{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1780",
  "content_hash": null,
  "slug": "total-coloring-conjecture-attempt-open-remainder",
  "type": "attempt",
  "title": "Work at the unresolved boundary",
  "summary": "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": "Turns the remaining uncertainty in The Total Coloring Conjecture into a checkable research target.",
  "relevance_source": "recorded",
  "body": "Research should address this boundary directly: 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 claimed resolution should satisfy every item below and preserve the statement's exact quantifiers and normalizations:\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": "open_strategy",
  "evidence_grade": "self_reported",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "attempt",
    "citation": {
      "url": "https://doi.org/10.1002/jgt.23268",
      "locator": "TheoremDB editorial route recorded 2026-08-01"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1002/jgt.23268",
    "locator": "TheoremDB editorial route recorded 2026-08-01"
  },
  "relations": [
    {
      "slug": "R1782",
      "title": "Current status and exact unresolved remainder",
      "object_type": "claim",
      "relation": "addresses",
      "direction": "outgoing"
    },
    {
      "slug": "total-coloring-conjecture",
      "title": "total coloring conjecture",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}