{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R365",
  "content_hash": null,
  "slug": "ham8-claim-exact-labeled-probability",
  "type": "claim",
  "title": "Exactly 151,676,112 labeled graphs on eight vertices are Hamiltonian",
  "summary": "The exact probability is 9,479,757 divided by 16,777,216, approximately 0.565037548542022705078125.",
  "relevance": "For Exact Hamiltonicity probability on eight labeled vertices, record ham8-claim-exact-labeled-probability (“Exactly 151,676,112 labeled graphs on eight vertices are Hamiltonian”) records a bound, answer, status fact, or structural consequence. The record states: The exact probability is 9,479,757 divided by 16,777,216, approximately 0.565037548542022705078125.",
  "relevance_source": "recorded",
  "body": "There are \\(\\binom82=28\\) possible edges, so the uniform sample space contains \\(2^{28}=268{,}435{,}456\\) labeled simple graphs. Exactly\n\\[\n151{,}676{,}112\n\\]\nof them contain a Hamilton cycle. Thus\n\\[\n\\Pr(G\\text{ is Hamiltonian})\n=\\frac{151{,}676{,}112}{268{,}435{,}456}\n=\\frac{9{,}479{,}757}{16{,}777{,}216}\n=0.565037548542022705078125.\n\\]\nThe complementary count is \\(116{,}759{,}344\\). The Hamiltonian count exceeds half of the sample space by \\(17{,}458{,}384\\).\n\nTwo exhaustive computations reproduce the numerator. The first takes the upward closure of all 2,520 labeled Hamilton-cycle edge masks. The second generates all 12,346 isomorphism classes on eight vertices, finds 6,196 Hamiltonian classes, and sums the orbit weights \\(8!/|\\operatorname{Aut}(G)|\\).",
  "status": "established",
  "evidence_grade": "reproduced",
  "scope": {
    "kind": "bounded",
    "statement": "every simple graph on the fixed labeled vertex set {0,1,2,3,4,5,6,7}",
    "bounds": {
      "vertices": {
        "min": 8,
        "max": 8
      },
      "possible_edges": {
        "min": 28,
        "max": 28
      },
      "graphs": {
        "min": 268435456,
        "max": 268435456
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://oeis.org/A326208",
      "locator": "Exact certificates in ham8-artifact-upward-closure and ham8-artifact-orbit-weighted-check"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://oeis.org/A326208",
    "locator": "Exact certificates in ham8-artifact-upward-closure and ham8-artifact-orbit-weighted-check"
  },
  "relations": [
    {
      "slug": "R363",
      "title": "Replayable 32 MiB upward-closure certificate",
      "object_type": "artifact",
      "relation": "reproduces",
      "direction": "incoming"
    },
    {
      "slug": "R362",
      "title": "Independent nauty orbit-weighted enumeration",
      "object_type": "artifact",
      "relation": "independently_reproduces",
      "direction": "incoming"
    },
    {
      "slug": "R364",
      "title": "OEIS records the labeled count and its complement",
      "object_type": "attempt",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "hamiltonian-graph-probability-eight",
      "title": "hamiltonian graph probability eight",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}