{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1387",
  "content_hash": null,
  "slug": "barnette-conjecture-claim-status-20260801",
  "type": "claim",
  "title": "Current status and exact unresolved remainder",
  "summary": "OPEN as checked on 2026-08-01. Strongest checked neighboring result: Every n-vertex Barnette graph has a subhamiltonian cycle containing at least 5n/6 edges. The conjecture has been verified through 90 vertices and proved when every face has size at most 8. Exact unresolved remainder: Prove that every Barnette graph has a spanning cycle, or exhibit a cubic, 3-connected, bipartite planar graph without one. TheoremDB corpus searches for Barnette returned no duplicate target.",
  "relevance": "This is the dated publication status for the canonical target Barnette’s conjecture.",
  "relevance_source": "recorded",
  "body": "The problem was checked as open on 2026-08-01.\n\nThe strongest neighboring result found in the cited sources is: Every n-vertex Barnette graph has a subhamiltonian cycle containing at least 5n/6 edges. The conjecture has been verified through 90 vertices and proved when every face has size at most 8.\n\nThe exact unresolved remainder is: Prove that every Barnette graph has a spanning cycle, or exhibit a cubic, 3-connected, bipartite planar graph without one. TheoremDB corpus searches for Barnette returned no duplicate target.\n\nA complete resolution must meet the following acceptance conditions:\n- Prove that every finite simple cubic, 3-connected, bipartite planar graph has a Hamiltonian cycle.\n- Or give an explicit graph satisfying all four hypotheses, together with a rigorous certificate that it has no Hamiltonian cycle.",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.4230/LIPIcs.GD.2025.6",
      "locator": "M. A. Bekos, M. Kaufmann, and M. Pfister, Approximating Barnette’s Conjecture, 33rd International Symposium on Graph Drawing and Network Visualization, LIPIcs 357, Article 6, pp. 6:1-6:7 (2025). Abstract, Introduction, Theorem 1, and Section 5"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.4230/LIPIcs.GD.2025.6",
    "locator": "M. A. Bekos, M. Kaufmann, and M. Pfister, Approximating Barnette’s Conjecture, 33rd International Symposium on Graph Drawing and Network Visualization, LIPIcs 357, Article 6, pp. 6:1-6:7 (2025). Abstract, Introduction, Theorem 1, and Section 5"
  },
  "relations": [
    {
      "slug": "R1386",
      "title": "Strongest checked neighboring result",
      "object_type": "claim",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "R1384",
      "title": "Dated source and duplicate audit",
      "object_type": "attempt",
      "relation": "evidences",
      "direction": "incoming"
    },
    {
      "slug": "R1385",
      "title": "Work at the unresolved boundary",
      "object_type": "attempt",
      "relation": "addresses",
      "direction": "incoming"
    },
    {
      "slug": "barnette-conjecture",
      "title": "barnette conjecture",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}