{
  "name": "graceful_graph_k2_p6__v7_h",
  "problem_type": "graceful_graph",
  "params": {
    "k": 2,
    "p": 6
  },
  "prompt": "Find a graceful labeling for the graph G_{2,6}: 6 disjoint K_2 cliques (numbered 0 through 5), where each pair of consecutive cliques (g, g+1) is connected by 2 edges that link vertex i of clique g to vertex i of clique g+1 for each i in 0..1. The graph has 12 vertices and 16 edges in total.\n\nA graceful labeling assigns each vertex a unique label from 0 to 16 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 2 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[0]=6, x[3]=4\n- d[10]=11, d[14]=1, d[15]=12\nReturn a complete solution consistent with these fixed assignments.",
  "satisfiable": true,
  "solution": {
    "x": [
      6,
      14,
      9,
      4,
      5,
      11,
      16,
      2,
      0,
      15,
      1,
      3
    ],
    "d": [
      8,
      5,
      6,
      14,
      15,
      2,
      3,
      10,
      4,
      7,
      11,
      9,
      16,
      13,
      1,
      12
    ]
  },
  "difficulty": {
    "solve_time_ms": 1225.8,
    "search_space": 582622237229761,
    "num_variables": 28,
    "num_constraints": 20,
    "num_edges": -1,
    "backend": "pycsp",
    "solve_tier": "hard",
    "solve_pct_global": 79.14,
    "solve_pct_type": 31.25
  },
  "partial_assignment": {
    "x": {
      "0": 6,
      "3": 4
    },
    "d": {
      "10": 11,
      "14": 1,
      "15": 12
    }
  }
}