{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R364",
  "content_hash": null,
  "slug": "ham8-attempt-enumeration-table-audit",
  "type": "attempt",
  "title": "OEIS records the labeled count and its complement",
  "summary": "A326208 gives 151,676,112, A326207 gives 116,759,344, and the classical unlabeled sequence gives 6,196 classes.",
  "relevance": "For Exact Hamiltonicity probability on eight labeled vertices, record ham8-attempt-enumeration-table-audit (“OEIS records the labeled count and its complement”) documents a concrete method, search boundary, or failed route. The record states: A326208 gives 151,676,112, A326207 gives 116,759,344, and the classical unlabeled sequence gives 6,196 classes.",
  "relevance_source": "recorded",
  "body": "OEIS A326208 is the labeled sequence. Its \\(n=8\\) term is 151,676,112, and its entry says the terms from orders 7 through 11 were added using Falk Hueffner's tinygraph at version `9766535`. OEIS A326207 independently records the complementary \\(n=8\\) term 116,759,344. Their sum is \\(2^{28}\\).\n\nOEIS A003216 records the unlabeled sequence and gives 6,196 at order 8. It cites J. P. Dolch's 1973 enumeration, Harary and Palmer's 1973 book, and later graph atlases. The nauty orbit computation in this fixture reproduces both the 6,196 unlabeled term and the labeled A326208 term.\n\nThe checked records settle this database target as a known finite enumeration. The two included computations supply local certificates with fixed encodings and checksums.",
  "status": "completed",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "bounded",
    "statement": "published and maintained enumeration records checked for Hamiltonian graphs on eight vertices in labeled and unlabeled conventions",
    "bounds": {
      "vertices": {
        "min": 8,
        "max": 8
      },
      "source_year": {
        "min": 1973,
        "max": 2019
      }
    },
    "exhaustive": false
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "attempt",
    "citation": {
      "url": "https://oeis.org/A326208",
      "locator": "OEIS A326208, Number of Hamiltonian labeled simple graphs with n vertices; OEIS A326207, Number of non-Hamiltonian labeled simple graphs with n vertices; OEIS A003216, Number of Hamiltonian graphs with n nodes; J. P. Dolch, Names of Hamiltonian Graphs, Congressus Numerantium 8 (1973), 259-271; Frank Harary and Edgar M. Palmer, Graphical Enumeration, Academic Press, 1973, p. 219; Brendan D. McKay and Adolfo Piperno, Practical Graph Isomorphism, II, Journal of Symbolic Computation 60 (2014), 94-112, DOI 10.1016/j.jsc.2013.09.003"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://oeis.org/A326208",
    "locator": "OEIS A326208, Number of Hamiltonian labeled simple graphs with n vertices; OEIS A326207, Number of non-Hamiltonian labeled simple graphs with n vertices; OEIS A003216, Number of Hamiltonian graphs with n nodes; J. P. Dolch, Names of Hamiltonian Graphs, Congressus Numerantium 8 (1973), 259-271; Frank Harary and Edgar M. Palmer, Graphical Enumeration, Academic Press, 1973, p. 219; Brendan D. McKay and Adolfo Piperno, Practical Graph Isomorphism, II, Journal of Symbolic Computation 60 (2014), 94-112, DOI 10.1016/j.jsc.2013.09.003"
  },
  "relations": [
    {
      "slug": "R365",
      "title": "Exactly 151,676,112 labeled graphs on eight vertices are Hamiltonian",
      "object_type": "claim",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "hamiltonian-graph-probability-eight",
      "title": "hamiltonian graph probability eight",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}