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"name": "graceful_graph_k3_p4__v3_h",
"problem_type": "graceful_graph",
"params": {
"k": 3,
"p": 4
},
"prompt": "Find a graceful labeling for the graph G_{3,4}: 4 disjoint K_3 cliques (numbered 0 through 3), where each pair of consecutive cliques (g, g+1) is connected by 3 edges that link vertex i of clique g to vertex i of clique g+1 for each i in 0..2. The graph has 12 vertices and 21 edges in total.\n\nA graceful labeling assigns each vertex a unique label from 0 to 21 such that all edge labels |label(u) - label(v)| are distinct.\n\nReturn a list of 12 integers giving the vertex labels, ordered by clique then vertex within clique (so the i-th block of 3 consecutive entries gives the labels for clique i), or state \"UNSATISFIABLE\" if no such labeling exists.\n\nPartial assignment (fixed values that must be respected):\n- x[2]=15, x[4]=0\n- d[6]=5, d[9]=11, d[15]=13, d[18]=9\nReturn a complete solution consistent with these fixed assignments.",
"satisfiable": true,
"solution": {
"x": [
3,
21,
15,
20,
0,
1,
7,
2,
17,
16,
5,
9
],
"d": [
18,
12,
6,
20,
19,
1,
5,
10,
15,
11,
7,
4,
17,
21,
14,
13,
2,
16,
9,
3,
8
]
},
"difficulty": {
"solve_time_ms": 1447.9,
"search_space": 12855002631049216,
"num_variables": 33,
"num_constraints": 25,
"num_edges": -1,
"backend": "pycsp",
"solve_tier": "hard",
"solve_pct_global": 86.28,
"solve_pct_type": 68.75
},
"partial_assignment": {
"x": {
"2": 15,
"4": 0
},
"d": {
"6": 5,
"9": 11,
"15": 13,
"18": 9
}
}
} |