{
  "name": "graceful_graph_k3_p5__v2_nh",
  "problem_type": "graceful_graph",
  "params": {
    "k": 3,
    "p": 5
  },
  "prompt": "Find a graceful labeling for the graph G_{3,5}: 5 disjoint K_3 cliques (numbered 0 through 4), 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 15 vertices and 27 edges in total.\n\nA graceful labeling assigns each vertex a unique label from 0 to 27 such that all edge labels |label(u) - label(v)| are distinct.\n\nReturn a list of 15 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.",
  "satisfiable": true,
  "solution": {
    "x": [
      5,
      24,
      14,
      23,
      3,
      7,
      1,
      6,
      18,
      25,
      12,
      26,
      2,
      27,
      0
    ],
    "d": [
      19,
      9,
      10,
      20,
      16,
      4,
      5,
      17,
      12,
      13,
      1,
      14,
      25,
      2,
      27,
      18,
      21,
      7,
      22,
      3,
      11,
      24,
      6,
      8,
      23,
      15,
      26
    ]
  },
  "difficulty": {
    "solve_time_ms": 1655.4,
    "search_space": 5097655355238390956032,
    "num_variables": 42,
    "num_constraints": 31,
    "num_edges": -1,
    "backend": "pycsp",
    "solve_tier": "hard",
    "solve_pct_global": 88.91,
    "solve_pct_type": 81.25
  },
  "partial_assignment": null
}