{"question": "In an 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: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an 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: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an 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: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an 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: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an 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: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an 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) (0,2) (0,6) (0,7) (0,1) (1,5) (2,3) (2,6) (2,5) (3,4) (3,7) (4,7) (4,6) (5,6) (5,7) (6,8) (7,8)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,1,5,7,8,6,4,3,2", "difficulty": "easy", "doc_id": "0"}
{"question": "In an 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: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an 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: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an 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: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an 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: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an 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: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an 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: (0,3) (0,1) (1,4) (1,6) (2,4) (3,5) (3,6) (4,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,1,6,3,5,4,2", "difficulty": "easy", "doc_id": "1"}
{"question": "In an 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: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an 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: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an 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: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an 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: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an 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: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an 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) (0,2) (0,9) (0,6) (0,4) (0,8) (1,3) (1,4) (1,7) (1,6) (1,9) (1,5) (2,7) (2,6) (2,4) (2,3) (3,4) (3,8) (3,5) (3,7) (3,9) (4,8) (5,7) (5,8) (6,7) (8,9)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,8,9,3,7,6,2,4,1,5", "difficulty": "easy", "doc_id": "2"}
{"question": "In an 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: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an 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: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an 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: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an 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: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an 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: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an 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) (0,3) (0,2) (0,13) (0,8) (0,11) (0,5) (0,10) (0,6) (1,5) (1,4) (1,12) (1,11) (1,10) (2,7) (2,5) (2,12) (2,8) (3,11) (3,9) (3,10) (3,13) (4,7) (4,5) (5,12) (5,6) (5,10) (5,8) (6,7) (6,11) (7,9) (7,13) (8,11) (9,11) (9,13) (9,10) (11,13)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,6,11,13,9,7,4,5,8,2,12,1,10,3", "difficulty": "hard", "doc_id": "3"}
{"question": "In an 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: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an 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: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an 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: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an 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: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an 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: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an 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) (0,2) (0,1) (0,8) (0,6) (0,7) (0,4) (1,8) (1,2) (1,4) (2,4) (2,5) (2,6) (2,8) (3,6) (3,5) (3,4) (4,6) (5,8) (6,8) (7,8)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,4,6,3,5,2,1,8,7", "difficulty": "easy", "doc_id": "4"}
{"question": "In an 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: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an 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: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an 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: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an 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: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an 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: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an 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: (0,4) (0,1) (0,5) (0,2) (0,6) (0,3) (1,4) (1,3) (2,3) (2,6) (3,4) (3,6) (4,6) (4,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,6,2,3,1,4,5", "difficulty": "easy", "doc_id": "5"}
{"question": "In an 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: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an 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: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an 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: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an 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: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an 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: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an 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) (0,3) (0,2) (1,7) (1,8) (1,5) (2,5) (2,3) (2,4) (3,8) (3,4) (3,5) (4,6) (4,7) (4,5) (5,7) (5,6) (5,8) (6,7)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,2,4,5,6,7,1,8,3", "difficulty": "easy", "doc_id": "6"}
{"question": "In an 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: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an 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: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an 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: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an 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: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an 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: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an 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: (0,2) (0,8) (0,3) (0,6) (0,4) (1,4) (1,8) (2,5) (2,3) (2,4) (3,6) (3,7) (3,4) (3,5) (3,8) (4,6) (4,5) (4,8) (5,7)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,6,4,2,5,7,3,8,1", "difficulty": "easy", "doc_id": "7"}
{"question": "In an 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: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an 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: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an 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: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an 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: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an 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: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an 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) (0,9) (1,5) (1,6) (1,8) (1,3) (1,4) (1,7) (2,9) (2,6) (2,8) (3,5) (3,8) (3,4) (3,6) (4,6) (7,8)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,9,2,8,7,1,4,6,3,5", "difficulty": "easy", "doc_id": "8"}
{"question": "In an 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: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an 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: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an 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: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an 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: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an 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: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an 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: (0,2) (0,1) (0,7) (1,2) (1,6) (1,4) (1,7) (2,4) (2,8) (2,5) (2,3) (2,7) (3,5) (3,6) (3,4) (4,8) (4,7) (5,8) (5,7) (6,8) (7,8)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,7,8,6,3,4,1,2,5", "difficulty": "easy", "doc_id": "9"}
{"question": "In an 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: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an 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: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an 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: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an 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: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an 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: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an 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) (0,8) (0,6) (0,5) (0,9) (0,7) (0,3) (1,9) (1,5) (1,4) (1,6) (1,7) (2,3) (2,6) (2,9) (2,7) (2,8) (3,5) (3,6) (3,8) (3,9) (3,7) (4,9) (4,7) (5,9) (5,6) (6,9) (7,8) (7,9) (8,9)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,3,7,9,8,2,6,5,1,4", "difficulty": "easy", "doc_id": "10"}
{"question": "In an 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: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an 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: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an 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: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an 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: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an 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: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an 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) (0,6) (0,12) (0,11) (0,16) (0,9) (0,1) (0,8) (0,10) (1,5) (1,10) (1,14) (1,9) (1,15) (1,16) (1,11) (1,3) (2,16) (2,14) (2,13) (2,8) (2,4) (2,9) (2,6) (2,10) (3,16) (3,12) (3,6) (3,5) (3,11) (3,7) (3,8) (4,7) (4,9) (4,10) (4,5) (4,8) (4,13) (4,11) (5,7) (5,6) (5,12) (5,8) (5,15) (6,14) (6,12) (6,7) (6,15) (6,10) (6,13) (6,16) (7,16) (7,12) (7,10) (7,13) (7,14) (7,8) (8,10) (8,11) (8,9) (8,14) (8,15) (8,12) (9,12) (9,16) (9,13) (9,15) (9,14) (10,13) (10,16) (10,11) (10,15) (10,14) (11,13) (11,12) (12,14) (12,16) (12,15) (13,15) (13,16)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,10,14,12,15,13,16,9,8,11,4,5,1,3,7,6,2", "difficulty": "hard", "doc_id": "11"}
{"question": "In an 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: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an 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: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an 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: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an 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: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an 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: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an 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: (0,11) (0,9) (0,4) (0,7) (0,6) (0,13) (0,5) (0,1) (0,18) (0,10) (0,3) (0,8) (1,14) (1,6) (1,12) (1,15) (1,10) (1,8) (1,13) (1,16) (1,18) (1,11) (1,19) (1,17) (1,9) (1,5) (2,13) (2,11) (2,15) (2,5) (2,8) (2,18) (2,19) (2,9) (2,4) (2,3) (3,19) (3,18) (3,10) (3,5) (3,4) (3,9) (3,17) (3,16) (4,16) (4,9) (4,11) (4,5) (4,12) (4,7) (4,19) (4,14) (4,18) (4,17) (5,18) (5,16) (5,15) (5,12) (5,6) (5,9) (5,11) (6,7) (6,18) (6,12) (6,10) (6,16) (6,8) (6,9) (6,19) (6,11) (6,15) (7,8) (7,11) (7,14) (7,16) (7,18) (7,10) (7,12) (8,14) (8,12) (8,15) (8,17) (8,11) (8,13) (8,10) (9,19) (9,14) (9,10) (9,12) (9,16) (9,13) (10,18) (10,16) (10,15) (10,14) (10,13) (10,12) (11,13) (12,14) (12,16) (12,18) (13,15) (13,19) (13,14) (13,17) (13,18) (14,15) (14,16) (14,19) (14,17) (15,19) (15,16) (16,18) (16,17) (17,18) (17,19)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,8,10,12,18,17,19,15,16,14,13,11,7,6,9,4,3,2,5,1", "difficulty": "hard", "doc_id": "12"}
{"question": "In an 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: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an 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: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an 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: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an 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: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an 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: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an 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: (0,5) (0,13) (0,10) (0,2) (1,13) (1,3) (1,11) (1,5) (1,2) (1,7) (1,8) (2,11) (2,10) (2,5) (2,9) (2,6) (3,5) (3,13) (3,12) (3,6) (4,9) (4,5) (5,8) (5,6) (6,11) (6,13) (6,7) (6,8) (6,9) (7,12) (7,13) (8,13) (8,10) (9,12) (9,11) (9,10) (10,13) (10,12) (11,12) (12,13)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,2,6,9,10,8,13,7,12,11,1,3,5,4", "difficulty": "hard", "doc_id": "13"}
{"question": "In an 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: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an 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: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an 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: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an 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: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an 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: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an 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) (0,6) (0,3) (1,8) (1,7) (2,3) (2,7) (2,5) (2,6) (2,4) (3,4) (3,6) (3,8) (4,8) (4,7) (7,8)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,6,3,8,1,7,4,2,5", "difficulty": "easy", "doc_id": "14"}
{"question": "In an 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: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an 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: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an 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: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an 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: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an 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: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an 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) (0,6) (0,2) (1,5) (1,2) (2,4) (3,7) (4,6) (5,6) (5,7) (6,7)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,6,4,2,1,5,7,3", "difficulty": "easy", "doc_id": "15"}
{"question": "In an 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: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an 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: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an 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: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an 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: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an 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: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an 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) (0,6) (0,5) (0,8) (1,5) (1,8) (2,3) (2,8) (2,7) (2,5) (3,8) (3,5) (4,7) (4,6) (4,5) (5,7) (5,6) (6,8) (7,8)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,8,6,4,7,2,3,5,1", "difficulty": "easy", "doc_id": "16"}
{"question": "In an 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: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an 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: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an 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: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an 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: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an 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: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an 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: (0,12) (0,3) (0,9) (0,13) (0,6) (0,5) (0,8) (0,16) (1,12) (1,14) (1,16) (1,4) (1,6) (1,9) (1,10) (1,17) (2,14) (2,13) (2,10) (2,3) (2,8) (2,15) (2,11) (2,9) (3,16) (3,6) (4,13) (4,6) (4,15) (4,9) (4,17) (4,10) (4,11) (5,9) (5,13) (5,11) (5,8) (5,7) (5,6) (6,11) (6,13) (6,7) (6,8) (6,12) (6,9) (7,15) (7,16) (7,13) (7,8) (7,11) (8,11) (8,14) (8,15) (8,12) (9,14) (9,15) (9,11) (10,14) (10,16) (11,17) (11,16) (12,15) (12,17) (12,14) (13,14) (13,16) (13,15) (13,17) (15,16) (15,17)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,16,15,17,13,14,12,8,11,9,5,7,6,4,1,10,2,3", "difficulty": "hard", "doc_id": "17"}
{"question": "In an 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: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an 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: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an 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: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an 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: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an 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: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an 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: (0,12) (0,11) (0,1) (0,4) (0,14) (0,10) (0,3) (0,2) (0,13) (0,8) (1,13) (1,11) (1,8) (1,14) (1,10) (1,12) (1,5) (1,9) (2,6) (2,14) (2,5) (2,10) (2,9) (2,7) (3,13) (3,8) (3,4) (3,14) (3,12) (3,7) (3,11) (3,10) (4,8) (4,7) (4,12) (4,13) (4,5) (4,6) (4,9) (5,6) (5,12) (5,13) (5,11) (5,9) (5,7) (6,14) (6,7) (6,13) (7,14) (7,10) (7,8) (7,11) (7,13) (8,9) (8,10) (8,14) (8,12) (8,11) (9,12) (9,13) (9,14) (9,10) (10,11) (10,14) (10,13) (10,12) (11,12) (11,13) (12,14) (13,14)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,8,11,13,14,12,10,9,4,3,7,6,2,5,1", "difficulty": "hard", "doc_id": "18"}
{"question": "In an 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: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an 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: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an 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: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an 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: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an 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: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an 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: (0,6) (0,14) (0,15) (0,8) (0,9) (0,1) (0,12) (0,7) (0,16) (0,3) (1,3) (1,7) (1,12) (1,4) (1,15) (1,10) (1,8) (1,11) (1,9) (2,16) (2,4) (2,3) (2,7) (2,6) (2,11) (2,9) (2,13) (3,4) (3,5) (3,16) (3,12) (3,9) (3,10) (3,13) (4,9) (4,16) (4,11) (4,12) (4,7) (4,13) (4,15) (4,14) (5,15) (5,11) (5,6) (5,12) (5,9) (5,8) (5,16) (5,10) (6,15) (6,9) (6,10) (6,8) (6,14) (7,14) (7,13) (7,10) (7,11) (7,12) (7,9) (8,13) (8,16) (8,14) (8,11) (8,12) (8,15) (9,12) (9,13) (9,16) (9,10) (10,11) (10,12) (10,14) (10,15) (10,13) (11,12) (11,13) (12,13) (12,14) (12,15) (12,16) (13,15) (14,16)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,3,13,15,12,16,14,10,11,8,6,5,9,7,2,4,1", "difficulty": "hard", "doc_id": "19"}
{"question": "In an 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: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an 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: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an 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: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an 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: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an 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: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an 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: (0,7) (0,4) (0,2) (0,8) (0,5) (0,12) (0,1) (0,10) (1,4) (1,11) (2,11) (2,3) (2,14) (2,15) (2,4) (3,5) (3,11) (3,12) (3,4) (3,10) (3,9) (3,6) (3,13) (4,13) (4,9) (4,7) (4,11) (4,10) (5,11) (5,7) (5,12) (6,7) (6,15) (6,11) (6,9) (7,14) (7,15) (7,9) (8,15) (8,9) (8,11) (9,13) (9,15) (9,10) (10,11) (10,14) (12,14) (12,15)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,10,14,12,15,8,11,5,7,6,9,13,3,2,4,1", "difficulty": "hard", "doc_id": "20"}
{"question": "In an 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: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an 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: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an 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: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an 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: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an 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: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an 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: (0,7) (0,1) (0,15) (0,11) (0,5) (0,10) (0,8) (1,12) (1,14) (1,11) (1,5) (1,10) (1,15) (2,4) (2,11) (2,12) (2,10) (2,6) (2,3) (2,9) (2,13) (2,8) (2,5) (3,5) (3,10) (3,6) (3,15) (4,15) (4,6) (4,10) (4,12) (4,5) (4,13) (4,14) (5,15) (5,13) (5,9) (5,11) (5,14) (5,6) (6,14) (6,11) (6,13) (6,9) (7,8) (7,13) (7,12) (7,9) (7,15) (8,13) (8,11) (8,14) (8,12) (8,9) (8,15) (9,10) (9,15) (9,11) (9,14) (10,15) (10,14) (10,13) (11,14) (11,12) (11,15) (12,15) (12,13) (13,15) (14,15)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,8,15,14,11,12,13,10,1,5,4,6,3,2,9,7", "difficulty": "hard", "doc_id": "21"}
{"question": "In an 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: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an 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: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an 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: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an 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: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an 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: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an 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: (0,6) (0,3) (0,9) (0,8) (0,11) (0,13) (0,1) (1,7) (1,8) (1,5) (1,6) (1,4) (1,9) (2,5) (2,12) (2,11) (2,10) (2,9) (2,3) (3,4) (3,7) (3,6) (3,11) (3,8) (3,12) (4,13) (4,10) (4,11) (4,5) (4,9) (4,8) (5,10) (5,13) (5,6) (5,11) (6,10) (6,8) (6,7) (6,9) (7,10) (7,13) (7,11) (7,12) (8,13) (8,9) (8,11) (8,12) (9,11) (9,10) (10,12) (11,12) (11,13) (12,13)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,1,9,10,12,13,11,8,6,7,3,4,5,2", "difficulty": "hard", "doc_id": "22"}
{"question": "In an 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: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an 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: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an 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: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an 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: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an 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: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an 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) (0,11) (0,14) (0,5) (0,7) (0,4) (0,8) (0,6) (0,10) (0,12) (1,12) (1,9) (1,10) (1,8) (1,3) (1,6) (1,14) (1,11) (1,5) (1,7) (1,13) (1,2) (2,8) (2,10) (2,7) (2,12) (2,3) (2,13) (2,4) (2,11) (2,5) (2,14) (2,9) (3,7) (3,12) (3,10) (3,4) (3,6) (3,13) (3,9) (3,5) (4,6) (4,11) (4,10) (4,7) (5,12) (5,11) (5,8) (5,6) (5,14) (5,9) (5,7) (6,7) (6,9) (6,8) (6,12) (6,14) (6,10) (7,8) (7,9) (7,10) (7,12) (7,14) (7,13) (8,14) (8,13) (8,12) (8,10) (8,11) (9,11) (9,10) (10,13) (10,12) (10,14) (11,14) (11,12) (11,13) (12,14) (12,13)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,12,13,11,14,10,9,7,8,6,5,3,4,2,1", "difficulty": "hard", "doc_id": "23"}
{"question": "In an 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: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an 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: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an 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: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an 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: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an 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: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an 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: (0,9) (0,6) (0,8) (0,4) (0,14) (0,1) (0,11) (0,3) (0,13) (0,10) (1,11) (1,14) (1,8) (1,4) (1,5) (1,2) (1,10) (1,15) (1,9) (2,9) (2,18) (2,11) (2,13) (2,12) (2,8) (2,15) (2,14) (3,4) (3,7) (3,18) (3,15) (3,17) (3,5) (3,16) (4,15) (4,13) (4,6) (4,9) (4,16) (5,10) (5,17) (5,18) (5,12) (5,11) (5,14) (5,16) (5,9) (6,14) (6,17) (6,13) (6,8) (6,11) (6,10) (6,18) (7,10) (7,16) (7,13) (7,15) (7,9) (7,17) (8,17) (8,11) (8,14) (8,15) (8,9) (8,10) (8,13) (9,16) (9,13) (9,12) (9,11) (9,17) (9,10) (10,16) (10,14) (10,13) (10,12) (10,15) (10,17) (11,13) (11,12) (11,18) (11,14) (12,15) (12,13) (12,16) (12,14) (13,15) (13,14) (13,16) (14,16) (14,17) (14,15) (15,16) (15,17) (16,18) (16,17)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,10,17,16,18,11,14,15,13,12,9,7,3,5,1,2,8,6,4", "difficulty": "hard", "doc_id": "24"}
{"question": "In an 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: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an 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: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an 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: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an 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: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an 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: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an 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: (0,3) (0,1) (0,5) (1,4) (1,2) (2,3) (2,5) (2,4) (3,4) (4,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,5,4,3,2,1", "difficulty": "easy", "doc_id": "25"}
{"question": "In an 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: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an 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: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an 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: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an 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: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an 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: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an 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) (0,9) (0,8) (0,2) (0,1) (0,3) (1,4) (1,2) (1,7) (1,9) (1,8) (1,3) (2,4) (2,6) (2,5) (2,9) (3,8) (3,4) (3,6) (3,7) (3,9) (3,5) (4,8) (4,5) (4,9) (4,6) (5,6) (6,7) (7,8)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,3,5,6,7,8,4,9,2,1", "difficulty": "easy", "doc_id": "26"}
{"question": "In an 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: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an 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: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an 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: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an 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: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an 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: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an 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: (0,8) (0,5) (0,11) (0,2) (0,3) (0,13) (0,6) (0,10) (0,7) (0,12) (1,7) (1,12) (1,2) (1,8) (1,5) (2,8) (2,11) (2,9) (2,4) (2,3) (2,10) (3,4) (3,6) (3,5) (3,11) (3,13) (3,10) (3,14) (4,5) (4,12) (4,11) (4,7) (4,10) (4,6) (5,7) (5,14) (5,13) (5,9) (5,11) (5,10) (5,12) (5,6) (6,8) (6,10) (6,14) (6,12) (6,13) (6,9) (7,11) (7,13) (7,8) (7,9) (7,14) (8,10) (8,13) (8,14) (9,11) (9,13) (9,12) (10,14) (10,12) (10,11) (12,13) (13,14)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,12,13,14,10,11,9,7,8,6,5,4,3,2,1", "difficulty": "hard", "doc_id": "27"}
{"question": "In an 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: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an 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: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an 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: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an 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: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an 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: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an 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: (0,8) (0,4) (0,2) (0,11) (0,7) (1,2) (1,3) (1,11) (1,4) (1,12) (2,6) (2,10) (2,7) (2,11) (3,12) (3,4) (3,8) (3,9) (3,10) (4,7) (4,9) (4,8) (5,10) (5,12) (5,9) (5,6) (6,9) (6,8) (6,12) (7,11) (7,10) (8,10) (8,11) (9,11) (9,10) (10,12) (11,12)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,7,10,12,11,9,5,6,8,4,3,1,2", "difficulty": "hard", "doc_id": "28"}
{"question": "In an 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: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an 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: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an 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: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an 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: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an 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: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an 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) (0,7) (0,5) (0,10) (0,9) (0,11) (1,5) (1,14) (1,12) (1,3) (1,10) (2,5) (2,6) (2,11) (2,13) (2,8) (3,9) (3,14) (3,13) (3,12) (3,8) (3,4) (4,6) (4,12) (4,5) (4,8) (4,10) (5,11) (5,7) (5,12) (5,14) (6,13) (6,8) (6,10) (6,7) (7,11) (7,9) (7,10) (8,13) (8,10) (8,11) (10,14) (10,13) (11,12) (11,13) (12,14) (12,13) (13,14)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,11,13,14,12,5,7,9,3,4,8,2,6,10,1", "difficulty": "hard", "doc_id": "29"}
{"question": "In an 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: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an 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: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an 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: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an 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: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an 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: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an 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: (0,1) (0,7) (0,9) (0,11) (1,2) (1,6) (1,11) (1,12) (1,8) (1,5) (2,9) (2,5) (3,11) (3,8) (3,10) (4,11) (4,9) (4,5) (4,8) (5,7) (5,6) (5,9) (6,9) (6,10) (7,9) (7,12) (7,8) (8,11) (8,10) (9,12) (10,11) (11,12)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,11,12,9,7,8,4,5,2,1,6,10,3", "difficulty": "hard", "doc_id": "30"}
{"question": "In an 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: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an 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: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an 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: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an 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: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an 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: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an 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) (0,2) (1,8) (1,9) (1,5) (1,6) (1,4) (2,6) (2,4) (2,7) (3,6) (4,8) (4,7) (4,5) (5,7) (5,6) (5,9) (6,7) (6,9) (6,8) (8,9)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,2,7,5,9,8,4,1,6,3", "difficulty": "easy", "doc_id": "31"}
{"question": "In an 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: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an 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: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an 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: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an 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: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an 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: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an 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) (0,1) (0,7) (0,3) (0,4) (1,7) (1,4) (1,5) (2,5) (2,4) (2,3) (3,6) (4,7) (4,6) (4,5) (6,7)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,4,5,2,3,6,7,1", "difficulty": "easy", "doc_id": "32"}
{"question": "In an 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: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an 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: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an 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: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an 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: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an 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: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an 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: (0,10) (0,8) (0,13) (0,5) (0,12) (0,11) (0,6) (0,1) (0,3) (1,6) (1,2) (1,5) (1,8) (1,12) (2,6) (2,13) (2,11) (2,8) (3,5) (3,10) (3,9) (3,8) (3,13) (3,6) (4,12) (4,11) (4,10) (4,6) (4,7) (5,9) (5,6) (5,10) (6,12) (6,9) (6,13) (6,11) (7,13) (7,10) (7,12) (8,13) (8,11) (9,12) (9,11) (9,13) (9,10) (10,11) (11,13)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,3,6,11,13,9,12,7,4,10,5,1,8,2", "difficulty": "hard", "doc_id": "33"}
{"question": "In an 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: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an 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: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an 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: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an 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: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an 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: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an 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: (0,4) (0,5) (0,1) (0,11) (0,3) (0,9) (0,7) (0,6) (0,15) (0,19) (0,10) (0,13) (1,2) (1,3) (1,12) (1,4) (1,11) (1,15) (1,19) (1,8) (1,13) (2,5) (2,7) (2,11) (3,15) (3,8) (3,16) (3,19) (3,18) (3,17) (4,7) (4,5) (4,16) (4,15) (4,8) (4,18) (4,6) (4,10) (4,17) (4,12) (4,13) (5,7) (5,16) (5,15) (6,14) (6,13) (6,17) (6,15) (6,18) (7,15) (7,11) (7,17) (7,19) (8,19) (8,9) (8,17) (8,12) (8,14) (8,10) (9,15) (9,11) (10,15) (10,16) (10,17) (10,11) (11,18) (11,16) (12,13) (12,19) (13,15) (13,16) (13,14) (13,19) (14,17) (14,19) (15,17) (17,18) (18,19)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,13,19,18,17,15,10,11,9,8,14,6,4,12,1,3,16,5,7,2", "difficulty": "hard", "doc_id": "34"}
{"question": "In an 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: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an 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: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an 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: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an 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: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an 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: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an 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: (0,12) (0,1) (0,8) (0,11) (1,6) (1,2) (1,9) (1,11) (1,4) (1,13) (2,3) (2,6) (2,12) (3,13) (3,12) (3,11) (3,10) (4,5) (4,9) (4,11) (4,12) (5,10) (5,11) (5,6) (7,12) (7,13) (7,10) (7,11) (8,11) (8,13) (9,13) (9,12) (10,13) (11,12)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,11,12,9,4,5,6,1,2,3,10,7,13,8", "difficulty": "hard", "doc_id": "35"}
{"question": "In an 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: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an 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: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an 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: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an 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: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an 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: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an 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: (0,6) (0,10) (0,14) (0,5) (0,3) (0,9) (0,1) (0,15) (0,8) (0,2) (1,11) (1,14) (1,15) (1,4) (1,6) (1,9) (1,3) (1,2) (2,12) (2,4) (2,13) (2,9) (2,3) (2,5) (2,8) (2,10) (3,7) (3,15) (3,13) (3,10) (3,8) (3,12) (4,9) (4,8) (4,12) (4,7) (4,13) (4,10) (5,7) (5,13) (5,10) (5,8) (6,13) (6,10) (6,7) (6,14) (7,15) (7,11) (7,14) (7,12) (7,10) (8,10) (8,12) (8,9) (8,11) (8,15) (9,10) (9,13) (9,11) (10,12) (10,15) (10,14) (11,13) (11,15) (11,12) (12,13) (12,15) (13,15) (14,15)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,2,10,14,15,13,12,11,9,4,7,6,1,3,8,5", "difficulty": "hard", "doc_id": "36"}
{"question": "In an 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: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an 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: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an 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: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an 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: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an 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: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an 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) (0,1) (0,6) (0,7) (1,5) (1,6) (1,7) (1,8) (2,7) (2,4) (2,5) (3,4) (3,5) (4,5) (7,8)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,6,1,8,7,2,5,4,3", "difficulty": "easy", "doc_id": "37"}
{"question": "In an 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: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an 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: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an 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: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an 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: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an 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: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an 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: (0,12) (0,5) (0,13) (0,1) (0,17) (0,10) (0,14) (1,17) (1,13) (1,6) (1,9) (1,16) (1,7) (1,15) (2,15) (2,4) (2,7) (2,11) (2,5) (2,9) (2,6) (2,10) (3,16) (3,11) (3,17) (3,12) (3,13) (3,4) (3,15) (4,16) (4,5) (4,12) (5,14) (5,8) (5,13) (5,17) (5,15) (5,11) (5,12) (6,11) (6,13) (6,9) (6,15) (6,12) (6,14) (6,10) (7,16) (7,13) (7,14) (7,8) (8,9) (8,13) (8,17) (8,10) (8,16) (9,12) (9,17) (10,17) (10,14) (10,12) (10,15) (10,13) (11,17) (11,13) (12,17) (13,15) (15,17) (16,17)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,14,10,13,15,17,16,8,9,12,6,11,3,4,5,2,7,1", "difficulty": "hard", "doc_id": "38"}
{"question": "In an 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: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an 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: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an 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: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an 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: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an 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: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an 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: (0,15) (0,10) (1,14) (2,14) (2,15) (2,10) (2,3) (2,6) (2,8) (3,7) (3,14) (3,4) (3,8) (3,5) (3,10) (3,9) (4,5) (4,14) (4,7) (4,9) (5,15) (5,12) (5,7) (5,13) (5,14) (5,6) (6,12) (6,8) (7,9) (7,10) (7,15) (7,14) (7,8) (8,9) (8,13) (9,13) (9,14) (9,11) (9,10) (10,11) (10,12) (11,15) (12,13) (14,15)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,10,12,13,9,11,15,7,8,6,5,4,3,2,14,1", "difficulty": "hard", "doc_id": "39"}
{"question": "In an 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: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an 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: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an 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: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an 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: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an 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: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an 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) (0,10) (0,5) (0,11) (0,12) (0,8) (0,1) (0,3) (1,9) (1,3) (1,6) (1,4) (1,13) (1,2) (1,11) (1,7) (1,12) (2,3) (2,13) (2,7) (2,8) (2,9) (2,6) (2,5) (2,11) (3,8) (3,13) (3,6) (3,9) (3,7) (3,5) (3,12) (3,11) (4,6) (4,8) (4,5) (4,13) (5,6) (5,10) (5,11) (5,7) (5,13) (5,8) (6,12) (6,9) (6,7) (6,10) (7,8) (7,11) (7,13) (7,10) (8,13) (8,12) (9,11) (9,10) (10,13) (10,12) (11,13) (11,12) (12,13)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,3,11,12,13,10,9,6,7,8,5,4,1,2", "difficulty": "hard", "doc_id": "40"}
{"question": "In an 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: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an 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: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an 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: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an 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: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an 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: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an 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) (0,11) (0,14) (0,6) (0,2) (0,13) (0,10) (1,6) (1,7) (1,10) (1,9) (1,12) (1,2) (1,13) (1,3) (2,5) (2,7) (2,4) (2,9) (2,14) (2,10) (2,8) (2,12) (2,3) (3,14) (3,11) (3,8) (3,6) (3,9) (3,13) (3,4) (4,8) (4,14) (4,6) (4,7) (4,11) (4,5) (4,13) (4,9) (5,6) (5,13) (5,8) (5,14) (5,12) (5,10) (6,7) (6,9) (6,8) (6,14) (6,10) (6,12) (7,13) (7,11) (7,12) (7,9) (7,14) (8,11) (8,10) (8,12) (8,13) (8,14) (8,9) (9,14) (10,11) (11,12) (11,13) (11,14) (12,13)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,10,11,14,9,8,13,12,7,6,5,4,3,2,1", "difficulty": "hard", "doc_id": "41"}
{"question": "In an 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: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an 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: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an 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: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an 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: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an 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: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an 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) (0,6) (1,3) (1,7) (2,6) (3,5) (4,6) (4,7) (4,5) (6,7)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,5,3,1,7,4,6,2", "difficulty": "easy", "doc_id": "42"}
{"question": "In an 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: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an 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: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an 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: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an 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: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an 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: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an 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) (0,2) (0,3) (0,7) (0,4) (1,6) (2,7) (2,4) (2,3) (2,6) (3,8) (3,4) (3,5) (4,7) (4,8) (4,6) (4,5) (5,7) (5,8) (6,7)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,4,5,8,3,2,7,6,1", "difficulty": "easy", "doc_id": "43"}
{"question": "In an 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: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an 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: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an 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: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an 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: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an 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: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an 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: (0,1) (0,13) (0,8) (0,6) (0,5) (0,3) (0,14) (0,4) (0,15) (0,2) (0,12) (1,13) (1,8) (1,6) (1,10) (1,4) (1,7) (1,2) (1,11) (1,14) (1,12) (1,5) (2,8) (2,15) (2,14) (2,11) (2,13) (3,6) (3,8) (3,5) (3,4) (3,7) (3,12) (3,11) (3,13) (3,15) (4,13) (4,14) (4,6) (4,11) (4,7) (4,12) (4,9) (5,7) (6,9) (6,10) (6,8) (6,12) (6,13) (7,10) (7,9) (7,12) (7,14) (7,15) (8,9) (8,13) (8,14) (8,12) (8,11) (9,13) (9,10) (9,14) (10,14) (10,12) (10,11) (10,13) (10,15) (11,15) (11,13) (11,14) (11,12) (12,15) (12,13)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,12,13,11,14,10,15,7,9,8,6,4,3,5,1,2", "difficulty": "hard", "doc_id": "44"}
{"question": "In an 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: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an 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: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an 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: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an 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: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an 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: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an 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: (0,8) (0,15) (0,11) (1,16) (1,6) (1,13) (1,10) (1,12) (1,5) (1,7) (1,9) (2,4) (2,8) (2,15) (2,12) (2,16) (3,15) (3,12) (3,10) (3,6) (3,13) (3,11) (4,15) (4,16) (4,11) (4,7) (4,6) (4,8) (5,11) (5,16) (5,9) (5,6) (6,12) (6,15) (6,9) (7,14) (7,8) (7,10) (7,15) (7,12) (8,14) (8,12) (9,15) (9,16) (9,14) (9,10) (10,11) (10,13) (11,15) (11,12) (11,14) (12,16) (12,13) (12,14) (13,14) (13,16) (14,16) (15,16)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,11,14,16,15,9,10,13,12,8,2,4,7,1,5,6,3", "difficulty": "hard", "doc_id": "45"}
{"question": "In an 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: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an 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: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an 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: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an 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: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an 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: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an 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: (0,11) (0,9) (0,15) (0,13) (0,16) (0,6) (0,2) (1,8) (1,14) (2,19) (2,17) (2,14) (2,8) (2,18) (2,13) (2,9) (2,7) (2,16) (3,16) (3,12) (3,19) (3,10) (3,15) (3,6) (3,13) (3,14) (4,18) (4,17) (4,9) (4,19) (4,7) (4,13) (4,14) (5,17) (5,13) (5,18) (5,7) (5,16) (5,15) (5,8) (5,11) (6,13) (6,11) (6,9) (6,17) (6,7) (7,12) (7,15) (7,11) (7,17) (7,16) (7,19) (7,10) (7,14) (8,11) (8,9) (8,13) (8,15) (9,19) (9,13) (9,15) (9,14) (10,16) (10,12) (11,14) (11,17) (11,12) (12,17) (12,18) (13,18) (13,14) (13,15) (14,15) (14,17) (14,19) (15,16) (15,18) (16,19) (17,18)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,2,16,19,14,17,18,15,13,9,4,7,10,12,3,6,11,5,8,1", "difficulty": "hard", "doc_id": "46"}
{"question": "In an 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: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an 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: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an 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: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an 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: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an 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: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an 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: (0,8) (0,16) (0,7) (0,10) (0,17) (0,2) (0,3) (0,13) (0,18) (0,11) (0,14) (0,4) (0,6) (0,19) (1,9) (1,5) (1,7) (1,11) (1,2) (1,6) (1,4) (1,14) (2,6) (2,14) (2,3) (2,16) (2,10) (2,19) (2,11) (2,8) (2,5) (2,13) (2,7) (2,15) (2,4) (3,13) (3,8) (3,7) (3,5) (3,17) (3,12) (3,4) (3,16) (3,9) (3,6) (3,19) (3,18) (4,9) (4,14) (4,10) (4,8) (4,12) (4,19) (5,15) (5,7) (5,16) (5,9) (5,8) (5,10) (5,11) (5,13) (5,12) (5,19) (6,7) (6,12) (6,15) (6,8) (6,16) (7,16) (7,19) (7,15) (7,8) (7,12) (7,11) (8,17) (8,16) (8,9) (8,12) (8,14) (8,18) (9,15) (9,16) (9,17) (9,19) (10,17) (10,16) (10,12) (10,11) (10,18) (10,14) (11,18) (11,16) (11,12) (11,17) (11,15) (12,13) (12,17) (12,19) (12,15) (12,14) (12,18) (13,19) (13,15) (13,14) (13,18) (14,16) (14,18) (14,17) (15,18) (15,19) (15,17) (17,18) (17,19)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,19,17,18,15,13,14,16,11,12,10,5,8,9,4,3,6,7,2,1", "difficulty": "hard", "doc_id": "47"}
{"question": "In an 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: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an 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: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an 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: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an 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: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an 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: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an 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: (0,15) (0,4) (0,13) (0,11) (0,2) (0,16) (0,14) (0,5) (0,9) (1,8) (1,3) (1,2) (1,6) (1,12) (1,11) (1,9) (1,4) (1,16) (2,10) (2,15) (2,8) (2,13) (2,6) (2,14) (2,12) (2,5) (3,15) (3,16) (4,9) (4,15) (4,8) (4,14) (5,10) (5,6) (5,12) (5,11) (5,7) (6,8) (6,7) (6,13) (6,11) (6,9) (6,16) (6,14) (7,15) (7,14) (8,13) (8,16) (9,12) (9,16) (9,10) (9,13) (9,15) (10,13) (11,12) (11,15) (12,13) (13,16) (13,14) (14,16)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,9,15,11,12,13,14,16,6,7,5,10,2,8,4,1,3", "difficulty": "hard", "doc_id": "48"}
{"question": "In an 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: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an 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: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an 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: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an 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: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an 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: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an 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) (0,5) (0,7) (0,2) (1,2) (1,8) (1,9) (1,3) (2,7) (2,5) (2,9) (3,7) (3,9) (3,5) (3,8) (4,8) (4,7) (4,9) (4,6) (5,6) (5,9) (6,7) (7,8) (7,9) (8,9)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,2,9,8,7,4,6,5,3,1", "difficulty": "easy", "doc_id": "49"}
{"question": "In an 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: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an 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: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an 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: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an 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: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an 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: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an 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: (0,7) (0,8) (0,1) (0,9) (1,6) (1,5) (1,9) (2,9) (2,8) (2,6) (2,4) (2,7) (3,8) (3,4) (3,9) (3,5) (4,9) (4,6) (5,6) (7,8) (7,9)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,9,7,8,3,5,1,6,4,2", "difficulty": "easy", "doc_id": "50"}
{"question": "In an 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: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an 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: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an 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: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an 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: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an 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: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an 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: (0,7) (0,8) (0,6) (1,6) (1,10) (1,11) (1,7) (1,3) (1,8) (2,11) (2,8) (2,5) (2,9) (2,7) (2,10) (3,6) (3,7) (4,10) (5,7) (5,8) (5,10) (6,11) (6,10) (7,11) (7,9) (9,11) (9,10)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,6,11,9,7,3,1,8,5,2,10,4", "difficulty": "hard", "doc_id": "51"}
{"question": "In an 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: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an 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: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an 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: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an 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: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an 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: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an 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: (0,6) (0,12) (0,7) (0,8) (0,10) (1,7) (1,5) (1,9) (1,3) (2,8) (2,11) (2,3) (2,5) (2,4) (2,12) (2,6) (3,4) (3,10) (3,12) (3,5) (4,12) (4,9) (4,8) (4,6) (5,11) (5,7) (5,12) (5,10) (5,9) (6,12) (6,9) (6,11) (6,10) (7,11) (7,12) (7,9) (7,8) (8,11) (8,12) (8,9) (8,10) (9,11) (9,10) (10,11) (10,12)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,10,12,8,9,11,7,5,2,6,4,3,1", "difficulty": "hard", "doc_id": "52"}
{"question": "In an 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: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an 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: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an 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: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an 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: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an 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: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an 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) (0,1) (0,2) (1,8) (1,5) (1,9) (1,6) (2,3) (2,7) (2,9) (2,6) (2,4) (3,8) (3,4) (3,7) (4,6) (4,8) (5,6) (5,9) (6,8) (6,9) (6,7) (7,8) (8,9)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,2,4,8,9,5,1,6,7,3", "difficulty": "easy", "doc_id": "53"}
{"question": "In an 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: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an 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: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an 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: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an 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: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an 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: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an 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: (0,7) (0,11) (0,4) (0,6) (0,2) (0,9) (1,7) (1,2) (2,10) (2,9) (2,6) (2,8) (2,3) (3,10) (3,4) (3,9) (3,13) (4,10) (4,5) (4,12) (4,11) (5,10) (5,8) (5,7) (5,13) (5,11) (5,6) (6,10) (6,7) (6,13) (7,13) (7,12) (7,11) (8,9) (8,12) (9,12) (10,11) (10,13) (11,12) (11,13) (12,13)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,9,12,13,11,10,6,7,1,2,3,4,5,8", "difficulty": "hard", "doc_id": "54"}
{"question": "In an 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: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an 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: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an 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: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an 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: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an 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: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an 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: (0,4) (1,3) (2,3) (2,5) (3,4) (3,5) (4,6) (5,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,4,6,5,2,3,1", "difficulty": "easy", "doc_id": "55"}
{"question": "In an 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: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an 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: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an 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: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an 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: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an 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: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an 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) (0,2) (0,4) (0,1) (1,4) (1,2) (1,6) (1,5) (2,3) (3,6) (3,5) (5,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,4,1,5,6,3,2", "difficulty": "easy", "doc_id": "56"}
{"question": "In an 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: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an 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: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an 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: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an 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: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an 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: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an 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) (0,5) (0,6) (1,2) (1,7) (1,8) (1,3) (1,11) (2,8) (2,3) (2,10) (3,7) (3,5) (4,5) (4,9) (5,11) (6,8) (7,11) (7,9) (8,9) (9,10) (10,11)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,6,8,9,10,11,7,1,2,3,5,4", "difficulty": "hard", "doc_id": "57"}