{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1782",
  "content_hash": null,
  "slug": "total-coloring-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 graph satisfies χ_T(G) ≤ Δ(G)+2⌈n/(Δ(G)+1)⌉. The conjectured Δ(G)+2 bound holds for sufficiently large graphs whose minimum degree exceeds half their order by a fixed proportion. Exact unresolved remainder: Remove all density and structural assumptions and prove the Δ(G)+2 bound for every finite simple graph, or find a graph requiring Δ(G)+3 colors. A 2020 arXiv manuscript claims a proof, while subsequent peer-reviewed papers continue to treat the unrestricted statement as open. TheoremDB corpus searches returned no duplicate target.",
  "relevance": "This is the dated publication status for the canonical target The Total Coloring 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 graph satisfies χ_T(G) ≤ Δ(G)+2⌈n/(Δ(G)+1)⌉. The conjectured Δ(G)+2 bound holds for sufficiently large graphs whose minimum degree exceeds half their order by a fixed proportion.\n\nThe exact unresolved remainder is: Remove all density and structural assumptions and prove the Δ(G)+2 bound for every finite simple graph, or find a graph requiring Δ(G)+3 colors. A 2020 arXiv manuscript claims a proof, while subsequent peer-reviewed papers continue to treat the unrestricted statement as open. TheoremDB corpus searches returned no duplicate target.\n\nA complete resolution must meet the following acceptance conditions:\n- Prove χ_T(G) ≤ Δ(G)+2 for every finite simple graph G.\n- Or give an explicit finite simple graph G and a rigorous lower-bound certificate showing χ_T(G) ≥ Δ(G)+3.",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1002/jgt.23268",
      "locator": "A. Dalal, J. McDonald, and S. Shan, Total Coloring Graphs With Large Maximum Degree, Journal of Graph Theory 110(3) (2025), 249-262. Abstract and main theorems"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1002/jgt.23268",
    "locator": "A. Dalal, J. McDonald, and S. Shan, Total Coloring Graphs With Large Maximum Degree, Journal of Graph Theory 110(3) (2025), 249-262. Abstract and main theorems"
  },
  "relations": [
    {
      "slug": "R1781",
      "title": "Strongest checked neighboring result",
      "object_type": "claim",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "R1779",
      "title": "Dated source and duplicate audit",
      "object_type": "attempt",
      "relation": "evidences",
      "direction": "incoming"
    },
    {
      "slug": "R1780",
      "title": "Work at the unresolved boundary",
      "object_type": "attempt",
      "relation": "addresses",
      "direction": "incoming"
    },
    {
      "slug": "total-coloring-conjecture",
      "title": "total coloring conjecture",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}