5,7) (0,7) (7,11) (10,6) (8,2) (0,8) (1,10) (1,11) (5,2) (6,3) (6,0) (7,1) (7,3) (9,2) (1,3) (7,9) (7,2) (1,8) (2,10) (9,11) (4,10) (5,8) (5,10) (9,10)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->7->5->2->8->1->3->6->11->9->10->4."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph258",
        "image": "data/hamilton/graph_visual/hard/graph258.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 14, and the edges are: (7,8) (7,14) (13,10) (4,8) (11,1) (13,8) (9,10) (13,9) (14,9) (0,3) (1,8) (7,2) (7,5) (13,1) (10,4) (13,12) (2,0) (7,1) (6,8) (5,11) (0,8) (6,5) (9,7) (11,6) (14,1) (12,11) (0,14) (0,7) (2,4) (9,8) (12,9) (8,10) (5,14) (6,14) (3,14) (5,10) (12,8) (9,11) (6,12) (3,11) (11,7) (13,7) (3,6) (5,12) (9,3) (9,0) (1,5)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->14->7->8->13->9->12->6->11->1->5->10->4->2."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph421",
        "image": "data/hamilton/graph_visual/hard/graph421.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 17, and the edges are: (8,16) (0,16) (3,9) (15,2) (5,10) (14,16) (13,11) (8,0) (10,14) (14,17) (16,1) (4,13) (8,12) (8,2) (3,8) (6,14) (0,1) (5,7) (4,14) (14,2) (7,17) (11,14) (9,16) (1,9) (11,10) (1,4) (17,0) (9,6) (4,16) (1,3) (3,13) (11,5) (2,5) (6,10) (16,5) (12,14) (8,10) (16,15) (12,5) (2,1) (3,6) (1,13) (13,9) (15,3) (15,10) (10,0) (6,1) (15,13) (14,7) (8,5) (7,12) (3,0) (2,11) (10,13) (3,11) (3,10) (12,11) (14,5) (17,11) (17,5) (6,8) (0,12) (9,7) (7,0) (8,11) (4,15) (12,16) (5,0) (16,7) (8,9)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->16->8->12->14->10->5->7->17->11->13->4->1->9->6->3->15->2."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph520",
        "image": "data/hamilton/graph_visual/hard/graph520.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (6,3) (0,11) (7,4) (9,16) (11,5) (16,0) (4,16) (18,15) (8,3) (8,17) (2,8) (4,11) (14,12) (1,2) (11,8) (17,3) (5,18) (13,9) (4,10) (10,15) (10,16) (2,14) (3,10) (2,4) (2,16) (11,12) (5,10) (9,8) (15,8) (12,15) (0,2) (15,1) (1,11) (11,7) (2,13) (8,10) (7,16) (6,2) (0,5) (14,6) (6,17) (11,15) (11,9) (11,3) (0,18) (10,17) (9,3) (11,18) (4,12) (1,3) (18,2) (7,2) (0,15) (13,4) (6,0) (13,5) (16,12) (10,13) (6,12) (7,6) (2,17) (14,10) (18,7) (13,8) (16,1) (13,7) (17,12) (9,10) (6,15) (8,16) (3,0) (10,12) (5,3) (7,12) (17,4) (14,11) (13,11) (16,17) (2,15) (7,14) (7,15) (0,14) (15,5) (4,8) (0,17) (13,12) (0,8) (6,9) (17,18) (6,10) (7,1) (7,3) (9,5) (5,12) (4,3) (11,6) (1,12) (15,13) (9,14) (1,6)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->11->5->18->15->10->4->7->16->9->13->2->8->3->17->6->14->12->1."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph192",
        "image": "data/hamilton/graph_visual/hard/graph192.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 13, and the edges are: (13,9) (0,5) (6,8) (5,3) (12,9) (12,13) (1,8) (13,8) (10,13) (13,4) (11,6) (5,12) (2,5) (0,13) (13,1) (0,7) (10,9) (1,2) (3,10) (9,8) (5,6) (9,1) (7,5) (4,7) (10,11) (11,12) (11,7) (2,8) (10,5) (4,8)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->5->3->10->13->12->9->1->2->8->6->11->7->4."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph379",
        "image": "data/hamilton/graph_visual/hard/graph379.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 16, and the edges are: (14,4) (2,15) (15,3) (12,3) (16,14) (2,3) (16,10) (11,16) (13,6) (16,6) (16,3) (9,7) (1,16) (6,3) (4,11) (11,13) (5,13) (0,10) (6,10) (13,1) (11,6) (6,12) (4,9) (15,6) (5,2) (13,0) (6,8) (2,1) (9,12) (8,9) (2,8) (5,14) (6,9) (9,15) (14,2) (14,13) (14,11) (5,9) (6,4) (2,0) (5,8) (0,12) (2,6) (2,9) (3,5) (10,5) (12,1) (16,8) (7,1) (7,2) (10,3) (15,5) (16,15)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->10->16->14->4->11->13->6->3->15->2->5->8->9->7->1->12."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph465",
        "image": "data/hamilton/graph_visual/hard/graph465.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 17, and the edges are: (13,17) (15,1) (5,3) (3,0) (16,6) (0,17) (17,10) (17,4) (9,11) (4,8) (7,4) (5,8) (4,13) (14,1) (7,10) (9,13) (0,12) (11,15) (9,0) (1,10) (12,15) (10,6) (5,14) (4,2) (16,9) (4,16) (6,0) (16,1) (2,10) (12,1) (4,5) (16,11) (15,7) (3,10) (2,8) (13,15) (15,14) (9,10) (5,11) (3,15) (2,1) (10,15) (0,2) (6,4) (1,5) (3,12) (3,2) (16,0) (10,4) (3,14) (4,14) (6,3) (2,6) (8,11) (1,9) (11,4) (13,3) (17,16) (7,13) (11,14) (2,15) (8,15) (7,9) (2,7) (14,7) (7,6) (13,12) (12,14) (3,1) (16,10) (3,9) (7,11) (3,4) (0,15) (13,14) (10,0) (6,9) (9,5) (1,13) (10,14) (14,9) (6,8) (4,15) (3,8) (2,9) (12,7) (14,8)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->5->8->4->17->13->9->11->15->1->14->12->7->10->2->6->16."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph313",
        "image": "data/hamilton/graph_visual/hard/graph313.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (3,14) (0,9) (1,5) (7,14) (10,14) (3,12) (13,14) (11,6) (8,5) (6,4) (1,0) (11,5) (14,8) (12,15) (12,14) (6,5) (0,10) (6,10) (1,6) (15,3) (10,12) (6,7) (11,13) (1,2) (0,15) (4,13) (13,0) (2,11) (12,9) (0,8) (5,3) (8,2) (13,7) (8,9) (11,0) (15,6) (7,15) (5,7) (12,4) (2,9) (13,5) (6,8) (2,13) (2,7) (2,12) (15,9) (5,2) (10,3) (6,13) (4,2) (3,2)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->9->12->3->15->7->14->10->6->11->13->4->2->1->5->8."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph298",
        "image": "data/hamilton/graph_visual/hard/graph298.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 14, and the edges are: (8,12) (8,7) (0,9) (11,6) (13,12) (11,7) (13,7) (14,3) (12,10) (5,13) (11,14) (13,10) (6,8) (7,2) (1,0) (10,5) (6,13) (9,10) (2,9) (4,6) (3,0) (8,3) (10,0) (3,1) (14,12) (8,10) (1,2) (1,7) (8,0) (2,5) (1,5) (10,2) (11,13) (12,5) (2,12) (14,5) (1,13) (5,11) (13,2) (14,2) (3,7) (7,6) (13,0) (8,1) (14,8) (2,11) (12,0) (14,7) (4,8) (13,14) (0,6) (8,11) (11,12) (4,0) (2,0) (4,10) (7,12) (5,0) (2,3) (1,6) (14,4)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->9->10->12->8->7->11->6->13->5->2->1->3->14->4."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph300",
        "image": "data/hamilton/graph_visual/hard/graph300.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (11,7) (0,13) (10,3) (8,12) (13,8) (2,13) (10,8) (5,6) (9,11) (4,5) (12,4) (4,10) (4,14) (5,15) (3,14) (2,7) (7,13) (0,15) (3,5) (0,2) (7,15) (5,11) (7,3) (7,14) (3,2) (8,15) (8,2) (1,7) (15,2) (3,8) (3,11) (2,14) (9,12) (14,12) (14,1) (13,10) (5,7) (7,10) (6,8) (2,12) (9,1) (11,12) (7,9) (4,11) (11,8) (6,14) (15,6) (13,12)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->13->8->12->4->5->6->14->2->15->7->1->9->11->3->10."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph145",
        "image": "data/hamilton/graph_visual/hard/graph145.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 12, and the edges are: (10,4) (9,2) (11,1) (3,12) (6,5) (10,0) (9,3) (2,1) (10,9) (12,8) (10,8) (9,8) (0,4) (11,3) (5,2) (1,0) (8,5) (0,7) (10,5) (5,11) (0,12) (7,9) (4,3) (12,1) (6,3) (11,0) (12,5) (6,11) (6,1) (6,9) (3,5)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->10->4->3->12->8->5->6->11->1->2->9->7."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph268",
        "image": "data/hamilton/graph_visual/hard/graph268.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 14, and the edges are: (10,3) (1,6) (13,6) (1,3) (0,14) (0,9) (13,3) (14,6) (0,8) (14,1) (1,10) (3,5) (7,4) (10,11) (7,14) (9,7) (9,6) (2,11) (10,0) (12,7) (9,3) (5,4) (2,8) (13,8) (11,1) (0,11) (12,14) (7,13) (2,13) (0,1) (1,5) (10,8) (2,10)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->14->6->9->3->10->8->13->2->11->1->5->4->7->12."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph277",
        "image": "data/hamilton/graph_visual/hard/graph277.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 14, and the edges are: (0,3) (2,8) (10,2) (8,7) (7,9) (6,4) (1,12) (4,11) (7,3) (12,5) (3,12) (6,7) (13,10) (10,4) (12,10) (9,1) (1,10) (11,9) (0,11) (14,0) (14,12) (7,2) (3,10) (10,14) (1,8) (5,0) (8,14) (0,7) (5,11) (4,0) (12,2) (3,4) (11,14) (8,13) (3,1) (11,12) (1,6) (8,0) (10,7) (4,7) (13,11) (6,0) (12,8) (9,10) (8,5) (12,7) (14,1) (3,2) (5,6) (9,6) (6,3) (6,8) (11,1) (3,13) (14,5) (13,2) (6,12) (13,12) (10,8) (8,11) (3,9) (14,7) (5,1) (2,4) (10,0) (3,5) (7,1) (7,13) (14,6) (11,2) (13,1) (2,5) (1,2) (5,9) (14,2) (0,12) (2,9) (6,10) (7,5)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->7->8->2->10->13->11->4->6->1->12->14->5->9."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph79",
        "image": "data/hamilton/graph_visual/hard/graph79.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 11, and the edges are: (10,3) (9,4) (6,8) (5,2) (2,7) (1,8) (11,7) (3,5) (1,9) (2,8) (3,6) (9,3) (5,8) (11,1) (4,0) (8,3) (7,1) (5,6) (6,0) (1,5) (2,3) (10,5) (9,2)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->4->9->1->8->6->3->10->5->2->7->11."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph50",
        "image": "data/hamilton/graph_visual/hard/graph50.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 10, and the edges are: (4,1) (2,3) (9,0) (2,0) (7,5) (4,7) (9,4) (2,6) (7,0) (5,8) (6,8) (1,8) (4,6) (0,4) (10,1) (1,3) (9,8) (10,6) (0,1) (1,2) (6,1) (7,1) (2,10) (9,7) (7,10) (8,7) (9,6) (9,10) (4,3) (3,9) (10,0) (4,2) (10,8)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->9->4->1->8->5->7->10->6->2->3."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph240",
        "image": "data/hamilton/graph_visual/hard/graph240.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 14, and the edges are: (1,5) (14,1) (1,12) (4,6) (13,6) (8,6) (0,6) (9,3) (11,7) (10,6) (12,4) (3,14) (4,5) (2,5) (11,5) (6,2) (13,3) (2,11) (3,12) (3,1) (8,13) (7,6) (3,8) (14,12) (4,8) (9,7) (7,10) (4,10) (10,14) (5,7) (7,0) (12,13) (12,11) (0,5) (13,14) (2,13) (0,10) (4,3) (2,8) (10,13) (12,5) (5,14) (8,10) (0,9) (8,11) (1,10) (0,11) (11,13)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->6->4->12->1->5->2->11->7->9->3->14->10->13->8."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph290",
        "image": "data/hamilton/graph_visual/hard/graph290.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 14, and the edges are: (10,6) (12,2) (12,3) (13,3) (9,1) (0,5) (3,11) (14,4) (2,14) (12,10) (7,13) (2,8) (1,7) (9,13) (0,2) (4,9) (6,11) (5,10) (14,8) (2,5) (0,4) (9,7) (11,7) (10,8) (3,8) (11,13) (10,11) (3,0) (3,9) (2,6) (12,11) (13,4) (9,10) (5,3) (5,1) (0,1) (1,2) (7,3) (14,3) (12,5) (3,2) (11,14) (14,0) (14,1) (1,12) (7,5) (7,10) (2,7) (10,13) (9,0) (6,9) (11,4) (8,7) (10,1) (8,4) (6,13) (5,13) (1,8) (9,11) (13,1) (5,4) (8,9)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->5->10->6->11->3->12->2->14->4->9->1->8->7->13."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph32",
        "image": "data/hamilton/graph_visual/hard/graph32.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 10, and the edges are: (6,0) (10,2) (0,5) (3,10) (2,4) (5,2) (6,9) (8,10) (5,9) (2,0) (9,0) (5,3) (4,0) (1,2) (6,2) (7,6) (0,7) (8,5) (10,0) (0,3) (4,5) (4,10) (1,6) (3,1) (7,3) (1,4) (8,4) (8,3) (8,1) (6,5) (2,8) (7,8) (3,6) (1,0) (7,2) (4,9)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->6->9->5->2->10->3->1->4->8->7."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph87",
        "image": "data/hamilton/graph_visual/hard/graph87.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 11, and the edges are: (9,10) (11,10) (8,11) (0,6) (4,0) (8,10) (5,0) (4,2) (3,4) (4,7) (1,11) (5,3) (1,4) (6,8) (7,5) (9,1) (7,1) (1,3) (9,3) (5,1) (4,6) (5,6) (2,6) (7,11) (0,1) (10,0) (8,4) (6,9) (3,0) (10,4) (5,4) (3,11)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->6->8->11->10->9->1->7->5->3->4->2."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph523",
        "image": "data/hamilton/graph_visual/hard/graph523.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (13,8) (1,16) (14,8) (16,9) (1,8) (8,10) (0,13) (4,15) (6,10) (5,0) (0,11) (6,1) (17,6) (6,11) (16,4) (2,1) (15,10) (11,9) (16,6) (12,5) (14,3) (10,11) (15,6) (10,0) (17,16) (9,12) (10,1) (1,7) (4,5) (2,4) (3,2) (8,16) (17,14) (5,2) (12,1) (0,12) (14,4) (9,14) (16,7) (15,1) (11,15) (16,3) (18,11) (9,18) (0,7) (4,9) (6,0) (15,8) (12,16) (3,4) (1,14) (13,12) (8,4) (14,13) (8,5) (13,2) (17,10) (16,0) (17,4) (6,4) (7,15) (13,18) (6,18) (9,5) (0,8) (17,7) (5,11) (2,17) (2,6) (5,18) (14,2) (4,11) (0,4) (11,16) (9,2) (18,4) (16,13) (5,1) (13,5) (18,12) (13,17) (0,14) (11,17) (12,10) (9,13) (1,0) (16,15) (15,12) (18,17) (15,5) (10,13) (3,1) (17,3) (7,11) (1,17) (2,12) (4,1) (17,0) (3,0) (15,9) (17,15) (13,15) (16,18) (12,3) (12,7) (18,14)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->13->8->14->3->2->1->16->9->11->6->10->15->4->5->12->18->17->7."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph351",
        "image": "data/hamilton/graph_visual/hard/graph351.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (8,3) (3,14) (6,12) (3,1) (14,12) (4,8) (6,8) (8,15) (13,2) (1,2) (7,11) (10,1) (7,2) (1,0) (6,10) (0,12) (11,13) (13,3) (3,6) (5,6) (2,5) (9,14) (12,11) (9,12) (9,13) (9,10) (8,9) (10,13) (7,0) (11,6) (1,15) (3,12) (6,9) (7,14) (4,6) (8,11) (9,7) (8,13) (9,1) (6,2) (15,14) (2,14) (15,11) (2,9) (2,4) (8,7) (12,10) (12,8) (13,6) (2,8) (0,5) (8,14) (3,10) (14,11) (10,7) (12,4) (0,10) (5,10) (9,5) (5,11) (0,9) (8,5) (9,3) (3,4) (13,7) (4,5) (15,0) (13,0) (8,0) (4,1) (7,4) (4,10) (9,11) (1,7)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->1->3->8->4->6->12->14->9->13->2->7->10->5->11->15."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph417",
        "image": "data/hamilton/graph_visual/hard/graph417.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 16, and the edges are: (9,12) (9,14) (1,15) (16,10) (10,12) (2,11) (7,2) (1,13) (3,1) (6,14) (8,3) (4,8) (5,6) (0,2) (1,12) (14,12) (0,16) (7,6) (9,15) (8,2) (3,10) (11,13) (2,6) (10,8) (12,6) (16,7) (1,0) (10,2) (15,11) (10,14) (9,6) (7,12) (8,7) (7,9) (10,13) (14,11) (8,14) (14,1) (3,11) (4,5) (3,4) (15,12) (4,1) (12,0) (15,16) (3,7) (13,4) (1,11) (2,1) (0,7) (10,15) (4,16) (13,14) (4,0) (2,16) (13,12) (2,5) (3,13) (11,8) (0,15) (2,3) (1,9) (5,8) (7,5) (14,5) (3,12) (5,0) (11,6) (7,4) (15,6) (14,15) (5,15) (9,13)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->11->13->1->15->9->12->10->16->7->6->14->8->3->4->5."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph31",
        "image": "data/hamilton/graph_visual/hard/graph31.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 10, and the edges are: (9,2) (10,0) (8,4) (7,8) (4,1) (4,9) (7,10) (1,10) (5,0) (8,6) (6,10) (10,9) (7,9) (7,4) (4,5) (10,5) (9,5) (1,2) (2,6) (1,0) (6,1) (1,7) (2,3) (8,3) (0,9) (6,0) (0,4) (9,3) (3,0)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->10->7->8->4->1->6->2->3->9->5."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph209",
        "image": "data/hamilton/graph_visual/hard/graph209.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 13, and the edges are: (9,11) (0,11) (10,3) (3,1) (10,5) (3,6) (4,7) (8,13) (3,8) (12,6) (0,6) (13,7) (12,2) (3,4) (2,7) (1,12) (4,11) (4,9) (0,7) (7,3) (3,9) (13,0) (4,10) (2,1) (7,12) (9,0) (12,13) (11,2) (0,2) (6,4) (12,8) (9,2) (7,11) (4,2) (2,13) (0,3) (9,6) (2,10) (11,13) (11,6) (12,9) (1,9) (5,7)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->11->9->4->7->13->8->3->6->12->1->2->10->5."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph141",
        "image": "data/hamilton/graph_visual/hard/graph141.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 12, and the edges are: (3,11) (5,8) (3,1) (11,1) (4,9) (7,4) (12,5) (0,1) (10,4) (4,6) (1,7) (12,1) (1,4) (10,2) (2,9) (11,10) (8,9) (5,4) (6,1) (2,1) (3,8) (9,10) (11,8) (6,9) (7,10) (3,10) (6,12) (2,4)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->1->3->11->10->2->9->8->5->12->6->4->7."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph13",
        "image": "data/hamilton/graph_visual/hard/graph13.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 10, and the edges are: (7,1) (6,5) (8,7) (9,1) (5,10) (5,8) (7,3) (0,7) (7,5) (4,3) (8,3) (1,4) (1,6) (9,5) (3,1) (7,6) (2,3) (5,0) (0,4) (9,3) (5,3) (4,10) (10,7) (10,3)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->4->10->5->8->7->6->1->9->3->2."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph590",
        "image": "data/hamilton/graph_visual/hard/graph590.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 19, and the edges are: (18,12) (19,18) (7,0) (8,6) (14,18) (12,9) (18,11) (0,14) (0,17) (15,7) (16,18) (12,16) (11,3) (6,2) (12,5) (6,14) (5,8) (16,5) (13,0) (2,15) (1,7) (6,5) (17,3) (4,1) (15,18) (9,18) (19,8) (2,17) (6,7) (5,18) (16,19) (13,7) (13,10) (17,7) (15,5) (8,3) (14,17) (3,12) (13,16) (18,8) (1,14) (18,3) (19,4) (19,15) (6,11) (5,19) (0,16) (2,7) (7,14) (19,9) (13,6) (19,17) (10,4) (4,6) (17,8) (0,6) (1,19) (11,16) (8,9) (5,4) (11,2) (10,2) (8,13) (12,17) (9,11) (3,0) (11,1) (13,3) (1,16) (8,4) (3,16) (0,15) (10,7) (1,3) (19,10) (17,11) (16,10) (3,2) (17,5) (15,11) (1,12) (7,16) (13,14) (5,11) (12,8) (17,1) (14,19) (17,18) (8,2) (2,1) (1,8) (10,3) (6,1) (4,14)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->7->15->2->6->8->5->12->18->19->16->13->10->4->1->14->17->3->11->9."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph464",
        "image": "data/hamilton/graph_visual/hard/graph464.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 17, and the edges are: (3,12) (17,4) (13,15) (7,4) (3,6) (14,10) (12,5) (1,3) (0,13) (17,3) (15,14) (7,11) (8,16) (6,8) (2,3) (2,13) (14,9) (16,7) (1,17) (5,0) (4,8) (17,14) (14,0) (15,4) (11,0) (0,17) (4,10) (1,12) (16,1) (9,15) (8,15) (8,11) (10,16) (0,3) (13,5) (10,6) (6,9) (9,17) (1,5) (17,6) (9,5) (6,16) (5,3) (13,10) (9,0) (2,16) (7,15) (10,12) (6,7) (15,1) (6,15) (10,7) (16,15) (10,17) (16,9) (7,9) (1,11) (17,12) (12,15) (2,6) (5,16) (16,0) (6,1) (9,12) (5,2) (0,15) (12,8) (4,2) (13,7) (1,4) (11,12) (4,9) (8,13) (17,11) (8,3) (11,13) (6,0) (4,5) (5,15) (11,2) (10,5) (4,3) (5,8) (16,4) (6,4) (4,11) (2,14) (7,14) (14,13) (4,13)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->13->15->14->10->4->17->3->12->5->1->16->8->6->9->7->11->2."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph212",
        "image": "data/hamilton/graph_visual/hard/graph212.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 13, and the edges are: (0,9) (0,12) (8,10) (4,0) (8,0) (13,2) (2,7) (5,11) (3,10) (7,5) (13,10) (9,7) (10,9) (5,0) (4,5) (10,1) (3,4) (11,6) (8,7) (12,13) (11,10) (2,0) (1,13) (2,11) (12,4) (10,0) (3,8) (7,0) (11,12) (11,0) (9,6) (5,9) (3,11) (5,13) (8,9) (6,5) (1,8) (3,0) (12,5) (2,4) (13,11) (3,1) (8,6) (3,7) (0,13) (3,9) (8,4)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->9->7->2->13->10->8->1->3->4->5->6->11->12."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph514",
        "image": "data/hamilton/graph_visual/hard/graph514.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (8,18) (17,5) (14,17) (0,15) (3,7) (2,18) (9,18) (6,16) (2,13) (5,15) (13,14) (2,8) (3,1) (9,4) (10,1) (9,15) (17,0) (5,3) (9,12) (18,4) (2,1) (11,14) (2,15) (1,16) (8,17) (4,10) (8,1) (12,3) (6,14) (14,5) (15,1) (7,0) (2,3) (7,10) (9,16) (6,18) (8,10) (16,5) (11,2) (18,3) (17,10) (13,12) (5,1) (7,18) (14,0) (3,8) (13,15) (4,17) (1,4) (7,9) (16,11) (3,9) (18,5) (10,5) (18,13) (11,10) (16,4) (10,6) (1,9) (13,9) (18,10) (8,15) (4,15) (11,9) (18,17) (18,16) (5,9) (14,18) (10,3) (7,1) (4,14) (16,14) (14,15) (15,16) (16,2) (8,6) (4,11) (13,16) (0,8) (10,15) (8,12) (6,4) (1,18) (3,15) (18,15) (14,12) (3,16) (3,0) (5,4) (1,17) (5,6) (17,2) (1,12) (11,13) (4,0) (4,8) (4,12) (15,6) (6,12) (3,11) (11,8) (8,5) (6,3) (5,7) (7,6) (4,13) (7,8) (9,17) (15,12)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->15->5->17->14->13->2->18->8->1->3->7->10->4->9->12->6->16->11."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph564",
        "image": "data/hamilton/graph_visual/hard/graph564.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 19, and the edges are: (1,9) (17,16) (10,15) (6,5) (7,16) (6,19) (7,2) (7,9) (6,16) (11,6) (12,0) (12,8) (1,18) (6,7) (10,2) (3,8) (4,16) (6,2) (7,8) (15,16) (4,12) (9,13) (13,7) (14,19) (0,17) (9,5) (17,6) (5,14) (10,5) (10,18) (16,1) (1,7) (7,4) (9,4) (1,15) (12,3) (2,13) (0,15) (5,16) (13,11) (12,15) (9,19) (6,3) (10,17) (12,6) (8,4) (11,19) (4,1) (0,4) (7,11) (18,3) (16,12) (4,14) (2,14) (13,18) (13,1) (10,11) (17,12) (16,2) (1,19) (17,5) (0,14) (7,14) (4,11) (15,7) (5,12) (9,6) (11,2) (10,1) (10,7) (3,10) (3,13) (1,5) (5,13) (12,1) (12,11) (18,6) (15,19) (1,17) (14,12) (10,8) (12,9) (17,8)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->12->8->3->6->5->9->1->18->10->15->7->2->13->11->19->14->4->16->17."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph325",
        "image": "data/hamilton/graph_visual/hard/graph325.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (2,6) (3,7) (7,14) (10,5) (13,14) (12,9) (3,15) (14,8) (2,4) (1,10) (15,5) (13,1) (9,11) (3,11) (3,13) (7,1) (3,4) (8,12) (11,6) (12,6) (3,0) (2,7) (15,7) (6,5) (9,1) (2,8) (8,9) (11,4) (5,9) (8,6) (12,7) (2,11) (2,12) (12,15) (6,13) (8,1) (4,7) (14,9) (4,13) (3,5) (3,8) (6,0) (1,0) (1,12) (6,3) (14,2) (8,13) (1,4) (15,10) (8,4) (3,2) (4,14) (0,2)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->7->14->13->1->10->5->15->12->9->11->6->2->4->8."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph443",
        "image": "data/hamilton/graph_visual/hard/graph443.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 17, and the edges are: (12,0) (4,10) (17,5) (3,0) (5,15) (16,2) (13,9) (12,17) (6,4) (10,12) (12,6) (2,5) (0,5) (8,10) (12,16) (9,14) (10,17) (12,1) (3,5) (13,12) (11,13) (14,11) (5,9) (10,2) (1,7) (0,7) (13,2) (7,4) (8,5) (1,2) (1,6) (2,6) (17,3) (8,6) (8,0) (12,5) (5,7) (1,13) (8,14) (11,9) (2,8) (10,1) (16,14) (16,4) (3,14) (2,12) (9,3) (12,15) (9,1) (6,7) (17,4) (16,10) (9,8) (1,5) (13,14) (3,12) (4,0) (8,11) (2,17) (3,2) (10,9) (15,7) (7,8)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->12->17->5->15->7->1->2->16->14->3->9->13->11->8->10->4->6."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph279",
        "image": "data/hamilton/graph_visual/hard/graph279.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 14, and the edges are: (3,2) (10,9) (5,10) (14,12) (0,7) (11,9) (2,7) (4,1) (13,7) (1,6) (8,2) (4,9) (12,13) (14,0) (13,11) (3,5) (7,9) (2,5) (8,0) (13,5) (2,4) (4,14) (11,14) (7,4) (13,8) (5,12) (10,3) (5,4) (9,2) (3,4) (4,11) (10,8) (0,9) (10,6) (0,1) (0,12) (3,7) (7,14) (4,6) (7,5) (0,5) (10,11) (9,8) (12,8) (7,10) (13,3) (2,13) (12,9) (0,11) (3,8) (9,3) (11,12) (2,14) (11,5) (12,1) (6,8) (4,0) (13,9) (6,11) (7,6) (14,3) (6,2) (9,6) (14,8) (14,6) (9,14) (7,12)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->7->2->3->5->10->9->11->13->12->14->4->1->6->8."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph45",
        "image": "data/hamilton/graph_visual/hard/graph45.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 10, and the edges are: (8,1) (10,2) (10,5) (10,7) (1,2) (3,10) (10,9) (4,3) (7,1) (9,4) (4,5) (1,0) (3,5) (5,6) (9,3) (3,0) (5,1) (7,2) (0,7) (10,4) (6,3) (0,10) (4,0) (7,8) (8,0) (7,5) (1,3) (7,4) (6,0) (8,4) (8,5) (10,1) (1,4) (5,2) (5,9)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->1->8->7->10->2->5->4->9->3->6."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph238",
        "image": "data/hamilton/graph_visual/hard/graph238.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 13, and the edges are: (5,7) (4,8) (13,4) (9,1) (11,7) (6,7) (3,1) (3,2) (1,10) (8,6) (0,10) (2,13) (10,11) (4,6) (3,13) (0,5) (12,13) (2,1) (12,9) (10,7) (8,7) (13,6) (12,6) (7,13) (12,11) (2,0) (7,3) (8,2) (12,4) (2,5) (12,8) (7,12) (4,10) (5,13) (11,1) (5,1) (4,9) (5,10) (4,0) (5,3) (11,3) (9,11) (9,5) (6,10) (8,13) (13,0) (3,8) (9,3) (12,5) (10,3) (6,0) (6,9) (8,10) (8,5) (11,13) (1,0) (2,9)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->10->1->9->12->13->4->8->6->7->5->2->3->11."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph353",
        "image": "data/hamilton/graph_visual/hard/graph353.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (8,13) (2,15) (13,4) (5,2) (11,8) (5,0) (4,9) (12,8) (3,4) (0,2) (9,5) (6,7) (8,2) (14,1) (11,12) (13,6) (11,10) (13,15) (10,3) (11,6) (9,3) (10,15) (14,5) (11,3) (2,13) (8,9) (9,0) (4,1) (12,4) (3,7) (13,5) (11,15) (6,4) (14,15) (12,13) (2,10) (2,6) (7,13) (10,5) (14,10) (10,4) (6,0) (7,5) (4,15) (4,11) (7,12) (10,12) (8,5) (15,1) (9,15) (12,0) (15,7) (13,10) (15,0) (1,11) (1,6) (2,11) (6,8) (3,5) (13,1) (12,14) (15,12) (12,1) (3,15) (0,1) (1,2) (11,7) (14,9) (1,5) (3,8) (6,14) (8,10) (11,9) (11,0) (4,7) (9,10) (10,1) (14,4) (7,9) (1,9)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->5->2->15->13->8->11->12->4->9->3->10->14->1->6->7."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph347",
        "image": "data/hamilton/graph_visual/hard/graph347.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (5,2) (11,5) (7,15) (7,6) (0,9) (10,1) (7,0) (1,4) (4,8) (6,3) (7,2) (7,14) (9,15) (11,15) (9,12) (13,2) (6,4) (1,15) (12,0) (14,3) (3,4) (4,7) (14,4) (1,3) (1,7) (12,4) (4,0) (11,1) (2,1) (10,4) (1,5) (13,7) (0,8) (10,5) (0,5) (3,7) (1,0) (8,12) (9,3) (3,15) (11,10) (9,8) (13,3) (11,13) (11,7) (0,14) (10,2) (10,12) (5,14) (6,10) (3,12) (2,4) (5,15) (2,8) (8,6) (0,3) (4,9) (0,11) (15,2) (6,2) (12,2) (8,3) (15,4) (8,15) (14,11) (8,1) (11,2) (14,12) (15,6) (8,13) (4,11) (12,11) (0,10) (7,9)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->9->15->7->6->3->14->4->1->10->5->2->13->11->12->8."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph159",
        "image": "data/hamilton/graph_visual/hard/graph159.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 12, and the edges are: (11,2) (6,10) (1,0) (3,2) (1,12) (7,2) (6,11) (12,10) (8,10) (10,4) (4,1) (8,7) (4,11) (7,12) (6,12) (1,6) (9,10) (1,10) (3,5) (9,0) (5,10) (9,1) (4,8) (6,5) (1,7) (4,9) (2,4) (4,5) (5,9) (3,7) (9,7) (11,10) (0,6) (4,12) (12,2) (8,3) (6,4) (11,12) (11,9) (6,3) (5,0) (3,11) (5,7) (4,3) (1,2) (0,4) (7,0) (8,9) (3,9) (6,8)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->1->12->10->6->11->2->3->5->4->8->7->9."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph44",
        "image": "data/hamilton/graph_visual/hard/graph44.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 10, and the edges are: (0,2) (6,4) (6,3) (9,0) (3,9) (9,8) (4,3) (7,10) (5,1) (7,4) (2,4) (3,0) (9,10) (5,9) (6,5) (0,7) (2,9) (10,6) (0,5) (2,6) (4,9) (8,7) (8,10) (8,2) (3,2) (8,5) (2,1) (0,4) (0,1) (2,5) (3,5) (3,7) (9,1) (4,8) (8,1)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->4->6->3->9->10->7->8->5->1."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph271",
        "image": "data/hamilton/graph_visual/hard/graph271.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 14, and the edges are: (0,6) (9,2) (10,5) (11,12) (5,12) (3,9) (2,12) (1,5) (5,14) (6,4) (10,11) (4,3) (2,6) (4,14) (12,4) (11,0) (14,2) (10,8) (7,10) (11,14) (12,10) (10,3) (8,13) (11,9) (7,6) (11,13) (3,13) (3,5) (7,13) (1,3) (10,0) (4,1) (9,13) (14,10) (8,4) (6,11) (3,0) (10,9) (5,13) (12,0) (4,0) (9,1) (0,7) (7,4) (6,10) (14,0) (9,8) (13,6) (5,8) (2,8) (8,7) (3,7) (2,5) (7,2) (10,2) (3,12) (12,9) (13,1) (13,4) (8,6) (9,4) (2,4) (1,11) (8,14)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->6->4->3->9->2->12->11->10->5->1->13->7->8->14."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph477",
        "image": "data/hamilton/graph_visual/hard/graph477.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 17, and the edges are: (5,8) (15,8) (3,14) (2,12) (0,9) (1,2) (3,4) (16,4) (10,0) (7,5) (11,15) (5,9) (10,6) (7,6) (0,13) (10,8) (13,1) (9,2) (16,3) (9,8) (13,11) (3,1) (14,8) (15,7) (2,13) (6,0) (7,9) (16,15) (1,9) (4,9) (17,16) (5,12) (4,8) (13,7) (4,1) (16,7) (3,17) (0,3) (5,0) (17,12) (4,7) (4,12) (8,12) (9,12) (0,16) (5,1) (5,6) (11,1) (6,8) (15,9) (3,2) (14,1) (6,1) (8,1) (7,11) (17,8) (0,1) (5,15) (6,12) (2,6) (14,17) (9,14) (0,17) (2,10) (14,0) (5,14) (1,7) (1,15) (5,17) (4,6) (10,14) (11,10) (4,14) (4,15) (12,15) (12,7) (13,14) (0,8) (11,17) (3,15) (13,10) (0,7) (7,10) (2,7) (9,11) (14,2) (16,13) (5,11) (3,13) (13,4) (3,7) (12,1) (16,2) (6,16)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->9->5->8->15->11->13->1->2->12->17->16->4->3->14->10->6->7."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph419",
        "image": "data/hamilton/graph_visual/hard/graph419.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 16, and the edges are: (1,8) (6,4) (10,5) (6,16) (12,0) (8,4) (0,2) (2,13) (13,7) (9,3) (16,11) (12,2) (3,5) (3,11) (10,2) (10,8) (5,1) (10,13) (16,9) (16,13) (11,1) (15,8) (9,12) (9,0) (1,9) (11,2) (13,8) (5,12) (8,9) (16,2) (8,5) (2,8) (7,9) (16,0) (0,1) (8,0) (15,12) (0,14) (3,0) (9,10) (16,10) (4,16) (16,7) (8,16) (4,13) (14,1) (5,4) (15,13) (7,15) (12,11) (16,12) (12,3) (15,0) (2,7) (15,6) (4,7) (13,6) (14,13) (3,1) (6,5) (15,5) (3,2) (10,12) (7,0) (10,0) (6,10) (7,12) (14,16) (12,1) (6,8) (6,2) (10,11) (6,11) (9,4) (0,13) (8,11) (6,0)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->12->2->13->7->9->3->5->10->8->4->16->14->1->11->6->15."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph241",
        "image": "data/hamilton/graph_visual/hard/graph241.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 14, and the edges are: (10,8) (13,14) (11,8) (2,3) (0,4) (12,6) (13,3) (12,7) (3,5) (4,13) (6,7) (4,8) (14,3) (12,5) (12,14) (9,2) (12,8) (9,7) (3,10) (4,5) (5,6) (8,13) (9,13) (13,5) (10,1) (9,14) (13,1) (1,2) (12,4) (8,1) (10,7) (9,12) (3,7) (10,14) (5,0) (14,7) (10,5) (11,12) (11,13) (8,14)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->4->13->14->3->2->9->7->12->6->5->10->1->8->11."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph448",
        "image": "data/hamilton/graph_visual/hard/graph448.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 17, and the edges are: (2,11) (16,7) (8,11) (12,6) (10,1) (11,10) (2,9) (6,5) (10,12) (10,6) (5,16) (2,6) (5,4) (15,17) (3,2) (2,4) (6,1) (4,3) (1,4) (6,7) (14,3) (11,4) (14,6) (3,1) (14,13) (5,12) (3,9) (17,0) (2,1) (13,4) (6,3) (4,6) (2,14) (11,9) (14,0) (3,13) (11,0) (1,14) (16,17) (8,15) (15,12) (17,13) (14,5) (17,8) (7,11) (7,1) (3,10) (0,2) (17,5) (12,7) (13,0) (13,11)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->17->15->8->11->2->9->3->4->5->16->7->6->12->10->1->14->13."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph193",
        "image": "data/hamilton/graph_visual/hard/graph193.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 13, and the edges are: (10,3) (12,5) (0,3) (10,13) (12,0) (10,12) (10,1) (7,12) (10,2) (0,5) (4,7) (9,10) (5,4) (10,7) (13,7) (6,10) (2,8) (6,7) (12,11) (0,8) (5,11) (2,11) (2,1) (13,0) (2,5) (4,10) (11,10) (13,8) (13,12) (7,5) (9,12) (12,4) (11,0) (6,11)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->10->1->2->8->13->7->6->11->5->4->12->9."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph393",
        "image": "data/hamilton/graph_visual/hard/graph393.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 16, and the edges are: (7,10) (16,14) (2,7) (5,2) (6,0) (1,5) (11,16) (1,3) (4,1) (12,13) (0,14) (15,6) (13,16) (8,0) (10,14) (8,1) (2,16) (4,0) (3,0) (3,6) (10,5) (10,8) (3,13) (5,6) (3,15) (7,5) (7,4) (2,12) (0,12) (7,16) (4,9) (6,8) (12,16) (10,11) (11,9) (10,12) (2,0) (8,15) (13,0) (13,9) (9,16) (2,15) (7,9) (5,15) (4,3) (10,0) (11,7) (8,2) (11,1) (14,3) (13,10) (2,3) (9,5) (10,9) (0,5) (6,2) (15,14) (2,9) (16,15) (4,6) (6,12) (6,9) (2,4) (11,13) (2,11) (16,0) (10,15) (8,16) (4,11) (13,8) (12,15) (9,12) (3,12) (4,15) (8,4) (1,6) (10,16) (13,15) (11,14) (4,16) (15,0)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->6->15->3->1->5->2->7->10->14->16->11->9->4->8->13->12."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph577",
        "image": "data/hamilton/graph_visual/hard/graph577.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 19, and the edges are: (1,7) (6,13) (9,14) (4,12) (19,10) (19,18) (6,9) (4,6) (14,13) (3,15) (15,10) (9,16) (5,19) (2,4) (11,7) (8,1) (3,19) (14,2) (16,3) (18,13) (19,9) (15,19) (10,14) (16,14) (19,11) (5,2) (15,12) (8,10) (2,8) (6,14) (15,4) (10,9) (9,4) (15,5) (7,6) (7,9) (10,16) (2,1) (9,18) (7,17) (0,2) (9,15) (0,11) (16,1) (16,7) (16,17) (1,13) (13,5) (3,13) (18,0) (7,12) (16,12) (12,1) (16,18) (15,8) (19,14) (12,9) (8,6) (6,19) (10,13) (0,6) (8,11) (8,12) (14,12) (7,0) (18,15) (11,12) (1,18) (14,0) (7,3) (2,7) (0,1) (12,2) (16,8) (14,17) (13,7) (5,4) (5,9) (0,19) (11,15) (1,4) (3,11) (3,10) (14,15) (18,5) (12,5) (10,17) (2,15) (19,1) (16,13) (18,8) (14,1) (4,16) (2,17) (9,1) (3,6) (4,11) (5,16) (12,19) (0,15) (13,15) (12,10) (1,5) (4,18) (2,13) (3,17) (8,19) (11,5) (1,3) (2,11) (0,13)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->4->12->15->3->19->10->14->9->6->13->18->16->17->7->1->8->11->5."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph264",
        "image": "data/hamilton/graph_visual/hard/graph264.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 14, and the edges are: (4,3) (0,8) (14,5) (1,14) (12,2) (10,13) (13,9) (12,9) (2,13) (5,3) (5,7) (7,13) (3,12) (6,1) (6,8) (14,13) (14,4) (0,5) (11,3) (11,14) (14,8) (10,2) (14,3) (8,11) (0,1) (12,0) (13,0) (2,4) (10,7) (7,11) (13,8) (11,12) (2,14) (6,5) (0,10) (1,9) (6,11) (1,11) (4,5) (1,8) (11,13) (1,13) (6,10) (12,6) (4,10) (8,4)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->8->6->1->14->5->3->4->2->12->9->13->10->7->11."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph507",
        "image": "data/hamilton/graph_visual/hard/graph507.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (8,16) (11,16) (1,12) (8,1) (14,12) (4,11) (18,2) (17,9) (13,1) (13,14) (3,12) (12,16) (18,8) (17,13) (12,8) (12,9) (12,7) (13,6) (11,2) (1,5) (5,12) (3,4) (16,14) (0,6) (10,1) (3,2) (4,5) (7,6) (10,7) (18,13) (9,2) (11,7) (10,8) (15,11) (11,5) (7,14) (10,14) (5,8) (3,10) (4,9) (0,17) (16,18) (7,17) (13,9) (5,18) (4,15) (2,14) (9,5) (2,8) (13,15) (10,15) (13,7) (17,12) (9,3) (4,2) (6,9) (8,3) (6,1) (18,9) (15,1) (14,18) (16,9) (13,4) (14,17) (14,1) (0,14) (17,10) (10,9) (0,7)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->6->13->1->12->14->16->8->18->2->11->7->17->9->5->4->3->10->15."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph224",
        "image": "data/hamilton/graph_visual/hard/graph224.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 13, and the edges are: (11,7) (5,1) (3,12) (1,2) (8,0) (12,4) (0,2) (4,13) (7,1) (8,12) (3,0) (8,9) (6,11) (1,10) (6,5) (11,10) (0,10) (3,5) (3,1) (5,10) (13,6) (5,2) (0,6) (4,11) (10,4) (1,8) (3,10) (2,12) (3,11) (9,10) (10,12) (8,13) (6,12) (9,4) (2,13) (7,13) (8,11) (4,2) (3,2) (11,9) (0,11) (4,7) (2,6) (0,5) (1,13) (6,4) (9,5) (3,13) (9,13) (9,3) (11,13) (10,7) (9,2) (0,7)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->8->12->3->5->1->2->13->4->9->10->7->11->6."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph173",
        "image": "data/hamilton/graph_visual/hard/graph173.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 12, and the edges are: (2,4) (11,10) (6,7) (2,1) (4,10) (2,12) (11,4) (4,3) (2,3) (9,2) (12,3) (0,1) (12,10) (10,0) (8,12) (5,9) (1,4) (1,7) (6,9) (3,7) (7,2) (0,9) (5,6) (9,1) (11,2) (12,6) (4,7) (4,0) (2,0) (6,8) (5,11) (9,4) (9,7) (7,11) (11,9) (8,10) (0,11) (0,6) (1,11) (3,9) (3,0) (12,1) (2,10) (9,12) (11,3) (0,8)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->1->2->4->10->11->5->9->6->7->3->12->8."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph433",
        "image": "data/hamilton/graph_visual/hard/graph433.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 17, and the edges are: (8,14) (11,9) (8,9) (5,14) (16,6) (11,2) (0,8) (4,1) (16,2) (1,3) (3,5) (4,11) (15,12) (15,13) (11,5) (7,1) (6,0) (5,4) (16,10) (4,7) (5,15) (12,11) (12,17) (1,16) (13,11) (2,4) (4,16) (3,4) (4,14) (15,11) (10,15) (17,3) (16,13) (11,3) (9,5) (15,7) (8,1) (8,12) (13,17) (16,11) (14,10) (17,0) (6,12) (4,13) (3,2) (16,8) (6,15) (0,4) (3,16) (12,5) (10,2) (4,12) (2,14) (9,7) (9,15) (8,3) (14,15) (0,16) (7,13) (7,11)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->8->14->5->3->1->4->11->9->7->15->13->17->12->6->16->2->10."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph151",
        "image": "data/hamilton/graph_visual/hard/graph151.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 12, and the edges are: (10,12) (8,6) (4,5) (3,2) (12,3) (12,8) (0,9) (9,5) (1,12) (2,0) (11,1) (0,8) (10,5) (7,8) (0,4) (6,11) (10,0) (2,4) (9,11) (4,6) (1,4) (9,1) (0,5) (1,3) (11,10) (8,11) (6,3) (10,6) (1,8) (7,4) (3,5) (12,2) (5,11) (2,7) (5,6) (0,12) (1,2)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->9->5->4->2->3->12->10->6->11->1->8->7."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph536",
        "image": "data/hamilton/graph_visual/hard/graph536.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (2,11) (7,14) (2,14) (2,0) (4,9) (4,13) (5,7) (6,15) (4,12) (1,16) (10,5) (14,4) (9,18) (0,12) (18,6) (12,11) (12,6) (2,18) (1,7) (11,3) (13,15) (11,0) (2,10) (16,8) (11,4) (15,2) (9,17) (16,11) (1,11) (12,8) (16,13) (6,14) (10,1) (5,9) (4,15) (0,5) (7,8) (6,5) (0,10) (15,9) (13,17) (6,11) (18,0) (7,3) (1,6) (1,14) (0,9) (18,5) (13,5) (2,6) (14,17) (15,14) (17,4) (15,12) (2,8) (17,3) (1,5) (12,16) (8,4) (9,14) (8,1) (17,12) (1,3) (16,4) (3,5) (3,6) (10,17) (11,5) (15,10) (4,5) (17,1) (18,12) (2,12) (17,0) (16,9) (4,1) (16,5) (12,7) (17,7) (5,17) (2,9) (12,10) (16,15) (10,18) (11,10) (15,7) (10,8) (0,3) (10,13) (18,17) (7,2) (4,10) (13,3) (17,6) (15,8) (15,3) (6,13) (15,1) (16,3) (11,17) (18,13) (7,9) (13,14) (7,0) (8,17) (12,9) (0,16)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->11->12->4->9->18->6->15->13->16->1->7->14->17->3->5->10->8."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph15",
        "image": "data/hamilton/graph_visual/hard/graph15.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 10, and the edges are: (2,0) (0,7) (4,10) (2,6) (1,8) (7,3) (3,0) (10,0) (5,2) (7,5) (2,9) (6,3) (6,0) (2,3) (2,7) (5,1) (10,6) (4,7) (8,2) (1,6) (9,5) (8,4) (9,1)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->6->3->7->5->9->1->8->4->10."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph233",
        "image": "data/hamilton/graph_visual/hard/graph233.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 13, and the edges are: (4,5) (10,3) (8,1) (5,10) (13,5) (2,11) (0,11) (2,7) (3,0) (9,3) (2,6) (13,0) (5,11) (3,13) (9,10) (1,10) (0,1) (5,7) (7,9) (1,6) (1,12) (6,10) (6,7) (6,0) (4,6) (13,2) (4,3) (2,9) (11,9) (1,11) (0,2) (2,3) (0,9) (1,5) (9,12) (8,7) (13,8) (8,3) (9,5) (9,1) (8,2) (8,6) (0,10) (8,11) (9,6) (7,12) (0,12) (12,10) (13,1) (4,9) (4,12) (7,11) (9,8)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->11->2->7->5->4->6->1->8->13->3->10->9->12."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph229",
        "image": "data/hamilton/graph_visual/hard/graph229.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 13, and the edges are: (7,0) (13,5) (7,6) (10,8) (8,13) (0,2) (0,4) (11,9) (5,10) (11,3) (1,11) (3,12) (0,10) (9,6) (2,4) (1,0) (0,12) (6,12) (11,13) (4,7) (10,7) (12,13) (6,3) (6,5) (11,0) (12,8) (6,11) (11,4) (0,3) (3,8) (7,13) (13,0) (10,11) (2,7) (5,8) (3,4) (2,1) (6,8) (5,0) (3,7) (4,5) (2,13) (7,9) (3,9) (4,6) (5,9) (2,9) (6,1) (0,9) (2,3) (12,5) (5,1) (1,9) (13,6) (3,13) (1,7) (4,9) (11,2) (12,9)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->7->6->9->11->3->12->13->8->10->5->4->2->1."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph4",
        "image": "data/hamilton/graph_visual/hard/graph4.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 10, and the edges are: (6,5) (7,5) (9,5) (0,2) (3,10) (5,8) (4,0) (5,10) (8,10) (9,10) (8,9) (9,1) (6,10) (2,10) (5,2) (1,4) (8,3) (7,0) (1,10) (4,9) (0,9) (7,4) (2,4) (7,6)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->10->3->8->5->6->7->4->1->9."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph176",
        "image": "data/hamilton/graph_visual/hard/graph176.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 12, and the edges are: (3,5) (9,7) (1,7) (8,9) (3,7) (12,3) (8,2) (12,0) (5,11) (7,10) (8,0) (8,10) (5,2) (1,4) (11,2) (1,2) (12,5) (9,10) (10,3) (6,12) (1,11) (9,3) (1,10) (0,4) (10,5) (5,8) (10,6) (4,9) (8,1) (5,9) (12,11) (7,4) (1,3) (11,3) (1,9) (6,3) (8,12) (2,6) (8,7) (7,2) (0,1)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->12->3->5->11->2->8->9->7->4->1->10->6."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph368",
        "image": "data/hamilton/graph_visual/hard/graph368.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 16, and the edges are: (15,10) (10,4) (1,2) (4,2) (0,13) (11,6) (3,1) (5,6) (6,14) (8,1) (6,10) (2,13) (3,9) (10,7) (0,8) (4,3) (12,13) (8,2) (5,16) (15,5) (3,13) (7,16) (15,1) (16,12) (15,3) (13,14) (14,16) (0,5) (12,3) (2,14) (14,5) (14,8) (4,1) (9,8) (15,0) (0,16) (5,7) (10,1) (4,16) (16,1) (10,11) (6,2) (15,9) (7,1) (14,7) (1,0) (8,7) (7,3) (1,13) (3,2) (11,5)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->13->2->1->3->9->8->7->14->6->11->5->15->10->4->16->12."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph385",
        "image": "data/hamilton/graph_visual/hard/graph385.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 16, and the edges are: (12,10) (4,8) (1,0) (13,8) (9,6) (15,3) (10,7) (3,1) (15,16) (15,9) (16,3) (5,7) (8,6) (9,14) (8,12) (11,3) (10,6) (4,13) (2,1) (10,13) (12,1) (11,10) (7,3) (11,5) (9,8) (4,12) (16,14) (14,10) (5,6) (14,2) (11,7) (5,8) (0,2) (7,13) (14,7) (14,8) (9,5) (8,0) (12,11) (16,11) (7,4) (2,3) (4,16) (12,5) (15,4) (9,3) (0,15) (4,9) (0,5) (1,7) (7,8) (13,2) (2,6) (9,0)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->1->3->15->16->14->9->6->8->4->12->10->11->5->7->13->2."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph17",
        "image": "data/hamilton/graph_visual/hard/graph17.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 10, and the edges are: (10,9) (0,4) (9,2) (10,4) (7,5) (9,4) (7,3) (2,3) (10,8) (1,3) (7,2) (8,4) (0,1) (0,7) (2,10) (6,3) (5,9) (8,2) (2,4) (3,0) (10,5) (9,3) (9,0) (1,2) (2,6) (8,9) (1,4) (3,4)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->4->10->8->9->5->7->3->1->2->6."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph531",
        "image": "data/hamilton/graph_visual/hard/graph531.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (18,7) (14,2) (16,18) (8,2) (7,11) (18,11) (8,6) (15,12) (18,6) (13,11) (12,17) (13,4) (6,1) (4,10) (16,13) (16,0) (14,5) (12,18) (5,8) (6,2) (10,14) (18,14) (9,4) (16,2) (7,13) (16,14) (12,13) (3,4) (9,15) (1,0) (13,18) (7,10) (6,4) (0,12) (0,10) (11,8) (11,12) (8,9) (4,12) (1,10) (18,8) (6,10) (6,17) (10,3) (17,2) (5,6) (5,15) (3,7) (15,13) (14,17) (14,11) (10,9) (7,9) (18,9) (1,12) (9,6) (9,0) (7,8) (15,2) (14,4) (1,8) (15,7) (11,0) (13,3) (8,3) (13,0) (8,0) (1,16) (16,12) (5,16) (6,15) (4,17) (17,5) (9,2) (5,3) (9,5) (12,14) (2,7) (8,14) (9,13) (1,14) (7,12) (13,6) (0,17) (8,15) (12,2) (18,5) (10,8) (14,7) (8,4) (4,0) (15,3) (17,18) (10,5) (3,16) (17,1) (2,13) (12,8) (18,3)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->16->18->7->11->13->4->10->14->2->8->6->1->12->17->5->3->15->9."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph541",
        "image": "data/hamilton/graph_visual/hard/graph541.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 19, and the edges are: (5,19) (0,11) (13,15) (12,11) (7,9) (16,2) (4,9) (19,17) (19,9) (18,9) (17,16) (6,13) (4,11) (15,10) (16,19) (4,15) (16,11) (1,6) (11,19) (16,12) (3,15) (16,7) (10,8) (2,13) (17,1) (7,15) (4,8) (15,8) (0,7) (8,14) (19,15) (19,12) (13,14) (5,14) (17,10) (6,4) (19,1) (2,1) (13,16) (17,11) (3,14) (19,14) (5,8) (18,1) (1,5) (19,4) (12,6) (0,4) (17,8) (16,1) (2,15) (4,1) (7,12) (8,16) (3,2) (15,9) (18,3) (14,1) (17,15) (6,15) (4,3) (7,11) (5,18) (12,18) (9,13) (0,17) (6,3) (5,7) (2,19) (5,15) (19,7) (17,5) (8,11) (11,5) (1,8) (7,18)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->11->12->16->2->13->15->10->8->4->9->7->5->14->19->17->1->6->3->18."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph27",
        "image": "data/hamilton/graph_visual/hard/graph27.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 10, and the edges are: (3,9) (2,8) (6,9) (8,1) (1,10) (10,2) (10,6) (5,1) (9,2) (4,9) (2,1) (5,3) (8,10) (5,6) (10,3) (4,10) (3,0) (7,5) (7,4) (7,6) (10,7) (0,9) (3,1) (1,6) (4,3) (5,4) (7,9) (2,3)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->9->6->10->2->8->1->5->7->4."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph592",
        "image": "data/hamilton/graph_visual/hard/graph592.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 19, and the edges are: (10,0) (5,6) (6,19) (16,3) (8,12) (19,13) (18,4) (2,10) (5,7) (7,17) (2,3) (4,2) (19,9) (10,17) (10,8) (15,1) (9,7) (19,17) (18,0) (3,18) (0,15) (4,3) (19,1) (19,10) (0,1) (13,17) (6,9) (5,0) (7,8) (18,14) (15,2) (11,9) (12,3) (13,10) (17,16) (17,1) (7,11) (2,14) (12,6) (3,14) (7,19) (7,6) (16,1) (6,18) (2,19) (6,3) (8,19) (1,14) (11,1) (17,18) (7,10) (1,12) (19,3) (13,6) (18,7) (4,6) (0,14) (11,18) (17,4) (5,11) (13,2) (10,11) (14,12) (11,17) (19,5) (4,0) (14,4) (9,3) (12,17) (15,16) (12,0) (6,2) (16,19) (11,14) (10,5) (2,1) (12,18) (15,11) (19,15) (12,2) (3,10) (4,16) (19,4) (0,6) (2,5) (8,3) (15,14) (10,12) (2,9) (17,8) (17,3) (5,15) (4,8) (1,18) (8,1) (0,8) (15,8) (18,13) (3,7) (15,17) (14,6) (8,11) (10,18) (2,11) (4,7) (17,0) (8,13) (9,18) (3,0) (4,11)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->10->2->3->16->17->7->5->6->19->13->18->4->14->1->12->8->15->11->9."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph599",
        "image": "data/hamilton/graph_visual/hard/graph599.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 19, and the edges are: (11,19) (13,9) (3,11) (6,19) (10,17) (19,17) (15,2) (0,14) (11,5) (3,14) (14,12) (19,2) (5,12) (15,18) (10,15) (2,13) (7,15) (18,7) (16,8) (4,17) (9,0) (9,15) (18,3) (1,15) (8,19) (11,14) (12,7) (8,12) (10,12) (2,12) (16,4) (13,19) (11,9) (9,18) (9,1) (5,9) (16,0) (15,0) (16,1) (3,15) (15,5) (12,9) (4,5) (7,5) (0,10) (5,3) (12,17) (16,5) (14,2) (16,3) (18,6) (14,15) (4,18) (9,7) (6,4) (1,17) (17,9) (9,14) (6,14) (5,18) (11,18) (11,10) (19,9) (10,16) (13,0) (1,10) (19,14) (1,6) (16,9) (4,3) (4,9) (9,10) (17,18) (16,19) (15,17) (7,2) (3,19) (12,19) (8,6) (17,16) (1,8) (5,8) (18,16) (12,16) (13,10) (6,7) (2,10) (7,16) (5,13) (2,0) (19,15) (2,18) (14,5) (3,2) (15,16) (5,17) (4,1) (13,15) (10,6) (19,10) (19,5) (4,2) (16,13) (16,11) (10,14) (16,2) (16,6) (11,15) (5,10) (5,6) (18,13) (14,1) (2,17) (7,14) (1,2) (3,13) (7,11) (12,1) (4,11) (5,1) (17,3) (0,4) (17,11) (0,12)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->14->3->11->19->6->18->15->2->13->9->1->16->8->12->7->5->4->17->10."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph309",
        "image": "data/hamilton/graph_visual/hard/graph309.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (10,11) (7,3) (15,5) (14,15) (14,2) (13,9) (7,9) (15,2) (12,6) (5,4) (12,5) (10,7) (3,14) (5,7) (5,13) (9,14) (4,14) (3,4) (2,10) (0,15) (3,8) (9,8) (11,15) (10,12) (5,3) (10,3) (8,6) (9,11) (14,5) (3,2) (4,7) (7,15) (6,2) (8,13) (0,10) (5,6) (9,3) (14,1) (9,10) (12,13) (8,2) (4,9) (7,14) (8,7)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->15->5->4->3->7->9->11->10->12->13->8->6->2->14->1."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph330",
        "image": "data/hamilton/graph_visual/hard/graph330.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (8,1) (4,3) (9,15) (4,11) (4,13) (6,3) (0,12) (0,2) (14,0) (10,4) (5,7) (0,1) (10,14) (0,3) (7,13) (10,0) (6,7) (8,2) (8,9) (5,14) (13,2) (9,14) (0,8) (8,4) (0,7) (14,13) (5,6) (10,7) (10,8) (5,12) (5,8) (6,10) (5,13) (15,11) (5,4) (9,10) (0,6) (8,15) (12,11) (9,12) (11,13) (2,6) (12,1) (15,14) (6,13) (8,11) (12,3) (2,7) (12,4) (7,3) (1,2) (4,6) (3,15) (1,9) (13,1) (12,10) (15,1) (7,1) (13,10) (3,13) (0,11) (6,14) (6,15) (2,12) (7,14) (2,10) (3,14) (11,2)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->12->5->7->13->4->3->6->10->14->9->15->11->8->1->2."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph511",
        "image": "data/hamilton/graph_visual/hard/graph511.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (18,1) (3,4) (4,17) (6,5) (15,0) (13,12) (6,13) (11,18) (18,0) (18,7) (6,4) (11,16) (10,7) (12,0) (3,13) (13,1) (9,11) (7,16) (7,11) (9,14) (0,5) (1,6) (17,2) (3,10) (11,8) (6,16) (13,9) (1,9) (3,7) (17,12) (2,3) (17,14) (11,2) (1,11) (14,18) (17,11) (3,1) (3,8) (7,8) (13,10) (8,4) (8,16) (1,8) (17,1) (3,6) (15,9) (15,18) (12,2) (4,9) (6,14) (5,7) (5,15) (2,7) (13,15) (5,17) (0,11) (8,17) (10,16) (15,16) (3,9) (2,5) (16,13) (0,4) (16,12) (17,15) (0,8) (18,8) (7,1) (1,0) (0,6) (5,16) (13,11) (3,15) (14,16) (9,2) (12,11) (14,2) (3,14) (14,10) (10,11) (4,2) (10,5) (9,7) (6,9) (14,4) (2,1) (13,5) (1,12) (14,15) (12,9) (8,2) (1,4) (4,16) (14,13)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->15->9->11->18->1->13->12->17->4->3->10->7->16->8->2->5->6->14."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph156",
        "image": "data/hamilton/graph_visual/hard/graph156.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 12, and the edges are: (4,12) (6,0) (4,3) (7,1) (10,11) (2,8) (11,8) (10,3) (12,8) (2,11) (6,12) (0,12) (4,9) (11,7) (9,8) (3,12) (5,1) (2,3) (2,5) (4,2) (10,12) (4,8) (4,6) (7,12) (3,5) (0,7) (12,2) (9,1) (8,0) (5,11) (5,7) (1,3) (12,5) (5,10) (9,7) (10,0) (6,9) (2,6) (9,5) (6,11) (10,6) (9,11) (7,8) (8,10) (10,9)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->6->12->4->3->10->11->8->2->5->1->7->9."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph399",
        "image": "data/hamilton/graph_visual/hard/graph399.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 16, and the edges are: (10,9) (14,3) (8,4) (13,3) (1,13) (3,10) (16,4) (9,1) (15,11) (16,10) (13,14) (15,7) (14,2) (4,9) (9,5) (1,11) (9,13) (2,8) (8,14) (9,8) (3,15) (14,16) (11,5) (1,8) (12,5) (1,16) (8,6) (9,15) (13,16) (10,8) (12,3) (4,2) (4,13) (12,8) (7,8) (9,0) (1,0) (16,0) (5,16) (13,2) (11,8) (4,0) (3,8) (3,16) (14,1) (1,15) (6,0) (7,0) (0,10) (3,5) (0,5) (7,12) (12,14) (0,14) (3,0) (16,15) (12,15) (12,0) (15,14) (3,9) (9,2) (15,13) (6,2) (12,4) (0,11) (3,1) (9,11) (7,6) (2,5) (16,2) (0,13) (5,15) (14,5) (14,11) (12,11) (16,8)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->9->10->3->14->13->1->11->15->7->8->4->12->5->16->2->6."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph460",
        "image": "data/hamilton/graph_visual/hard/graph460.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 17, and the edges are: (4,10) (2,15) (16,14) (5,1) (15,14) (6,13) (9,14) (11,10) (12,3) (0,16) (14,10) (6,0) (7,4) (10,0) (4,3) (15,4) (12,1) (17,0) (11,13) (13,15) (0,2) (12,16) (15,1) (15,17) (6,10) (15,3) (7,12) (17,11) (5,13) (2,8) (6,3) (1,8) (14,11) (12,11) (13,4) (15,12) (2,6) (15,8) (3,9) (3,2) (16,9) (6,5) (5,16) (0,4) (11,6) (13,8) (12,5) (5,4) (9,2) (17,6) (8,7) (8,12) (8,9) (14,5) (6,7) (9,15) (5,11) (12,14) (6,14) (3,13) (1,0) (12,6) (10,8) (16,6) (0,14) (17,5) (1,4) (7,2) (17,13) (1,16) (7,13) (1,10) (7,17) (10,3) (12,4) (6,1) (11,0) (3,1) (12,17) (2,10) (7,5) (8,14) (9,10) (4,2) (12,0) (17,9) (8,5) (0,15)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->16->14->15->2->8->1->5->13->6->10->4->7->12->3->9->17->11."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph341",
        "image": "data/hamilton/graph_visual/hard/graph341.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (2,10) (0,13) (3,2) (1,11) (7,3) (7,8) (1,12) (8,11) (4,1) (9,6) (2,14) (9,10) (8,2) (15,9) (2,12) (7,9) (3,4) (9,14) (9,3) (0,9) (10,11) (11,13) (2,15) (4,5) (6,1) (9,2) (3,14) (2,1) (12,13) (2,5) (5,7) (10,15) (10,4) (13,9) (5,12) (6,11) (7,4) (12,11) (6,5) (9,5) (6,8) (14,11) (15,14) (13,14) (6,4) (11,5) (12,9) (7,13) (0,14) (14,10) (13,15) (3,5) (5,8) (2,13) (1,14) (5,15) (10,6) (2,11) (7,10) (3,6) (12,15) (11,15) (7,0) (9,11) (1,7) (14,8) (5,14) (15,4) (12,7) (0,5) (12,0) (6,2) (13,6) (9,8) (12,6) (3,1)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->13->11->1->12->2->10->9->6->5->4->3->7->8->14->15."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph106",
        "image": "data/hamilton/graph_visual/hard/graph106.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 11, and the edges are: (3,7) (1,10) (0,10) (3,2) (2,5) (8,6) (5,6) (1,8) (5,1) (4,3) (11,4) (4,9) (1,6) (10,8) (2,9) (2,10) (6,4) (5,8) (7,4) (1,11) (7,8) (3,5) (11,0) (7,1) (0,8) (6,2) (10,4) (11,10) (2,11) (9,10) (3,9) (3,6) (0,5) (6,11) (3,11) (2,8) (7,10) (9,1) (5,10) (6,0) (4,1) (7,9) (2,0)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->10->1->8->6->5->2->3->7->9->4->11."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph103",
        "image": "data/hamilton/graph_visual/hard/graph103.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 11, and the edges are: (9,2) (4,9) (6,0) (7,2) (8,6) (6,3) (3,10) (0,5) (10,5) (1,6) (10,7) (0,3) (10,8) (6,5) (0,2) (7,0) (0,8) (4,7) (9,6) (5,7) (3,1) (11,8) (4,2) (7,6) (5,9) (4,0) (10,4) (6,10) (5,11) (4,6) (3,7) (0,11) (8,5) (11,9) (9,8) (7,8) (2,6) (10,9) (1,7) (1,4) (2,1) (11,7) (1,0) (11,10)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->6->8->10->3->1->7->2->4->9->5->11."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph316",
        "image": "data/hamilton/graph_visual/hard/graph316.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (7,14) (14,9) (1,8) (9,1) (7,0) (2,13) (9,8) (1,0) (5,9) (11,5) (7,9) (8,7) (1,5) (6,12) (3,14) (3,4) (4,2) (11,12) (1,7) (2,6) (12,13) (15,7) (3,12) (10,2) (4,13) (12,5) (3,2) (10,1) (3,9) (0,13) (14,4) (10,5) (14,0) (3,6) (11,10) (8,6) (0,5) (11,0)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->1->8->9->5->11->10->2->13->12->6->3->4->14->7->15."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph377",
        "image": "data/hamilton/graph_visual/hard/graph377.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 16, and the edges are: (12,14) (6,3) (10,13) (10,4) (0,7) (6,10) (12,11) (14,10) (12,4) (0,6) (12,5) (3,13) (3,8) (13,5) (0,3) (5,10) (8,0) (9,10) (11,4) (5,2) (7,9) (0,4) (15,14) (1,13) (8,4) (14,9) (11,10) (6,14) (4,2) (15,6) (14,3) (14,13) (13,15) (6,8) (9,13) (8,5) (6,7) (9,3) (15,10) (2,15) (0,13) (7,15) (10,8) (1,14) (4,13) (0,14) (12,15) (16,8) (2,10) (9,5) (4,16) (8,12) (14,4) (15,11) (11,1) (12,0) (16,15) (4,3) (15,8) (1,9) (13,2)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->7->9->10->13->3->6->14->12->4->2->5->8->16->15->11->1."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph342",
        "image": "data/hamilton/graph_visual/hard/graph342.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (5,2) (12,4) (4,0) (9,8) (14,13) (13,9) (15,14) (5,7) (10,2) (3,4) (5,15) (7,2) (6,14) (1,4) (8,15) (3,12) (13,7) (0,7) (11,12) (5,13) (0,5) (5,14) (9,4) (7,9) (11,14) (2,11) (6,7) (15,11) (15,0) (2,15) (3,5) (13,0) (13,4) (4,14) (7,1) (6,8) (11,9) (10,13) (1,12) (9,6) (8,14) (10,0) (11,5) (8,10) (9,15) (14,3) (1,6) (2,1) (12,14) (10,11) (3,8) (1,3) (9,12) (4,15) (4,8) (14,0) (6,5) (6,12) (5,10) (7,14) (9,14) (8,0) (10,1) (4,11) (12,15) (2,3) (11,6)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->4->12->3->5->2->10->13->14->15->8->9->7->1->6->11."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph201",
        "image": "data/hamilton/graph_visual/hard/graph201.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 13, and the edges are: (1,12) (4,2) (13,11) (6,4) (3,2) (3,4) (1,3) (7,0) (7,4) (10,0) (1,10) (0,3) (5,6) (6,11) (6,9) (13,2) (3,12) (13,9) (7,5) (10,13) (1,9) (10,6) (2,7) (8,6) (4,9) (7,3) (8,11) (9,5) (12,2) (8,13) (7,6)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->7->4->2->3->12->1->10->13->11->8->6->5->9."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph566",
        "image": "data/hamilton/graph_visual/hard/graph566.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 19, and the edges are: (4,7) (13,2) (16,9) (11,6) (10,1) (15,18) (19,14) (4,18) (5,15) (0,18) (13,12) (6,16) (19,0) (3,12) (7,19) (0,15) (6,14) (5,17) (12,7) (0,17) (18,7) (3,17) (1,5) (1,16) (3,15) (14,9) (17,10) (8,17) (14,11) (5,14) (10,5) (0,2) (12,14) (0,6) (4,15) (18,10) (4,0) (17,15) (4,6) (18,12) (0,7) (8,14) (17,7) (7,11) (16,4) (8,15) (8,10) (13,3) (13,14) (18,1) (19,6) (3,6) (13,0) (3,0) (16,12) (8,2) (0,10) (13,1) (11,2) (3,4) (12,15) (4,14) (3,7) (13,7) (12,5) (2,17) (12,10) (3,14) (18,19) (14,15) (12,19) (10,2) (10,15) (1,7) (16,14) (12,2) (10,3) (3,8) (15,9) (7,15) (13,11)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->18->15->5->17->3->12->13->2->8->10->1->16->9->14->19->7->4->6->11."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph449",
        "image": "data/hamilton/graph_visual/hard/graph449.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 17, and the edges are: (15,16) (14,12) (6,16) (3,11) (5,6) (3,8) (13,6) (2,1) (0,5) (10,16) (2,13) (8,13) (3,12) (12,8) (8,4) (11,13) (13,15) (16,1) (14,10) (13,0) (1,5) (11,0) (1,0) (14,9) (15,0) (3,0) (0,2) (3,1) (7,6) (11,16) (3,15) (8,17) (13,3) (5,9) (11,10) (5,12) (15,5) (17,13) (12,2) (17,7) (4,13) (7,9) (8,2) (10,12) (1,4) (7,16) (11,9) (8,1) (11,8) (0,10) (15,10) (4,12) (3,10) (7,11)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->5->6->16->15->13->2->1->3->11->10->14->12->4->8->17->7->9."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph372",
        "image": "data/hamilton/graph_visual/hard/graph372.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 16, and the edges are: (2,5) (15,8) (2,8) (4,10) (14,13) (6,11) (5,10) (13,16) (7,1) (11,5) (2,13) (15,5) (13,8) (3,9) (11,1) (3,10) (11,2) (0,12) (7,16) (13,11) (12,3) (0,9) (3,16) (8,16) (9,2) (10,7) (1,13) (9,12) (0,10) (5,7) (16,11) (12,1) (12,7) (9,15) (14,10) (1,3) (14,15) (16,4) (1,14) (6,0) (7,2) (12,10) (14,7) (6,5) (13,6) (15,12) (15,0) (0,4) (8,0) (5,4) (0,7) (11,12) (11,14) (1,4) (14,3) (3,2) (14,0) (16,14) (11,0) (3,5) (12,13)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->12->3->9->2->5->10->4->16->13->8->15->14->7->1->11->6."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph133",
        "image": "data/hamilton/graph_visual/hard/graph133.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 12, and the edges are: (11,8) (9,6) (3,11) (11,4) (4,5) (4,6) (9,2) (12,11) (7,1) (5,1) (4,0) (10,2) (6,3) (12,5) (0,2) (2,7) (9,0) (8,0) (3,4) (2,4) (7,6) (5,3) (2,3) (1,0) (7,12)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->8->11->12->7->1->5->4->3->6->9->2->10."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph521",
        "image": "data/hamilton/graph_visual/hard/graph521.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (7,1) (6,1) (0,11) (6,12) (13,1) (5,8) (7,13) (12,5) (8,12) (0,3) (12,14) (4,5) (5,15) (13,10) (3,11) (9,18) (8,18) (7,9) (4,18) (2,9) (16,6) (6,0) (8,7) (10,18) (16,12) (16,5) (17,18) (1,9) (8,9) (17,10) (11,1) (9,3) (6,18) (12,9) (6,13) (14,0) (12,3) (0,12) (0,8) (17,12) (11,17) (5,10) (1,2) (5,14) (15,0) (4,17) (8,6) (15,9) (5,13) (14,15) (6,2) (3,14) (3,7) (18,15) (3,5) (5,11) (7,17) (11,16) (16,14) (7,11) (2,7) (11,13) (7,15) (5,0) (14,10) (16,4) (11,9) (10,2) (13,16) (6,14) (0,17) (15,10) (17,5) (10,0) (14,4) (1,0) (17,3) (11,18) (5,6) (18,0) (7,6) (12,4) (10,1) (7,5) (14,13) (6,4) (16,15) (18,16) (14,9) (4,0) (18,13) (13,12) (16,9) (10,7) (10,11) (11,15) (1,3) (12,11) (8,2) (3,15) (2,18) (15,6)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->11->3->9->18->8->5->12->6->1->7->13->16->15->14->4->17->10->2."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph562",
        "image": "data/hamilton/graph_visual/hard/graph562.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 19, and the edges are: (17,5) (11,16) (1,14) (8,19) (0,17) (16,18) (9,2) (13,10) (7,16) (14,19) (14,17) (9,6) (8,14) (14,9) (6,19) (4,11) (17,13) (4,8) (10,5) (13,7) (17,12) (3,1) (2,3) (11,1) (14,18) (15,10) (14,16) (9,18) (2,15) (18,19) (5,3) (1,5) (15,1) (5,18) (11,10) (1,18) (6,1) (5,2) (14,12) (15,8) (1,17) (12,1) (4,16) (16,17) (8,16) (9,19) (10,8) (1,2) (19,5) (5,16) (6,13) (14,10) (4,14) (11,14) (15,7) (16,0) (0,7) (1,10) (12,2) (13,16) (16,1) (7,12) (3,18) (11,3)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->17->5->10->13->7->16->11->4->8->19->14->12->1->15->2->3->18->9->6."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph39",
        "image": "data/hamilton/graph_visual/hard/graph39.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 10, and the edges are: (4,5) (0,3) (1,2) (5,10) (3,4) (2,6) (8,0) (1,3) (2,3) (3,5) (9,0) (4,0) (6,1) (7,0) (7,9) (8,6) (4,8) (0,2) (6,9) (9,8) (1,9) (10,7) (0,10) (9,3) (0,5) (7,5) (8,10) (2,9) (6,7) (2,7) (5,6) (4,7)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->4->5->10->7->9->8->6->2->1."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph248",
        "image": "data/hamilton/graph_visual/hard/graph248.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 14, and the edges are: (0,5) (10,5) (12,5) (14,4) (12,11) (4,10) (9,6) (2,3) (14,8) (2,11) (4,8) (2,7) (12,7) (6,13) (2,13) (6,4) (2,14) (10,1) (8,3) (0,13) (11,0) (13,10) (6,14) (8,5) (1,9) (9,7) (11,1) (3,14) (7,0) (14,5) (13,7) (0,3) (7,11) (5,4) (0,6) (4,9) (0,1) (8,0) (3,6) (11,5) (10,9) (13,11) (14,1) (12,2) (7,10) (7,3)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->5->10->4->14->8->3->2->11->12->7->13->6->9->1."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph274",
        "image": "data/hamilton/graph_visual/hard/graph274.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 14, and the edges are: (9,12) (2,8) (2,7) (3,5) (0,10) (7,14) (12,11) (13,12) (9,11) (12,14) (11,0) (1,10) (0,2) (4,13) (7,10) (6,9) (8,3) (12,0) (10,14) (1,4) (14,6) (3,10) (5,6) (5,4) (5,9) (3,14) (14,9) (11,6) (10,6) (7,1) (3,2) (6,4) (10,4) (2,1) (2,9) (13,14) (11,13) (13,0) (11,8) (14,11) (13,7) (6,13) (5,10) (0,5) (0,8) (0,9) (4,9) (3,11) (6,0) (4,2) (2,10) (8,13) (5,12) (12,8) (3,7) (9,7) (14,5) (4,12) (4,7) (3,9) (7,8) (3,6) (7,11) (9,1)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->10->1->4->13->12->9->11->6->14->7->2->8->3->5."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph506",
        "image": "data/hamilton/graph_visual/hard/graph506.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (0,3) (6,11) (9,5) (9,10) (12,17) (5,7) (17,14) (10,3) (14,5) (12,13) (14,11) (15,11) (9,3) (4,17) (2,9) (3,4) (4,13) (8,18) (10,12) (16,2) (15,9) (6,4) (12,18) (17,13) (9,7) (5,15) (13,1) (18,0) (5,12) (2,4) (18,2) (11,17) (1,16) (11,5) (14,10) (0,9) (15,8) (17,7) (11,8) (10,15) (6,1) (13,7) (14,9) (2,14) (15,1) (1,7) (0,14) (16,5) (1,18) (2,5) (15,12) (7,10) (8,12) (17,0) (6,10) (5,1) (11,10) (18,6) (9,6) (1,2) (15,4) (5,18) (0,12) (2,7) (4,1) (4,12) (8,16) (16,7) (6,0) (8,6)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->10->9->5->7->17->12->13->4->6->11->14->2->16->1->15->8->18."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph382",
        "image": "data/hamilton/graph_visual/hard/graph382.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 16, and the edges are: (16,7) (11,9) (12,8) (4,16) (0,16) (11,6) (11,4) (2,16) (0,15) (13,16) (14,13) (6,1) (6,12) (11,3) (15,5) (3,4) (4,12) (6,3) (2,15) (13,7) (3,12) (5,0) (9,14) (14,2) (10,14) (8,16) (3,5) (5,2) (8,4) (6,8) (3,14) (1,5) (14,11) (11,5) (1,14) (11,15) (2,12) (6,10) (4,1) (4,9) (16,14) (6,16) (11,10) (2,8) (14,8) (12,16) (3,1) (3,9)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->16->7->13->14->9->11->4->3->12->8->2->15->5->1->6->10."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph308",
        "image": "data/hamilton/graph_visual/hard/graph308.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (5,2) (6,5) (1,7) (1,5) (14,0) (13,11) (6,8) (6,13) (6,7) (11,10) (8,12) (10,0) (11,0) (8,14) (1,8) (4,15) (12,15) (7,10) (6,11) (4,13) (11,15) (10,12) (12,11) (4,14) (7,5) (14,10) (5,9) (8,13) (13,3) (13,15) (0,8) (5,3) (10,8) (0,15) (7,4) (5,12) (0,5) (3,11) (2,3) (9,8) (6,10) (0,4) (9,13) (4,9) (14,13) (10,5) (6,9) (1,13) (15,6)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->14->8->6->5->2->3->13->9->4->15->12->11->10->7->1."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph272",
        "image": "data/hamilton/graph_visual/hard/graph272.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 14, and the edges are: (8,4) (4,14) (12,13) (2,5) (7,13) (6,5) (1,6) (2,7) (14,3) (5,13) (7,6) (4,6) (7,1) (9,6) (8,5) (2,4) (4,7) (10,1) (8,11) (3,11) (2,9) (14,2) (0,5) (0,11) (11,4) (6,8) (0,14) (3,8) (9,1) (11,7) (11,12) (5,14) (1,12) (10,11) (2,10) (5,4) (14,9) (1,2) (6,0) (12,7) (3,6) (13,1) (2,8) (2,0) (3,9) (13,3) (13,0) (2,12) (7,9) (7,14) (13,4) (12,5) (8,10) (3,4) (14,6) (0,10) (2,3) (11,13) (8,12) (3,1) (8,13) (10,6) (14,8) (11,14) (12,6) (9,8) (10,5) (9,4)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->5->2->7->13->12->11->8->4->14->3->6->9->1->10."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph147",
        "image": "data/hamilton/graph_visual/hard/graph147.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 12, and the edges are: (5,1) (0,9) (2,9) (9,4) (8,10) (3,10) (12,9) (3,0) (2,8) (4,7) (6,9) (4,5) (6,0) (4,0) (0,10) (9,5) (10,5) (5,6) (5,8) (6,8) (4,8) (7,9) (12,0) (11,12) (4,1) (12,5) (8,3) (5,0) (7,6) (10,11) (12,10) (12,8) (10,6) (8,7) (12,1) (7,5) (0,7) (11,0)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->9->2->8->5->6->7->4->1->12->11->10->3."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph557",
        "image": "data/hamilton/graph_visual/hard/graph557.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 19, and the edges are: (17,6) (3,14) (4,16) (18,1) (16,11) (0,8) (3,12) (1,15) (15,19) (3,17) (2,5) (11,17) (7,18) (13,3) (0,7) (7,5) (0,14) (6,8) (5,14) (5,8) (11,7) (3,11) (2,17) (1,12) (19,3) (12,11) (17,14) (2,10) (10,7) (15,0) (3,5) (15,8) (9,1) (6,19) (8,2) (6,3) (12,0) (4,19) (4,12) (18,13) (1,8) (16,17) (5,1) (4,1) (14,18) (6,2) (17,18) (7,14) (7,9) (18,5) (1,13) (3,15) (9,4) (11,14) (9,15) (2,9) (6,4) (10,0) (10,5) (12,17) (14,15) (0,1) (13,16) (5,9) (9,19) (6,15) (17,7) (19,14) (10,11) (18,3) (14,8)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->8->6->17->3->14->5->2->10->7->18->1->15->19->9->4->12->11->16->13."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph376",
        "image": "data/hamilton/graph_visual/hard/graph376.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 16, and the edges are: (2,12) (5,4) (3,5) (3,9) (0,10) (8,0) (4,8) (12,3) (1,5) (9,12) (16,0) (7,11) (5,10) (5,2) (9,5) (1,15) (6,5) (2,15) (8,2) (3,11) (13,12) (6,12) (12,11) (7,8) (8,3) (6,14) (9,15) (4,9) (2,16) (13,10) (0,3) (15,3) (11,10) (4,7) (9,6) (7,13) (16,8) (16,13) (3,2) (16,12) (7,6) (14,11) (4,3) (6,2) (9,7) (0,5) (5,8) (8,6) (3,13) (13,15) (1,11) (0,12) (1,7) (12,1) (16,4) (2,4) (11,15) (11,6) (11,16) (13,0) (11,13)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->10->5->4->8->2->12->3->9->15->1->11->14->6->7->13->16."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph291",
        "image": "data/hamilton/graph_visual/hard/graph291.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 14, and the edges are: (3,10) (7,6) (2,12) (11,8) (9,10) (11,4) (4,12) (1,7) (14,8) (7,12) (14,13) (12,1) (7,8) (12,9) (3,13) (5,10) (3,7) (5,9) (9,3) (4,13) (10,12) (2,13) (6,5) (8,12) (3,2) (14,10) (3,6) (14,12) (0,5) (2,6) (8,1) (3,12) (2,5) (11,14) (4,1) (9,4) (11,9) (0,2) (10,2) (10,1) (10,4) (11,2) (6,11) (2,1) (6,14) (5,8) (9,2) (12,0) (0,14) (6,10) (3,11) (13,7) (4,6) (9,0) (2,14) (9,13) (14,3) (7,4) (13,8) (1,13) (0,13) (7,0) (2,7)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->5->10->3->13->14->8->11->4->12->1->7->6->2->9."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph476",
        "image": "data/hamilton/graph_visual/hard/graph476.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 17, and the edges are: (9,1) (2,0) (12,17) (9,17) (0,1) (8,11) (11,9) (9,10) (10,17) (8,5) (10,0) (4,16) (3,10) (6,15) (5,1) (16,1) (8,13) (15,10) (4,1) (17,1) (16,7) (16,2) (10,14) (8,10) (1,11) (13,11) (12,14) (12,9) (14,4) (6,17) (11,17) (9,2) (3,0) (0,15) (2,10) (1,3) (14,17) (17,15) (16,5) (4,6) (8,9) (14,7) (16,8) (0,8) (1,8) (11,14) (3,11) (9,13) (7,2) (5,14) (7,1) (5,10) (3,13) (17,8) (5,9) (16,3) (4,8) (16,6) (10,12) (5,12) (6,1) (12,3) (13,1) (0,5) (17,2) (7,0) (7,8) (9,14) (7,12) (12,4) (7,15) (12,15) (14,0) (11,6) (4,17) (16,15) (5,3) (3,8) (8,6) (10,11) (13,2) (7,11) (17,3) (14,2) (12,1) (7,17) (10,6) (0,17) (14,13) (6,0) (10,4) (3,14)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->16->4->1->9->17->12->14->10->3->13->11->6->15->7->8->5."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph234",
        "image": "data/hamilton/graph_visual/hard/graph234.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 13, and the edges are: (6,3) (9,5) (0,1) (10,12) (7,12) (3,11) (10,13) (6,1) (1,4) (10,8) (1,9) (3,13) (7,13) (6,11) (10,3) (4,11) (11,1) (6,9) (1,3) (3,5) (6,12) (7,9) (5,7) (11,10) (10,5) (4,7) (0,4) (6,0) (7,6) (2,6) (11,9) (7,2) (8,4) (8,0) (1,7) (3,0) (6,8) (8,7) (4,6) (13,2) (10,4) (1,10) (5,13) (5,6) (9,13) (7,0) (11,8) (12,13) (0,5) (2,10) (0,13) (10,6) (9,12) (12,4) (8,5) (9,2) (8,13) (5,11) (2,0) (12,5)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->1->6->3->11->4->7->12->10->13->2->9->5->8."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph499",
        "image": "data/hamilton/graph_visual/hard/graph499.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (7,8) (8,11) (2,13) (6,9) (8,12) (14,0) (11,12) (8,10) (5,12) (7,5) (18,12) (13,18) (9,1) (8,0) (4,9) (13,5) (0,1) (18,3) (11,15) (18,9) (16,17) (6,3) (18,14) (18,11) (10,17) (16,6) (4,6) (13,1) (14,1) (4,18) (10,13) (2,5) (15,9) (7,12) (14,2) (6,14) (1,17) (8,9) (9,7) (7,1) (13,6) (2,18) (12,3) (11,16) (17,0) (6,5) (7,14) (10,18) (3,13) (2,1) (17,13) (0,7) (16,12) (1,4) (4,3) (18,15) (6,1) (14,5) (16,9) (2,17) (11,13) (10,15) (9,14) (12,10) (0,16) (2,4) (14,13) (15,2) (17,12)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->14->18->12->8->7->5->13->2->1->9->4->3->6->16->17->10->15->11."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph397",
        "image": "data/hamilton/graph_visual/hard/graph397.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 16, and the edges are: (4,15) (3,12) (1,13) (16,8) (7,15) (6,10) (7,5) (5,13) (12,13) (0,13) (14,16) (14,0) (4,7) (0,1) (0,2) (1,2) (13,16) (14,13) (2,8) (5,1) (12,10) (5,12) (1,3) (11,5) (8,7) (14,1) (4,10) (8,11) (8,3) (9,6) (1,4) (11,3) (13,3) (6,14) (9,13) (12,0) (9,7) (13,7) (12,16) (16,0) (12,9) (3,15) (0,4) (8,6) (6,13) (1,15) (8,4) (3,10) (10,7) (16,1) (8,15) (0,3) (2,4) (16,5) (8,12) (14,3) (5,10) (14,2) (2,10) (2,5) (11,1) (11,9) (8,10) (2,12) (1,12) (8,1) (3,4) (9,2) (6,7) (0,9) (5,6) (15,9) (2,15) (14,10) (1,7) (0,6) (5,8) (8,9) (12,4) (11,14) (6,4) (11,6)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->13->1->2->8->16->14->6->10->12->3->11->5->7->4->15->9."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph69",
        "image": "data/hamilton/graph_visual/hard/graph69.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 11, and the edges are: (9,8) (11,7) (8,6) (3,7) (2,1) (4,5) (2,8) (11,5) (7,1) (3,2) (8,1) (9,10) (3,5) (0,8) (11,10) (5,0) (9,7) (9,4) (3,1) (2,10) (0,6) (11,1)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->5->4->9->10->11->7->3->2->1->8->6."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph496",
        "image": "data/hamilton/graph_visual/hard/graph496.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (12,0) (13,16) (16,2) (17,0) (9,15) (7,0) (17,1) (15,11) (7,12) (5,12) (9,1) (14,12) (11,8) (8,16) (4,11) (10,3) (18,2) (13,12) (7,6) (16,18) (18,8) (2,14) (2,13) (15,2) (2,7) (5,17) (9,6) (14,13) (12,2) (11,6) (9,0) (4,14) (1,6) (0,14) (6,5) (7,8) (3,5) (7,15) (8,10) (17,16) (9,16) (8,9) (0,10) (18,9) (15,1) (13,10) (4,16) (12,6) (6,17) (12,3) (3,15) (3,18) (13,9) (11,2) (1,13) (14,1) (10,12) (5,13) (18,6) (8,2) (3,4)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->12->7->6->9->15->11->8->16->2->18->3->10->13->5->17->1->14->4."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph14",
        "image": "data/hamilton/graph_visual/hard/graph14.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 10, and the edges are: (7,10) (1,8) (0,10) (4,2) (9,4) (1,4) (9,2) (7,1) (6,10) (7,3) (4,0) (9,1) (6,1) (2,7) (1,0) (10,2) (3,5) (4,7) (7,9) (9,0) (1,10) (10,8) (5,7) (0,3) (6,5) (10,5) (9,8)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->10->7->3->5->6->1->8->9->4->2."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph231",
        "image": "data/hamilton/graph_visual/hard/graph231.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 13, and the edges are: (1,8) (1,10) (13,5) (12,6) (8,7) (9,12) (4,2) (7,12) (7,13) (2,3) (2,0) (11,9) (4,8) (9,4) (5,8) (12,2) (0,3) (4,12) (9,13) (5,9) (12,10) (12,13) (2,1) (6,3) (3,12) (2,5) (6,11) (4,13) (6,13) (9,7) (5,4) (13,3) (7,4) (5,11) (4,11) (7,1) (7,0) (6,0) (3,10) (1,4) (1,6) (7,5) (12,0) (3,1) (2,13) (10,8) (13,0) (10,13) (3,9) (0,8) (5,6) (10,2) (0,9) (7,10) (1,13) (10,0) (11,13) (9,10) (4,0) (3,7) (10,6) (7,6) (9,1)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->4->8->1->10->12->6->3->13->5->11->9->7."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph493",
        "image": "data/hamilton/graph_visual/hard/graph493.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (16,8) (6,4) (16,7) (18,15) (12,8) (10,5) (3,10) (7,3) (10,15) (15,4) (17,4) (18,0) (11,2) (2,12) (8,7) (17,7) (8,6) (11,17) (3,1) (6,10) (18,5) (10,17) (3,17) (9,10) (12,6) (18,16) (12,10) (17,9) (16,12) (12,9) (12,17) (12,4) (4,16) (5,0) (3,5) (14,12) (2,15) (10,14) (15,17) (13,4) (4,8) (16,0) (18,10) (7,9) (18,1) (18,8) (13,6) (13,0) (3,16) (8,3) (15,11) (12,5) (4,1) (1,0) (2,18) (1,10) (13,10) (0,10) (13,3) (2,5) (18,6) (7,2) (8,9) (0,11) (7,13) (4,14) (9,1) (16,13) (0,6) (11,18)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->18->15->10->5->3->7->16->8->6->13->4->1->9->17->11->2->12->14."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph137",
        "image": "data/hamilton/graph_visual/hard/graph137.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 12, and the edges are: (6,3) (7,3) (4,0) (7,12) (2,6) (9,3) (10,11) (1,12) (0,6) (11,7) (10,6) (9,8) (11,1) (5,10) (0,9) (9,10) (4,2) (1,9) (11,4) (12,4) (12,10) (5,11) (0,12) (1,3) (8,0) (12,3) (7,1) (5,2) (3,4) (5,4) (2,9) (2,1)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->4->2->6->3->7->12->1->11->5->10->9->8."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph453",
        "image": "data/hamilton/graph_visual/hard/graph453.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 17, and the edges are: (17,9) (7,0) (10,13) (2,16) (15,8) (2,8) (12,3) (12,13) (14,16) (2,9) (8,5) (6,11) (0,17) (14,6) (16,0) (10,7) (1,0) (9,5) (7,14) (5,16) (16,12) (14,11) (13,16) (11,10) (16,4) (6,12) (6,8) (5,13) (1,10) (13,15) (2,12) (1,4) (8,13) (4,17) (1,17) (2,13) (9,0) (17,5) (6,13) (5,2) (8,9) (4,7) (0,2) (7,2) (13,7) (0,15) (0,4) (4,12) (10,15) (5,7) (12,11) (6,3) (6,17) (16,8) (15,1) (7,17) (15,7) (10,9) (9,15) (16,15) (6,16) (1,2) (14,13) (3,16) (11,13) (13,0) (10,14) (13,17) (13,4) (8,12) (1,14) (7,8) (17,8) (3,8) (9,1) (3,13) (6,9) (1,6) (1,12) (0,8) (7,3) (1,16) (0,11) (5,15) (13,9) (16,11) (0,14) (10,0) (2,6) (11,9) (7,9)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->7->10->13->12->3->6->11->14->16->2->8->15->1->4->17->9->5."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph187",
        "image": "data/hamilton/graph_visual/hard/graph187.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 13, and the edges are: (11,3) (4,1) (11,9) (7,1) (7,2) (4,0) (3,4) (5,12) (1,13) (11,2) (13,6) (2,8) (11,4) (7,9) (8,4) (4,2) (10,5) (7,11) (10,13) (5,9) (11,1) (0,11) (10,0) (1,10) (8,5) (13,0) (11,6) (10,8) (3,7) (13,9) (8,7) (7,5)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->4->1->10->8->2->7->3->11->6->13->9->5->12."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph369",
        "image": "data/hamilton/graph_visual/hard/graph369.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 16, and the edges are: (0,2) (9,0) (15,2) (12,14) (8,12) (3,11) (0,12) (2,8) (13,1) (8,14) (14,13) (6,0) (10,14) (5,13) (15,11) (9,5) (2,3) (11,6) (12,2) (8,9) (10,15) (9,16) (8,15) (9,3) (10,6) (12,7) (10,11) (7,9) (3,8) (7,3) (2,16) (3,6) (0,5) (0,10) (8,6) (2,4) (13,16) (6,13) (14,6) (13,9) (9,15) (13,10) (8,10) (13,4) (3,1) (7,2) (13,7) (7,15) (8,5) (14,11) (12,9) (1,2) (4,9) (14,9) (6,7) (1,16)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->15->11->3->7->12->14->10->6->8->5->13->1->16->9->4."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph487",
        "image": "data/hamilton/graph_visual/hard/graph487.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (0,5) (2,12) (9,18) (9,5) (14,6) (7,18) (5,4) (5,16) (10,13) (18,4) (0,6) (4,15) (7,0) (10,6) (12,15) (2,10) (10,7) (9,7) (16,8) (4,7) (1,0) (14,4) (11,16) (6,9) (3,10) (17,12) (17,8) (16,18) (5,18) (8,6) (16,7) (2,1) (18,11) (11,14) (4,16) (11,2) (8,10) (12,10) (6,11) (14,12) (0,11) (15,13) (9,0) (12,5) (12,8) (6,16) (15,9) (11,3) (16,10) (3,16) (13,14) (13,11) (4,2) (13,0) (8,1) (0,14) (16,14) (4,17) (9,4) (4,0)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->5->9->18->7->10->13->15->4->14->6->11->3->16->8->17->12->2->1."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph219",
        "image": "data/hamilton/graph_visual/hard/graph219.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 13, and the edges are: (10,7) (4,3) (0,6) (4,13) (7,13) (2,5) (6,10) (10,5) (3,0) (13,8) (1,7) (0,9) (11,12) (13,5) (8,0) (11,9) (8,9) (8,1) (12,2) (11,13) (5,6) (3,7) (6,8) (3,6) (11,2) (9,10) (13,12) (7,11) (5,11) (11,3) (4,10) (4,11) (5,4) (11,8) (10,2) (0,11) (9,4) (10,12) (8,12) (9,2) (13,0) (7,6) (0,1) (5,1) (3,2) (1,6) (7,12) (8,3) (3,12) (1,4) (8,4) (9,1) (9,6)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->6->10->7->13->4->3->11->12->2->5->1->8->9."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph456",
        "image": "data/hamilton/graph_visual/hard/graph456.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 17, and the edges are: (5,16) (7,15) (11,8) (13,14) (6,11) (6,3) (8,2) (1,13) (3,10) (16,12) (17,12) (13,11) (12,13) (12,14) (13,4) (10,5) (4,0) (10,7) (6,7) (7,0) (13,3) (13,10) (11,16) (15,5) (8,1) (13,7) (1,7) (0,15) (0,9) (5,17) (0,3) (12,0) (7,16) (8,0) (12,1) (4,1) (3,17) (9,5) (9,2) (9,8) (10,1) (1,9) (1,16) (3,2) (8,6) (17,8) (0,1) (8,3) (6,2) (6,13) (1,5) (4,2) (15,3) (6,14) (14,8) (14,10) (2,17) (16,2) (4,14) (5,12) (10,12) (4,12) (6,12) (3,1) (16,0) (9,14) (16,15) (16,14) (7,2) (5,13) (3,11) (16,6) (15,11) (17,15) (4,10) (10,9) (3,7) (9,17) (16,8) (10,0) (10,15) (9,3) (3,4) (6,15) (13,8) (13,15) (4,6) (15,9) (0,2) (12,7)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->4->13->14->12->16->5->10->3->6->11->8->2->9->1->7->15->17."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph373",
        "image": "data/hamilton/graph_visual/hard/graph373.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 16, and the edges are: (16,11) (4,0) (15,13) (9,6) (1,11) (3,12) (16,4) (7,11) (5,1) (5,2) (16,3) (14,11) (14,12) (10,12) (8,10) (10,6) (4,15) (7,12) (6,3) (12,4) (4,2) (11,13) (16,7) (14,15) (12,16) (1,10) (5,7) (9,10) (11,12) (13,5) (15,12) (4,8) (6,11) (14,8) (15,11) (5,8) (8,3) (0,13) (7,4) (8,1) (6,1) (2,15) (13,9) (8,2) (14,6) (16,8) (9,0) (2,3) (6,0) (12,8) (4,11)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->4->16->11->1->5->2->15->13->9->10->8->14->6->3->12->7."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph24",
        "image": "data/hamilton/graph_visual/hard/graph24.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 10, and the edges are: (3,9) (9,10) (2,0) (6,2) (5,10) (0,6) (4,9) (1,8) (9,0) (10,4) (1,0) (4,5) (7,4) (8,6) (10,7) (4,8) (2,10) (9,6) (4,6) (2,8) (3,7) (4,1) (6,1)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->6->8->1->4->9->3->7->10->5."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph386",
        "image": "data/hamilton/graph_visual/hard/graph386.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 16, and the edges are: (11,6) (11,14) (5,0) (10,2) (3,2) (3,8) (16,6) (13,0) (2,6) (0,3) (5,1) (11,9) (9,1) (1,10) (16,3) (2,8) (2,5) (15,14) (13,4) (13,11) (0,14) (15,4) (13,7) (0,8) (5,13) (13,12) (4,11) (1,2) (15,7) (3,5) (12,5) (9,12) (11,3) (7,9) (7,11) (8,15) (3,9) (14,6) (14,16) (5,6) (6,12) (16,7) (15,2) (12,11) (6,13) (11,1) (16,11) (8,6) (12,3) (16,4) (10,3) (7,3) (8,5) (5,4) (15,1) (13,9) (10,12) (7,0) (9,4) (11,8) (10,7) (6,9) (10,15) (14,5)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->5->1->9->11->6->16->14->15->4->13->7->3->8->2->10->12."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph237",
        "image": "data/hamilton/graph_visual/hard/graph237.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 13, and the edges are: (6,11) (12,7) (7,10) (4,1) (1,5) (0,6) (7,6) (3,10) (11,5) (7,4) (4,3) (0,9) (11,12) (11,2) (13,4) (10,8) (11,7) (5,2) (6,5) (8,4) (4,0) (11,0) (3,9) (4,12) (8,12) (3,0) (8,11) (1,11) (6,8) (4,5) (12,9) (6,1) (13,1) (0,13) (0,2) (9,8) (4,10) (1,0) (3,5) (9,6) (8,3) (9,4) (2,4) (11,10) (3,1) (7,2) (1,9) (1,2) (3,11) (3,13) (8,5) (2,9) (1,12) (11,9) (10,2)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->6->11->5->1->4->7->12->8->10->2->9->3->13."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph228",
        "image": "data/hamilton/graph_visual/hard/graph228.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 13, and the edges are: (12,9) (6,11) (4,2) (5,13) (9,5) (10,11) (12,10) (2,0) (1,6) (10,7) (8,6) (6,9) (1,12) (12,11) (2,12) (8,7) (10,5) (8,5) (8,2) (2,1) (11,4) (3,8) (6,4) (2,3) (1,8) (0,9) (12,5) (3,12) (7,4) (0,1) (10,8) (5,2) (3,10) (10,0) (8,11) (6,2) (13,7) (10,13) (0,5) (9,4) (0,7) (13,11) (4,5) (13,1) (3,6) (3,13) (3,7) (2,13) (8,4) (7,5) (13,12) (11,1) (0,11) (12,0) (3,9) (10,9) (4,12) (4,3) (3,5) (0,13) (4,13) (9,8) (9,11)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->4->11->6->1->12->9->5->13->7->10->8->3."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph101",
        "image": "data/hamilton/graph_visual/hard/graph101.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 11, and the edges are: (2,4) (2,11) (10,4) (6,3) (5,7) (6,1) (9,11) (6,7) (8,1) (2,1) (6,10) (0,2) (3,4) (0,8) (1,10) (2,6) (1,0) (3,2) (0,5) (9,5) (11,7) (0,6) (6,9) (9,7) (10,11) (11,6) (4,7) (0,4) (8,3) (8,9) (3,11)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->4->10->6->3->11->7->5->9->8->1."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph585",
        "image": "data/hamilton/graph_visual/hard/graph585.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 19, and the edges are: (1,8) (2,14) (18,12) (13,2) (10,0) (6,9) (6,11) (11,8) (5,0) (13,10) (11,0) (13,4) (8,9) (0,6) (11,4) (9,19) (5,13) (16,19) (16,5) (5,17) (1,3) (19,1) (2,10) (12,9) (4,7) (5,4) (8,0) (9,5) (14,4) (15,19) (13,15) (10,11) (7,18) (3,12) (16,10) (17,4) (6,10) (11,9) (8,17) (12,5) (12,11) (16,13) (17,14) (2,7) (13,11) (5,14) (17,16) (6,8) (9,1) (19,11) (0,2) (17,1) (0,1) (9,10) (2,18) (18,10) (7,8) (19,17) (13,18) (1,6) (18,9) (8,2) (12,0) (6,2) (6,16) (3,2) (14,0) (19,4) (13,8) (13,0) (18,17) (5,2) (16,0) (15,6) (18,19) (13,14) (2,4) (1,16) (9,7) (3,17) (1,15) (4,12) (4,6) (16,9) (11,1) (10,14) (13,7) (7,17) (12,16) (2,19) (10,5) (9,14) (8,12) (17,11) (7,1) (5,7) (10,15) (14,15) (13,19) (7,3) (13,12) (16,15) (6,7) (4,15) (11,2) (3,15) (18,11) (4,18) (7,0) (5,3) (10,8) (3,11) (19,0) (16,7)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->10->13->2->14->4->11->6->9->8->1->3->12->18->7->17->5->16->19->15."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph383",
        "image": "data/hamilton/graph_visual/hard/graph383.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 16, and the edges are: (6,0) (5,7) (10,6) (2,10) (16,7) (4,6) (9,2) (10,5) (13,7) (7,9) (4,7) (0,8) (10,1) (14,13) (5,4) (12,14) (5,13) (15,9) (12,3) (6,16) (2,13) (2,16) (15,16) (3,7) (15,2) (5,14) (5,8) (16,9) (16,10) (1,7) (0,13) (15,12) (9,4) (15,6) (7,6) (7,10) (0,12) (5,6) (3,14) (8,11) (13,4) (11,15) (0,1) (16,14) (13,15) (9,13) (5,1) (9,12) (2,0) (11,7)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->6->10->2->9->16->7->4->13->14->3->12->15->11->8->5->1."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph432",
        "image": "data/hamilton/graph_visual/hard/graph432.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 17, and the edges are: (17,3) (14,5) (3,1) (6,8) (10,5) (13,8) (2,10) (3,12) (11,4) (14,1) (17,16) (9,15) (7,13) (15,12) (8,10) (1,7) (12,11) (10,3) (7,15) (0,16) (13,12) (1,17) (13,17) (12,2) (4,1) (7,6) (9,3) (10,4) (13,3) (13,14) (12,8) (9,13) (8,5) (4,17) (4,12) (17,12) (0,11) (14,9) (13,16) (9,10) (7,11) (3,5) (6,15) (11,9) (9,1) (8,16) (8,2) (10,7) (3,0) (0,13) (0,1) (16,15) (3,2) (9,17) (15,0)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->16->17->3->1->14->5->10->2->12->15->9->13->8->6->7->11->4."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph301",
        "image": "data/hamilton/graph_visual/hard/graph301.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (7,12) (1,13) (6,10) (12,9) (7,2) (11,12) (8,10) (6,11) (9,10) (3,11) (0,4) (0,6) (9,1) (12,4) (7,5) (11,1) (1,7) (5,6) (5,1) (15,13) (10,14) (7,11) (8,14) (15,7) (5,8) (13,0) (1,4) (10,7) (14,13) (11,8) (1,14) (3,7) (6,3) (9,14) (2,6) (9,6) (14,5) (0,2) (0,5) (5,13) (5,12) (9,3) (13,6) (8,15) (8,7) (0,10) (10,5) (13,10) (4,8) (5,3)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->4->12->7->2->6->10->8->14->9->1->11->3->5->13->15."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph491",
        "image": "data/hamilton/graph_visual/hard/graph491.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (12,0) (8,5) (10,0) (2,9) (18,2) (18,10) (4,3) (0,6) (6,2) (9,8) (7,14) (16,14) (17,2) (2,13) (2,0) (14,6) (8,4) (17,1) (16,11) (10,7) (7,6) (15,8) (13,4) (1,0) (2,15) (9,5) (9,11) (10,3) (11,0) (16,4) (10,11) (17,18) (5,4) (12,9) (18,15) (18,1) (13,17) (13,7) (18,3) (13,9) (0,5) (1,13) (14,10) (16,6) (3,16) (14,5) (14,0) (5,1) (8,0) (16,1) (8,7) (5,18) (18,11) (4,17) (14,17) (2,14) (10,4) (2,10) (13,15) (14,15) (2,11) (17,16) (5,11) (2,7) (16,8) (3,8) (9,18) (4,1) (1,12) (9,3) (12,15) (12,18)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->12->9->2->18->10->7->14->6->16->11->5->8->15->13->17->1->4->3."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph513",
        "image": "data/hamilton/graph_visual/hard/graph513.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (2,1) (8,10) (12,3) (10,15) (2,4) (13,7) (16,12) (5,18) (6,13) (5,8) (14,9) (18,3) (10,16) (15,7) (16,1) (8,14) (2,16) (2,0) (4,18) (9,18) (6,7) (6,1) (7,8) (15,16) (17,12) (17,6) (15,3) (14,3) (11,5) (10,18) (12,1) (14,13) (2,14) (6,12) (13,16) (6,0) (11,7) (5,2) (0,15) (14,12) (5,9) (1,17) (11,10) (10,5) (10,2) (6,14) (11,6) (10,13) (11,9) (10,17) (15,14) (7,12) (18,16) (9,16) (5,1) (16,5) (11,15) (3,1) (10,9) (15,5) (1,0) (4,15) (5,6) (14,7) (4,6) (7,2) (2,12) (17,18) (13,9) (18,15) (15,12) (1,18) (13,2) (1,11) (14,18) (9,7) (17,9) (12,18) (9,15) (18,8) (2,9) (12,8) (0,4) (8,16) (2,17) (16,11) (14,17) (8,1) (10,0) (6,3) (3,5) (7,1) (12,5) (7,5) (6,9) (3,16)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->1->16->12->3->18->5->8->10->15->7->13->14->17->9->11->6->4."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph561",
        "image": "data/hamilton/graph_visual/hard/graph561.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 19, and the edges are: (4,3) (1,11) (2,6) (5,16) (8,17) (19,13) (1,2) (10,1) (17,7) (11,19) (9,5) (17,16) (2,9) (18,3) (0,2) (16,19) (17,11) (11,0) (19,2) (12,15) (5,4) (17,13) (1,19) (18,6) (7,19) (2,3) (8,19) (15,8) (7,16) (3,11) (6,4) (5,2) (12,2) (18,12) (19,18) (1,14) (10,15) (11,14) (1,8) (11,9) (12,10) (19,10) (13,15) (4,14) (18,11) (14,3) (5,13) (18,13) (18,9) (15,14) (1,15) (6,8) (7,12) (6,11) (8,11) (7,3) (8,10) (2,16) (6,5) (9,3) (4,15) (6,0) (11,5) (4,11) (3,0) (4,2) (3,8) (6,16) (12,1) (3,10) (9,12) (8,16) (5,14) (6,19) (5,19) (9,7)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->6->18->3->4->5->16->17->8->19->13->15->12->10->1->14->11->9->7."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph114",
        "image": "data/hamilton/graph_visual/hard/graph114.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 11, and the edges are: (6,9) (0,5) (4,1) (1,8) (2,4) (8,11) (11,1) (0,9) (11,2) (9,11) (7,9) (5,6) (11,3) (8,9) (3,4) (7,8) (4,9) (8,10) (6,8) (3,0) (3,7) (0,1) (5,3) (3,6) (10,1) (2,3) (5,11) (0,2) (9,10) (8,3) (3,10) (4,6) (11,4) (5,4) (7,0) (0,8) (7,10) (8,2) (2,5) (5,10) (5,1) (4,0) (10,6)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->5->6->9->11->8->1->4->2->3->7->10."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph510",
        "image": "data/hamilton/graph_visual/hard/graph510.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (8,11) (16,8) (6,4) (14,12) (18,12) (8,5) (4,13) (10,7) (10,8) (15,17) (12,7) (6,2) (0,1) (8,2) (15,0) (16,4) (8,4) (8,0) (16,2) (12,2) (3,7) (9,11) (12,8) (0,14) (11,17) (18,13) (2,14) (15,10) (1,17) (15,2) (10,5) (18,2) (12,6) (18,1) (5,6) (7,11) (4,3) (3,13) (16,18) (0,18) (18,3) (5,1) (1,16) (0,3) (14,16) (6,0) (9,2) (16,5) (10,1) (9,17) (16,7) (9,18) (11,12) (5,17) (18,17) (3,10) (6,16) (12,16) (13,2) (1,9) (14,6) (14,17) (5,4) (0,12) (15,1) (3,8) (16,0) (11,16) (14,1) (3,12) (14,4) (11,18) (5,18) (18,15) (6,9) (12,13) (13,1) (14,5) (3,15) (14,9) (13,17) (4,18) (5,15) (0,10) (9,8) (10,4) (8,1) (11,3) (7,0) (14,11) (5,2) (14,15) (10,18) (7,4) (18,8) (16,13) (17,0) (2,11) (17,4) (17,8) (16,9) (1,2) (9,15)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->1->17->15->10->7->12->14->2->6->4->13->18->16->5->8->3->11->9."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph356",
        "image": "data/hamilton/graph_visual/hard/graph356.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (14,8) (3,10) (4,13) (9,13) (0,5) (15,8) (14,3) (4,6) (2,14) (6,0) (14,6) (1,8) (12,1) (0,9) (14,13) (9,4) (14,4) (5,12) (6,5) (10,8) (9,15) (9,7) (10,2) (12,13) (2,15) (15,6) (9,14) (13,11) (11,14) (8,7) (12,7) (3,8) (14,5) (8,0) (0,7) (6,2) (11,10) (4,15) (13,5) (13,10) (14,10) (15,13) (3,9) (2,12) (0,11) (8,12) (1,7) (6,3) (3,1) (7,6) (10,1) (5,3) (10,15) (9,6) (15,3) (2,13) (12,11) (13,1) (5,4) (3,7) (0,1) (4,8) (5,15) (8,2) (12,14) (9,10) (3,13) (11,3) (10,0) (14,1) (12,4) (6,12) (3,0)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->5->12->1->8->14->3->10->2->15->9->7->6->4->13->11."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph64",
        "image": "data/hamilton/graph_visual/hard/graph64.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 11, and the edges are: (6,8) (5,1) (3,0) (3,1) (6,0) (3,10) (3,7) (9,2) (3,2) (1,6) (10,0) (7,11) (2,5) (8,10) (4,2) (1,2) (4,6) (1,9) (8,1) (2,11) (3,11) (1,4) (6,5) (7,0) (6,11) (7,2) (0,4)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->7->11->3->10->8->6->4->2->9->1->5."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph529",
        "image": "data/hamilton/graph_visual/hard/graph529.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (18,16) (12,9) (17,11) (11,5) (14,12) (2,10) (5,18) (13,11) (6,8) (1,0) (13,4) (18,14) (0,12) (5,8) (17,10) (16,2) (11,2) (15,2) (17,5) (10,13) (3,7) (15,10) (1,5) (17,13) (0,10) (6,5) (0,7) (2,6) (13,12) (8,0) (3,16) (2,3) (8,4) (4,5) (11,4) (17,16) (1,11) (3,9) (6,18) (3,11) (7,2) (7,18) (10,6) (18,13) (18,9) (1,6) (16,13) (4,14) (3,12) (14,11) (14,15) (13,14) (9,11) (1,4) (14,0) (0,6) (12,2) (15,13) (16,11) (3,8) (9,13) (12,10) (3,6) (15,0) (7,9) (15,7) (3,0) (1,7) (9,0) (1,3) (18,11) (18,1) (15,17) (1,15) (15,18) (2,17) (8,11) (12,1) (2,0) (4,2) (16,6) (13,3) (5,16) (1,8) (17,12) (2,5) (15,11) (4,7) (9,15) (8,13) (1,9) (7,8) (4,9) (0,18) (10,3) (11,12) (14,3) (14,8) (6,7) (6,9) (8,16) (15,6) (2,9) (11,10) (4,12) (0,5)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->1->5->11->17->10->2->16->18->14->12->9->3->7->15->13->4->8->6."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph199",
        "image": "data/hamilton/graph_visual/hard/graph199.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 13, and the edges are: (0,8) (4,12) (6,10) (10,4) (2,5) (2,11) (8,6) (2,0) (9,8) (9,10) (0,1) (12,10) (1,11) (3,8) (3,4) (11,3) (2,7) (7,1) (12,2) (2,9) (1,3) (9,4) (12,0) (6,4) (9,13) (3,7) (6,13) (2,8) (1,4) (5,0) (5,13) (3,12) (13,11) (10,3) (8,1) (6,0) (10,5) (11,4) (8,7) (13,10) (7,0) (4,0)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->8->6->10->4->12->3->11->1->7->2->5->13->9."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph398",
        "image": "data/hamilton/graph_visual/hard/graph398.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 16, and the edges are: (4,5) (13,3) (15,13) (10,13) (11,9) (7,0) (8,15) (1,10) (2,8) (4,15) (8,16) (1,8) (9,13) (8,3) (3,2) (14,5) (6,13) (13,12) (12,0) (16,6) (0,15) (0,5) (12,2) (13,7) (3,15) (12,9) (5,15) (0,9) (6,1) (9,2) (9,15) (7,5) (11,2) (0,14) (7,2) (4,8) (8,11) (16,1) (16,5) (6,10) (8,7) (2,15) (13,2) (10,8) (15,11) (3,7) (14,8) (1,7) (1,4) (0,2) (1,11) (13,11) (10,9) (0,3) (16,14) (15,1) (12,14) (8,9) (12,1) (5,1) (6,3) (5,13) (4,13) (14,9) (6,5) (0,13) (2,1) (6,2) (11,0) (16,0) (16,12) (12,15) (16,2) (7,15) (16,10) (5,11)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->7->13->3->8->15->4->5->14->16->6->1->10->9->11->2->12."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph117",
        "image": "data/hamilton/graph_visual/hard/graph117.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 11, and the edges are: (4,3) (7,10) (0,9) (5,2) (3,9) (11,5) (3,7) (2,7) (6,11) (6,0) (11,10) (2,3) (11,9) (5,4) (0,1) (0,10) (6,2) (4,8) (10,6) (9,8) (11,0) (0,7) (10,4) (9,1) (6,8) (8,10) (10,3) (0,2) (0,4) (8,2) (5,8) (6,9) (1,10) (1,5) (5,10) (4,11) (0,5) (4,9) (5,6)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->9->3->4->5->11->6->8->2->7->10->1."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph22",
        "image": "data/hamilton/graph_visual/hard/graph22.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 10, and the edges are: (5,4) (9,0) (10,9) (0,4) (7,9) (10,7) (8,7) (1,5) (2,10) (6,1) (4,7) (10,3) (0,5) (7,1) (7,5) (2,6) (9,8) (7,2) (1,2) (6,4) (9,5) (4,9) (8,3) (6,5) (6,3) (3,9) (0,6)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->9->10->7->8->3->6->2->1->5->4."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph527",
        "image": "data/hamilton/graph_visual/hard/graph527.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (1,5) (7,8) (8,13) (9,17) (0,7) (12,10) (2,15) (6,11) (7,1) (5,6) (12,13) (9,6) (3,16) (15,9) (9,5) (0,18) (4,15) (8,6) (2,12) (0,1) (16,10) (12,15) (2,11) (18,11) (17,7) (2,6) (13,6) (2,18) (3,12) (9,4) (12,16) (11,8) (13,14) (7,18) (7,3) (1,16) (7,11) (17,6) (7,12) (3,6) (18,6) (17,14) (8,9) (6,16) (8,3) (16,15) (18,4) (15,11) (9,12) (5,15) (1,9) (9,2) (10,5) (14,6) (13,10) (9,18) (16,14) (14,3) (2,13) (16,2) (15,1) (4,12) (4,16) (0,6) (15,3) (1,4) (17,15) (14,2) (4,3) (16,18) (15,8) (0,13) (0,15) (11,5) (14,18) (8,5) (2,5) (11,1) (7,16) (2,0) (4,5) (0,12) (7,14) (7,13) (4,8) (11,10) (17,18) (5,16) (2,10) (14,1) (11,16) (6,12) (3,13) (8,17) (13,11) (8,10) (12,17) (1,8) (11,9) (11,3) (16,13) (17,5) (6,1) (10,7)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->7->8->13->12->10->16->3->6->11->2->15->9->17->14->18->4->1->5."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph71",
        "image": "data/hamilton/graph_visual/hard/graph71.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 11, and the edges are: (5,3) (10,5) (9,1) (3,2) (11,4) (4,10) (2,11) (3,11) (3,8) (9,8) (9,3) (3,4) (1,7) (3,7) (8,2) (7,9) (4,0) (10,6) (2,1) (6,7) (0,11) (1,3) (6,4) (9,5) (1,6)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->4->11->2->3->5->10->6->7->1->9->8."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph160",
        "image": "data/hamilton/graph_visual/hard/graph160.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 12, and the edges are: (9,1) (4,7) (4,5) (1,5) (6,4) (4,1) (2,4) (1,11) (4,9) (11,7) (1,7) (9,3) (2,9) (4,11) (1,3) (2,12) (5,3) (3,10) (9,5) (10,0) (5,7) (3,6) (1,12) (9,12) (2,6) (7,12) (11,10) (10,4) (8,1) (0,12) (10,2) (11,2) (7,8) (11,12) (8,0) (7,2) (11,3) (0,1) (6,0) (10,9) (3,0)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->10->3->9->1->5->4->6->2->12->11->7->8."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph98",
        "image": "data/hamilton/graph_visual/hard/graph98.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 11, and the edges are: (6,8) (2,3) (1,3) (5,6) (10,7) (0,7) (2,9) (8,5) (6,10) (3,6) (10,11) (9,10) (4,8) (11,5) (8,2) (8,11) (9,0) (10,3) (2,11) (1,8) (4,2) (9,6) (8,9) (9,7) (1,4) (10,2) (2,6) (5,0) (11,9) (10,0) (7,3) (5,2) (7,11) (11,3) (6,0) (9,5) (2,1) (1,0) (10,5) (6,1) (3,9) (7,4) (11,0) (7,6) (1,5)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->7->10->6->8->5->11->2->9->3->1->4."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph430",
        "image": "data/hamilton/graph_visual/hard/graph430.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 17, and the edges are: (0,13) (10,2) (3,1) (3,14) (12,17) (3,8) (7,13) (1,9) (11,3) (1,14) (1,16) (6,4) (8,12) (10,7) (11,10) (11,9) (5,6) (0,5) (13,1) (11,13) (17,9) (0,8) (6,13) (17,5) (5,9) (2,1) (3,5) (1,10) (16,2) (4,5) (14,0) (6,12) (12,3) (6,7) (1,4) (16,0) (17,3) (0,2) (14,15) (5,15) (14,7) (15,17) (0,4) (8,4) (1,17) (13,9) (16,7) (4,12) (10,0) (17,0) (5,12) (7,3) (15,7) (11,17) (3,9) (10,14) (16,5) (8,9) (8,15) (15,1) (8,7)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->13->7->10->2->1->3->14->15->17->11->9->8->12->6->4->5->16."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph532",
        "image": "data/hamilton/graph_visual/hard/graph532.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (17,6) (10,12) (9,4) (14,17) (5,3) (4,18) (17,5) (1,4) (17,10) (18,10) (1,11) (9,13) (9,5) (16,6) (16,17) (1,8) (14,5) (7,4) (4,11) (9,16) (12,4) (12,7) (3,8) (13,7) (14,2) (17,7) (2,5) (3,0) (1,9) (3,4) (10,1) (0,1) (16,18) (7,8) (0,14) (3,12) (2,7) (11,3) (7,15) (3,1) (0,11) (13,14) (18,17) (4,10) (16,13) (1,18) (17,2) (5,6) (11,7) (2,4) (12,15) (6,0) (5,4) (15,14) (6,8) (13,2) (6,14) (15,2) (0,8) (13,10) (9,0) (9,17) (10,11) (15,13) (17,12) (11,18) (12,16) (16,5) (2,11) (12,5) (17,8) (18,13) (10,14) (11,14) (1,6) (10,5) (2,0) (7,6) (0,7) (15,1) (11,5) (9,14) (16,7) (7,14) (11,15) (9,7) (15,10) (2,10) (16,11) (6,13) (3,9) (6,9) (1,5) (12,8) (12,6) (13,3) (18,7) (3,18) (2,12) (8,16)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->5->17->6->16->9->4->18->10->12->7->13->14->2->15->11->1->8."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph246",
        "image": "data/hamilton/graph_visual/hard/graph246.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 14, and the edges are: (12,10) (1,5) (2,4) (8,1) (7,11) (3,4) (1,10) (0,13) (9,6) (14,5) (14,0) (4,13) (7,6) (0,5) (0,9) (6,3) (0,1) (9,1) (8,12) (2,0) (10,13) (7,12) (1,11) (7,9) (9,2) (2,3) (9,5) (3,10) (7,3) (7,10) (8,7) (4,14) (14,12) (11,13) (7,0) (3,8) (4,10) (5,8) (3,12) (5,13) (13,9) (2,7) (8,10)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->13->4->2->9->6->7->11->1->5->14->12->10->3->8."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph269",
        "image": "data/hamilton/graph_visual/hard/graph269.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 14, and the edges are: (11,4) (5,10) (11,0) (12,8) (3,8) (1,10) (1,6) (3,11) (5,12) (8,1) (12,14) (1,7) (8,2) (0,4) (13,12) (9,11) (11,8) (12,0) (0,10) (14,0) (12,1) (3,13) (13,4) (3,1) (5,6) (5,1) (12,7) (13,8) (0,6) (14,2) (14,5) (6,8) (1,13) (5,13) (12,3) (3,0)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->10->5->6->1->7->12->14->2->8->3->13->4->11->9."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph163",
        "image": "data/hamilton/graph_visual/hard/graph163.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 12, and the edges are: (11,4) (4,6) (11,10) (8,7) (12,2) (12,5) (10,6) (0,8) (10,1) (6,7) (8,5) (12,0) (12,6) (6,11) (6,2) (12,1) (5,11) (4,7) (6,8) (9,2) (5,6) (8,3) (2,11) (3,9) (0,1) (9,0) (5,4) (2,8) (1,2) (9,10) (7,0) (9,8) (11,3) (9,1)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->8->7->6->4->11->5->12->2->1->10->9->3."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph97",
        "image": "data/hamilton/graph_visual/hard/graph97.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 11, and the edges are: (11,8) (10,9) (3,1) (11,7) (6,5) (5,10) (2,3) (9,11) (8,2) (0,9) (8,6) (7,9) (0,2) (2,7) (7,10) (2,1) (0,1) (2,5) (10,0) (4,0) (6,0) (4,1) (2,10) (10,11) (11,0) (9,4) (3,4) (0,7) (1,11) (5,3)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->9->10->5->6->8->11->7->2->3->1->4."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph462",
        "image": "data/hamilton/graph_visual/hard/graph462.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 17, and the edges are: (16,14) (8,0) (14,11) (12,8) (0,17) (7,15) (12,13) (2,13) (5,8) (4,6) (15,17) (1,0) (16,6) (5,17) (6,11) (12,11) (17,11) (12,15) (2,15) (3,9) (16,17) (17,1) (16,4) (4,2) (9,8) (8,3) (2,14) (4,11) (13,5) (11,10) (7,12) (6,3) (11,9) (16,1) (12,2) (15,14) (8,16) (8,6) (5,11) (13,9) (0,3) (5,4) (17,3) (1,5) (17,2) (10,12) (9,0) (15,10) (4,17) (7,0) (7,6) (4,10) (15,16) (11,3) (12,4) (0,4) (0,12) (11,13) (16,3) (13,4) (9,10) (9,5) (17,7) (16,2) (13,3) (14,13) (14,10) (14,6) (16,7) (10,5) (10,6) (3,5) (12,16) (11,1) (2,6) (6,15) (8,4) (5,12) (1,15) (0,11) (17,13) (0,13) (11,15) (15,5) (1,8) (2,0) (0,6) (6,5)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->8->12->13->2->15->7->6->4->16->14->11->17->1->5->10->9->3."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph49",
        "image": "data/hamilton/graph_visual/hard/graph49.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 10, and the edges are: (3,2) (9,3) (3,4) (5,4) (0,5) (4,6) (5,6) (8,1) (9,2) (0,10) (7,2) (9,0) (0,2) (9,8) (3,10) (8,3) (6,9) (10,2) (2,8) (6,10) (6,3) (5,7) (6,0) (4,7) (0,8) (6,2) (1,6) (0,7) (5,8) (1,10) (10,9)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->5->4->3->9->8->1->6->10->2->7."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph182",
        "image": "data/hamilton/graph_visual/hard/graph182.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 13, and the edges are: (12,7) (13,7) (5,10) (13,3) (12,3) (0,12) (1,6) (1,0) (2,3) (11,12) (10,13) (11,8) (11,5) (10,7) (5,6) (5,4) (2,1) (2,6) (9,1) (11,1) (9,4) (4,11) (0,8) (4,1) (13,8) (13,9) (11,3) (0,11) (1,13) (12,2) (3,10) (4,12) (9,12) (7,11)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->12->7->13->3->10->5->6->2->1->9->4->11->8."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph479",
        "image": "data/hamilton/graph_visual/hard/graph479.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 17, and the edges are: (13,17) (10,2) (7,17) (1,0) (16,6) (6,8) (3,4) (6,2) (13,4) (6,13) (13,0) (16,11) (17,9) (8,11) (12,7) (11,0) (3,1) (17,16) (15,1) (10,4) (8,1) (1,17) (4,1) (6,17) (5,10) (11,6) (2,11) (5,1) (11,1) (15,5) (9,12) (3,15) (9,7) (16,10) (2,12) (4,8) (11,5) (8,13) (4,9) (3,10) (0,3) (0,5) (9,8) (10,11) (12,15) (11,13) (15,2) (15,8) (16,13) (0,2) (9,14) (8,7) (2,17) (12,13) (8,3) (10,8) (2,3) (1,10) (6,3) (10,0) (11,3) (6,14) (17,0) (7,15) (3,12) (17,12) (14,3) (7,6) (12,10) (11,15) (15,9) (13,7) (9,13) (16,4) (1,2) (13,10) (14,13) (16,7) (14,17) (16,8) (2,8) (9,6) (5,3) (2,16) (3,9) (7,0) (12,11) (15,0) (5,17) (1,13) (7,3) (2,13) (0,14) (5,4) (14,4) (5,16) (16,1) (7,2) (17,3) (13,5)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->1->3->4->13->17->7->12->9->8->11->16->10->5->15->2->6->14."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph336",
        "image": "data/hamilton/graph_visual/hard/graph336.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (7,0) (10,9) (2,4) (14,6) (5,15) (1,0) (11,2) (13,5) (2,12) (15,10) (14,10) (8,13) (12,1) (8,7) (5,3) (6,11) (0,15) (11,0) (13,7) (10,3) (3,6) (15,3) (2,10) (15,13) (14,1) (0,5) (6,2) (15,4) (12,15) (7,12) (3,2) (7,9) (6,4) (10,4) (9,2) (12,13) (15,7) (11,8) (13,6) (1,11) (15,9) (14,8) (5,9) (12,4) (14,11) (13,2) (10,0) (5,1) (5,11) (9,11) (13,10) (14,5) (2,8) (8,12) (12,11) (9,14) (5,4) (9,8) (13,4) (1,10) (5,6) (15,8) (6,9) (5,2) (15,1) (15,11) (14,15) (8,0) (14,4)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->7->8->13->5->15->10->9->2->4->12->1->14->11->6->3."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph418",
        "image": "data/hamilton/graph_visual/hard/graph418.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 16, and the edges are: (5,3) (4,2) (5,12) (16,5) (15,7) (4,5) (2,5) (9,15) (2,7) (11,7) (11,6) (3,8) (11,12) (7,5) (16,11) (14,0) (16,12) (1,5) (9,2) (3,1) (8,9) (16,1) (13,1) (13,16) (10,11) (15,1) (1,2) (0,1) (14,3) (2,11) (4,9) (7,13) (9,0) (13,5) (2,15) (2,14) (7,3) (14,13) (12,3) (3,4) (8,11) (0,11) (8,2) (2,3) (5,6) (0,15) (10,1) (13,9) (3,9) (13,2) (4,8) (2,0) (6,3) (8,14) (5,10) (8,1) (9,10) (13,12) (8,13) (6,16) (14,5) (4,10) (4,0) (14,15) (13,4) (2,6) (8,0) (9,11) (15,5) (9,14) (12,10) (0,12) (16,4) (5,0) (13,11) (14,12) (14,7) (14,16) (4,1) (7,1) (12,4) (2,16) (4,14) (15,12)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->14->3->5->12->11->7->15->9->2->4->10->1->8->13->16->6."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph172",
        "image": "data/hamilton/graph_visual/hard/graph172.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 12, and the edges are: (2,4) (5,12) (3,5) (6,8) (11,9) (9,3) (0,3) (4,5) (1,6) (8,9) (9,12) (1,12) (8,4) (7,8) (8,0) (1,10) (8,11) (7,6) (0,5) (9,6) (10,0) (5,2) (4,1) (4,3) (11,2) (3,6) (12,8) (11,3) (5,7) (5,9) (2,1) (2,12) (0,11) (1,9) (8,3) (9,7) (7,10) (12,11) (0,1) (12,0) (1,11) (5,10)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->5->12->9->11->8->6->7->10->1->4->2."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph115",
        "image": "data/hamilton/graph_visual/hard/graph115.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 11, and the edges are: (5,7) (8,10) (11,4) (6,8) (9,11) (6,3) (0,10) (4,1) (11,3) (9,5) (5,0) (8,3) (5,3) (6,2) (3,2) (9,3) (3,0) (4,7) (11,1) (7,6) (2,4) (5,10) (2,7) (1,8) (4,0) (10,2) (1,0) (1,7) (6,11) (5,2) (2,9) (8,2) (7,3) (10,1) (1,3) (7,11) (0,9) (6,5)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->10->8->6->3->11->4->1->7->5->9->2."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph70",
        "image": "data/hamilton/graph_visual/hard/graph70.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 11, and the edges are: (2,11) (4,1) (4,5) (7,1) (3,7) (3,2) (4,9) (7,8) (7,4) (4,6) (6,11) (11,1) (1,10) (7,9) (11,10) (7,6) (4,8) (2,9) (11,3) (8,2) (0,5) (5,8) (5,11)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->5->8->7->1->10->11->6->4->9->2->3."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph19",
        "image": "data/hamilton/graph_visual/hard/graph19.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 10, and the edges are: (4,9) (1,10) (8,2) (9,6) (2,9) (3,8) (0,8) (10,9) (7,6) (4,1) (8,10) (4,10) (6,5) (10,5) (8,7) (0,6) (8,5) (9,0) (8,1) (3,1) (7,2)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->8->2->7->6->5->10->9->4->1->3."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph206",
        "image": "data/hamilton/graph_visual/hard/graph206.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 13, and the edges are: (5,7) (12,11) (7,11) (3,10) (8,7) (2,13) (9,4) (0,4) (1,11) (4,7) (4,10) (9,1) (12,0) (13,7) (13,0) (12,3) (1,13) (3,0) (8,0) (10,11) (5,8) (5,11) (9,3) (7,2) (4,5) (1,10) (9,11) (1,3) (1,8) (1,6) (0,9) (9,10)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->4->9->10->3->12->11->5->8->7->2->13->1->6."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph403",
        "image": "data/hamilton/graph_visual/hard/graph403.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 16, and the edges are: (16,12) (15,13) (9,11) (15,9) (5,14) (12,9) (3,7) (6,1) (0,5) (6,3) (12,10) (2,16) (6,7) (13,11) (1,0) (13,0) (13,14) (0,9) (3,4) (15,10) (14,1) (14,11) (6,11) (7,8) (2,9) (12,0) (16,5) (4,13) (6,13) (12,6) (8,0) (8,14) (6,2) (1,7) (7,5) (0,14) (2,13) (9,13) (9,5) (14,3) (2,4) (14,12) (14,7) (15,6) (16,1) (3,9) (11,12) (4,16) (15,5) (6,10) (7,15) (10,11) (13,12) (8,5) (14,2) (8,15) (16,13) (2,10) (2,1) (13,7) (5,2) (4,5) (12,1) (1,3) (13,5) (3,11) (1,10) (10,7) (4,14) (10,3) (15,4) (0,3) (10,5) (8,9) (15,14) (4,9) (11,1) (10,16) (5,1) (14,9) (0,16) (13,1) (16,9) (11,0) (0,4) (14,6)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->5->14->13->15->9->11->6->1->16->12->10->2->4->3->7->8."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph400",
        "image": "data/hamilton/graph_visual/hard/graph400.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 16, and the edges are: (10,2) (5,6) (9,1) (14,8) (4,3) (11,13) (15,2) (8,3) (12,13) (4,15) (13,14) (15,16) (3,6) (11,6) (15,13) (7,2) (14,16) (13,6) (15,1) (8,0) (6,12) (3,16) (4,14) (2,1) (7,0) (16,1) (7,10) (14,5) (2,11) (8,16) (11,14) (13,0) (1,6) (4,13) (1,5) (7,3) (1,3) (12,5) (10,16) (13,8) (7,8) (12,0) (16,9) (15,9) (4,8) (13,1) (0,10) (12,11) (12,1) (0,16) (10,6) (9,8) (9,13) (9,7) (5,11) (15,7) (11,3) (14,7) (10,14) (1,11) (11,16) (8,15) (14,1) (9,0) (6,14) (6,7) (10,8) (12,10) (1,10) (3,15) (10,11) (14,12) (4,10) (4,5) (16,13) (13,5) (6,8) (3,10)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->8->14->13->11->6->5->1->9->16->15->2->7->3->4->10->12."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph597",
        "image": "data/hamilton/graph_visual/hard/graph597.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 19, and the edges are: (19,16) (0,3) (7,16) (18,6) (8,5) (1,15) (5,17) (7,17) (18,15) (14,16) (1,8) (10,8) (6,10) (3,7) (10,15) (18,12) (12,3) (12,4) (15,11) (7,6) (2,5) (16,9) (17,0) (17,3) (10,2) (1,3) (6,11) (4,16) (6,16) (18,10) (11,1) (4,15) (17,8) (11,4) (15,8) (7,1) (16,17) (19,9) (14,1) (2,16) (10,1) (11,7) (9,4) (7,10) (5,4) (11,5) (11,16) (5,9) (18,0) (3,9) (11,2) (2,18) (7,15) (9,6) (9,12) (5,15) (5,16) (8,9) (13,19) (6,4) (5,6) (7,4) (11,9) (17,15) (13,7) (9,2) (12,1) (10,12) (16,0) (15,14) (5,1) (15,12) (0,5) (16,1) (15,19) (19,5) (12,8) (2,0) (16,18) (6,15) (1,18) (13,10) (14,19) (1,0) (14,17) (7,5) (6,3) (9,10) (0,12) (1,13) (10,17) (0,10) (14,5) (0,11) (4,8) (5,18) (0,19) (9,14) (14,18) (17,1) (10,19) (16,3) (3,19) (8,7) (3,5) (6,17) (14,12) (19,1) (13,11) (2,15) (2,3) (3,10) (8,19) (17,2)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->7->16->19->9->4->12->18->6->10->8->5->17->15->14->1->13->11->2."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph422",
        "image": "data/hamilton/graph_visual/hard/graph422.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 17, and the edges are: (15,3) (12,7) (9,10) (13,15) (3,16) (12,4) (16,17) (9,6) (11,10) (0,12) (5,15) (6,15) (15,12) (8,0) (4,5) (4,17) (15,1) (5,8) (5,0) (2,1) (11,1) (14,0) (2,0) (13,10) (11,12) (17,7) (10,16) (1,3) (16,11) (10,4) (10,3) (7,10) (3,7) (14,4) (3,13) (0,9) (10,8) (17,12) (0,10) (15,0) (16,7) (14,6) (1,10) (17,9) (6,5) (16,9) (11,2) (11,4) (13,8) (17,3) (17,8) (17,10) (14,16) (7,0) (0,16) (13,17) (8,9)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->12->7->17->16->3->15->13->10->9->8->5->6->14->4->11->1->2."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph2",
        "image": "data/hamilton/graph_visual/hard/graph2.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 10, and the edges are: (10,2) (1,7) (6,8) (2,9) (8,0) (6,5) (1,4) (1,6) (6,0) (9,8) (9,6) (7,6) (2,6) (8,1) (0,4) (2,8) (9,10) (3,6) (5,8) (0,10) (3,1) (1,9) (7,4) (8,10) (6,4) (10,1) (5,4)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->8->9->2->10->1->7->4->5->6->3."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph390",
        "image": "data/hamilton/graph_visual/hard/graph390.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 16, and the edges are: (13,11) (1,8) (8,14) (1,5) (11,15) (15,7) (10,16) (5,0) (7,8) (7,13) (1,6) (0,2) (15,4) (7,6) (1,2) (12,9) (12,14) (11,2) (2,6) (7,11) (6,3) (1,3) (4,14) (15,16) (12,11) (4,11) (12,2) (3,4) (16,2) (7,5) (16,13) (4,8) (6,14) (7,12) (13,10) (14,2) (12,13) (5,6) (3,7) (14,1) (3,10) (15,8) (2,7) (12,6) (16,6) (6,10) (13,8) (1,10) (0,16) (13,9) (9,8) (15,3) (15,5) (13,2) (1,9) (4,5) (16,1) (0,11) (10,12) (12,3) (13,6) (5,13) (9,0) (11,6) (2,5) (9,11) (5,12) (0,12) (4,1) (11,1) (6,0) (13,1) (7,1) (16,12) (16,8) (9,14) (5,3) (4,0) (2,8) (8,11) (4,6) (5,11) (7,4) (7,10) (5,10) (15,13)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->5->1->8->14->12->9->13->11->15->7->6->2->16->10->3->4."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph148",
        "image": "data/hamilton/graph_visual/hard/graph148.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 12, and the edges are: (2,0) (7,3) (9,11) (2,4) (4,1) (9,12) (0,5) (9,6) (8,0) (8,10) (12,7) (1,3) (1,11) (11,5) (6,7) (1,5) (5,7) (1,0) (3,10) (3,12) (10,11) (1,2) (9,0) (7,4) (0,7) (8,12) (7,11) (6,4) (5,8) (6,8) (0,12) (4,5)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->4->1->3->7->12->9->6->8->10->11->5."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph388",
        "image": "data/hamilton/graph_visual/hard/graph388.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 16, and the edges are: (5,12) (1,5) (10,0) (13,14) (0,8) (4,16) (1,10) (12,16) (1,4) (11,10) (3,9) (10,5) (13,11) (15,0) (2,9) (8,10) (8,5) (6,3) (10,13) (9,11) (7,11) (11,14) (11,2) (2,8) (13,5) (3,8) (1,14) (6,12) (4,15) (13,7) (2,13) (16,5) (10,7) (0,2) (11,16) (6,2) (4,12) (1,8) (13,1) (1,2) (4,10) (6,0) (3,11) (15,5) (7,12) (12,15) (15,16) (9,4) (14,2) (12,0) (13,15) (6,7) (1,7) (4,5) (16,8) (5,3)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->10->1->5->12->16->4->15->13->14->11->9->3->8->2->6->7."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph583",
        "image": "data/hamilton/graph_visual/hard/graph583.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 19, and the edges are: (18,5) (10,14) (14,1) (3,6) (15,13) (5,7) (13,10) (9,15) (15,17) (13,18) (18,6) (5,15) (14,19) (11,15) (9,16) (17,16) (12,8) (10,3) (6,5) (10,12) (8,14) (15,14) (12,19) (4,2) (7,11) (17,4) (1,6) (19,6) (6,7) (13,2) (19,18) (19,9) (6,4) (12,11) (4,15) (0,8) (6,8) (1,17) (15,12) (8,3) (7,12) (6,14) (18,3) (0,15) (7,10) (5,14) (16,2) (16,6) (12,13) (5,0) (3,15) (4,13) (5,1) (12,14) (6,17) (13,9) (4,7) (17,12) (12,4) (6,15) (0,1) (10,5) (0,14) (7,1) (18,1) (11,17) (2,10) (16,14) (14,4) (14,7) (14,11) (13,3) (2,9) (0,2) (9,5) (18,2) (3,4) (18,9) (4,18) (16,7) (16,12) (8,2) (1,9) (10,18) (9,11) (2,7) (13,8) (17,9) (9,12) (1,15) (19,2) (1,11) (15,2) (1,10) (18,0) (14,9) (16,1) (6,2) (1,8) (0,11) (11,5) (5,19) (13,0) (16,4) (10,4) (14,3) (1,2) (5,8) (10,16) (16,0)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->8->12->10->14->1->6->3->18->5->7->11->15->13->2->4->17->16->9->19."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph343",
        "image": "data/hamilton/graph_visual/hard/graph343.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (3,11) (1,10) (15,0) (8,0) (13,10) (8,2) (9,15) (8,5) (9,0) (5,12) (2,15) (12,9) (10,15) (2,1) (3,7) (14,3) (6,1) (2,11) (6,10) (5,1) (4,15) (11,4) (14,8) (10,2) (8,6) (10,12) (4,12) (14,9) (1,14) (9,13) (6,3) (7,8) (4,2) (8,12) (1,4) (1,11) (2,3) (12,13) (7,11) (0,6) (10,3) (13,6) (0,2) (0,5) (1,15) (8,10) (8,9) (14,13) (4,3) (14,2) (5,15) (6,11) (7,15) (9,2) (9,10) (0,14) (6,12) (6,4) (7,0) (14,12) (6,2) (13,5) (9,11) (14,10) (3,15) (13,0) (11,5) (3,0) (11,15) (7,9) (14,5) (9,3) (8,1) (8,3) (9,4) (3,5)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->15->9->12->5->8->2->1->10->13->6->4->11->7->3->14."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph83",
        "image": "data/hamilton/graph_visual/hard/graph83.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 11, and the edges are: (5,4) (2,1) (4,11) (5,1) (8,11) (2,7) (0,4) (3,9) (2,4) (4,1) (5,8) (7,1) (6,2) (2,8) (11,0) (6,1) (3,4) (8,10) (11,3) (1,8) (7,5) (8,0) (11,2) (10,7) (10,11) (10,4) (7,6) (8,9) (10,9) (5,11)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->4->5->1->2->6->7->10->8->11->3->9."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph519",
        "image": "data/hamilton/graph_visual/hard/graph519.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (1,11) (5,10) (8,17) (16,9) (4,3) (14,1) (15,4) (6,14) (8,1) (1,4) (6,17) (11,8) (0,9) (13,6) (16,18) (12,15) (1,5) (13,12) (15,13) (10,7) (7,3) (9,13) (9,2) (16,10) (18,3) (9,12) (13,4) (6,0) (14,8) (0,8) (2,1) (16,12) (13,14) (6,8) (16,13) (17,16) (4,0) (16,14) (16,7) (18,2) (7,13) (9,11) (6,4) (11,2) (13,11) (2,13) (15,8) (9,4) (12,2) (5,17) (11,6) (4,16) (17,14) (10,6) (0,14) (11,12) (10,1) (17,9) (15,7) (5,18) (1,0) (9,8) (14,12) (10,8) (9,10) (14,15) (10,14) (3,15) (10,13) (15,16) (8,13) (12,5) (6,18) (17,15) (2,8) (18,11) (7,9) (11,5) (12,10) (7,17) (14,5) (15,2) (17,3) (16,5) (11,14) (15,10) (11,0) (15,1) (0,3) (5,9) (17,10) (14,2) (9,1) (5,3) (3,16) (13,0) (10,0)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->9->16->18->3->4->15->12->13->6->14->1->11->2->8->17->5->10->7."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph250",
        "image": "data/hamilton/graph_visual/hard/graph250.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 14, and the edges are: (8,9) (5,7) (2,3) (9,14) (7,13) (4,14) (8,11) (6,0) (6,1) (1,3) (12,1) (9,1) (8,0) (10,7) (9,0) (7,14) (10,8) (6,7) (13,8) (4,5) (0,1) (9,4) (2,6) (3,9) (14,10) (5,3) (3,12) (0,3) (5,10) (2,1) (10,0) (7,3) (5,9) (6,12) (11,1) (4,13) (12,14) (10,3) (6,3) (13,12)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->6->1->2->3->9->14->12->13->4->5->7->10->8->11."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph304",
        "image": "data/hamilton/graph_visual/hard/graph304.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (14,15) (2,8) (9,3) (4,9) (11,4) (1,10) (5,15) (3,10) (2,12) (8,4) (1,5) (2,5) (15,11) (7,9) (9,12) (0,2) (8,5) (4,13) (7,0) (7,3) (10,14) (0,1) (1,8) (12,14) (11,14) (15,4) (4,1) (8,10) (9,1) (14,9) (2,3) (13,1) (7,11) (3,5) (13,12) (13,0) (11,5) (7,2) (11,10) (1,12) (1,7) (9,0) (1,3) (6,0) (2,11) (15,13) (7,10) (6,13) (8,9) (12,11) (6,5) (3,11) (8,6) (7,13) (10,4)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->2->8->4->9->3->10->1->5->15->14->12->11->7->13->6."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph440",
        "image": "data/hamilton/graph_visual/hard/graph440.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 17, and the edges are: (1,4) (7,15) (2,5) (12,10) (12,4) (7,10) (17,6) (14,0) (14,6) (10,5) (17,13) (17,8) (4,5) (3,0) (2,6) (6,8) (16,17) (7,6) (13,14) (10,15) (4,9) (0,1) (11,10) (14,8) (0,11) (11,13) (14,16) (15,6) (8,0) (4,6) (7,16) (11,14) (4,8) (10,17) (3,12) (17,4) (16,2) (5,1) (4,3) (15,12) (14,5) (8,10) (9,13) (3,15) (16,0) (4,2) (9,3) (13,10) (0,17) (3,5) (3,8) (14,9) (0,15) (7,3) (6,5) (7,9) (13,7) (2,11) (10,0) (1,7) (1,11) (8,11) (14,12) (16,8) (16,1) (17,2) (13,4) (1,13)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->14->6->17->13->11->10->12->4->1->5->2->16->8->3->15->7->9."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph74",
        "image": "data/hamilton/graph_visual/hard/graph74.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 11, and the edges are: (5,8) (0,3) (10,11) (9,6) (1,4) (3,1) (5,2) (11,5) (8,9) (5,1) (0,10) (9,11) (8,10) (7,9) (6,2) (3,5) (0,2) (7,1) (3,6) (9,3) (4,7) (10,2) (0,7) (7,3) (8,4) (4,6) (9,4)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->3->1->4->7->9->6->2->5->8->10->11."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph595",
        "image": "data/hamilton/graph_visual/hard/graph595.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 19, and the edges are: (18,19) (9,3) (2,5) (16,5) (13,7) (12,14) (13,16) (15,9) (12,1) (3,18) (17,0) (12,15) (15,0) (11,2) (17,18) (19,1) (18,15) (16,9) (8,19) (9,19) (15,4) (19,12) (0,6) (13,6) (5,0) (12,16) (19,3) (9,5) (14,11) (11,1) (5,1) (15,13) (14,6) (13,18) (6,2) (7,4) (1,8) (4,13) (12,11) (18,1) (11,19) (10,17) (13,1) (0,4) (1,16) (16,17) (8,5) (17,15) (11,3) (5,4) (1,2) (1,6) (17,13) (1,17) (12,7) (4,6) (3,13) (2,19) (4,16) (9,18) (4,8) (12,0) (18,7) (15,7) (11,0) (7,8) (15,5) (5,13) (10,13) (7,2) (8,15) (3,17) (5,10) (6,11) (3,12) (9,17) (18,0) (6,12) (18,4) (16,6) (5,17) (18,2) (9,11) (13,11) (4,14) (11,4) (5,7) (5,19) (8,11) (15,16) (17,7) (1,10) (16,2) (12,8) (8,13) (5,11) (16,3) (8,17) (10,14) (6,9) (10,19) (7,16) (3,10) (2,4) (9,14) (15,1) (3,5) (15,11) (6,10) (2,14)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->17->18->19->1->12->14->11->2->5->16->13->7->4->6->10->3->9->15->8."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph28",
        "image": "data/hamilton/graph_visual/hard/graph28.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 10, and the edges are: (0,6) (4,10) (4,1) (0,10) (8,4) (8,10) (0,9) (1,7) (3,0) (1,9) (5,3) (5,10) (4,0) (1,0) (4,2) (6,9) (7,10) (8,6) (2,6) (6,4) (4,3) (7,3) (7,8) (4,9) (3,6)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->6->9->1->7->3->5->10->8->4->2."
            }
        ]
    },
    {
        "id": "hamilton-hard-graph452",
        "image": "data/hamilton/graph_visual/hard/graph452.png",
        "conversations": [
            {
                "from": "human",
                "value": "<image>\nIn this undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 17, and the edges are: (7,13) (3,6) (13,17) (17,14) (13,0) (10,13) (0,4) (7,16) (11,1) (16,3) (13,6) (17,6) (0,1) (13,5) (13,8) (15,2) (9,3) (3,0) (16,4) (15,5) (2,0) (8,7) (1,2) (7,10) (15,12) (12,1) (9,16) (3,1) (2,7) (11,2) (12,17) (7,17) (15,4) (6,2) (1,15) (11,14) (7,14) (5,10) (12,2) (16,0) (4,12) (6,14) (3,17) (17,1) (6,1) (17,8) (6,12) (0,15) (3,4) (9,5) (1,8) (4,2) (2,13) (16,14) (11,9) (14,2) (0,11) (8,4) (1,7) (7,12) (5,3) (16,6) (15,7) (7,6) (11,5) (10,15) (8,6) (1,16) (5,12) (4,6) (3,12) (1,5) (16,8) (8,12) (12,13) (17,9) (14,10) (0,8) (16,17) (8,3) (16,15) (2,16) (7,11) (15,13) (9,13) (16,5) (10,9)\nQ: Begin with node 0,  Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges."
            },
            {
                "from": "gpt",
                "value": "0->13->7->16->3->6->17->14->11->1->2->15->4->8->12->5->10->9."
            }
        ]
    }
]