{
  "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
    }
  }
}