{
  "name": "graceful_graph_k2_p3__v0_h",
  "problem_type": "graceful_graph",
  "params": {
    "k": 2,
    "p": 3
  },
  "prompt": "Find a graceful labeling for the graph G_{2,3}: 3 disjoint K_2 cliques (numbered 0 through 2), 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 6 vertices and 7 edges in total.\n\nA graceful labeling assigns each vertex a unique label from 0 to 7 such that all edge labels |label(u) - label(v)| are distinct.\n\nReturn a list of 6 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[2]=7\n- d[6]=3\nReturn a complete solution consistent with these fixed assignments.",
  "satisfiable": true,
  "solution": {
    "x": [
      0,
      2,
      7,
      3,
      1,
      6
    ],
    "d": [
      2,
      4,
      5,
      7,
      1,
      6,
      3
    ]
  },
  "difficulty": {
    "solve_time_ms": 1025.2,
    "search_space": 262144,
    "num_variables": 13,
    "num_constraints": 11,
    "num_edges": -1,
    "backend": "pycsp",
    "solve_tier": "hard",
    "solve_pct_global": 72.37,
    "solve_pct_type": 18.75
  },
  "partial_assignment": {
    "x": {
      "2": 7
    },
    "d": {
      "6": 3
    }
  }
}