{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1781",
  "content_hash": null,
  "slug": "total-coloring-conjecture-claim-literature-frontier",
  "type": "claim",
  "title": "Strongest checked neighboring result",
  "summary": "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.",
  "relevance": "Locates the present research frontier immediately below The Total Coloring Conjecture.",
  "relevance_source": "recorded",
  "body": "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\nThis leaves the following boundary unresolved: 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. The distinction is retained here so a restricted theorem, finite computation, or neighboring case is not presented as a solution of the full target.",
  "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": "R1782",
      "title": "Current status and exact unresolved remainder",
      "object_type": "claim",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "total-coloring-conjecture",
      "title": "total coloring conjecture",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}