{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1779",
  "content_hash": null,
  "slug": "total-coloring-conjecture-attempt-dated-source-audit",
  "type": "attempt",
  "title": "Dated source and duplicate audit",
  "summary": "The exact target, equivalent terminology, and 2025-2026 status evidence were checked on 2026-08-01. Strongest checked 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. 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": "Documents why The Total Coloring Conjecture was treated as a distinct open target on 2026-08-01.",
  "relevance_source": "recorded",
  "body": "The audit ran on 2026-08-01 across the cited primary literature and problem collections, Crossref, arXiv, and stable publisher records, TheoremDB published, prospecting, packet, formal, and retired corpus. It checked the exact statement, its named or normalized variants, recent status evidence, and the controlled TheoremDB corpus.\n\nQueries included:\n- Total Coloring Conjecture Delta plus 2 general graph open 2026\n- total coloring large minimum degree 2025\n- total chromatic number Delta plus 2 conjecture\n\nThe exact target, equivalent terminology, and 2025-2026 status evidence were checked on 2026-08-01. Strongest checked 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. 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.",
  "status": "completed",
  "evidence_grade": "sourced",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "attempt",
    "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": "R1782",
      "title": "Current status and exact unresolved remainder",
      "object_type": "claim",
      "relation": "evidences",
      "direction": "outgoing"
    },
    {
      "slug": "total-coloring-conjecture",
      "title": "total coloring conjecture",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}