{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:kotzig-perfect-1-factorisation-asymptotic",
  "title": "Kotzig's Perfect 1-Factorisation Conjecture, Asymptotically",
  "canonical_statement": "Kotzig conjectured that for every even $n \\ge 4$ the complete graph $K_n$ decomposes into $n-1$ perfect matchings such that every pair of them forms a Hamilton cycle. An asymptotic version holds: $K_n$ decomposes into $n-1$ perfect matchings of which $(1-o(1))n$ have the property that any pair forms a Hamilton cycle.",
  "plain_summary": "VibeMathed reports this item as partial. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
  "current_status": "partial",
  "solution_events": [
    {
      "id": "vibemathed:kotzig-perfect-1-factorisation-asymptotic:event",
      "problem_id": "vibemathed:kotzig-perfect-1-factorisation-asymptotic",
      "type": "proved",
      "status": "partial",
      "occurred_at": "2026-07-10T00:00:00.000Z",
      "title": "arXiv:2607.09459 - The perfect 1-factorisation conjecture holds asymptotically",
      "summary": "asymptotic form only; Kotzig's conjecture itself remains far from solved",
      "ai_contribution": "ai_assisted",
      "attempt_ids": [
        "vibemathed:kotzig-perfect-1-factorisation-asymptotic:attempt"
      ],
      "method_family_ids": [
        "vibemathed:kotzig-perfect-1-factorisation-asymptotic:method"
      ],
      "source_assertion_ids": [
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      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:kotzig-perfect-1-factorisation-asymptotic:attempt",
      "problem_id": "vibemathed:kotzig-perfect-1-factorisation-asymptotic",
      "model": "ChatGPT 5.4",
      "provider": "OpenAI",
      "attempted_at": "2026-07-10T00:00:00.000Z",
      "human_collaborators": [
        "Yangyang Cheng",
        "Amedeo Sgueglia"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The authors state that the construction in Theorem 1.3, a generalisation of the one in their introduction, was generated by ChatGPT 5.4, which also supplied a correct but long proof of it. The proof they present is a cleaner and substantially different one of their own, and the proof of the main theorem, Theorem 1.2, was obtained entirely by the authors. The model's construction is nonetheless load-bearing: the main result is built by deleting a random subset of the matchings it produces.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "arXiv:2607.09459 - The perfect 1-factorisation conjecture holds asymptotically",
          "url": "https://arxiv.org/abs/2607.09459",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Kotzig's Perfect 1-Factorisation Conjecture, Asymptotically",
          "url": "https://vibemathed.com/problem/kotzig-perfect-1-factorisation-asymptotic",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:kotzig-perfect-1-factorisation-asymptotic:verification",
      "problem_id": "vibemathed:kotzig-perfect-1-factorisation-asymptotic",
      "solution_event_id": "vibemathed:kotzig-perfect-1-factorisation-asymptotic:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "arXiv preprint that routes through a robust-expander result of Kuhn and Osthus; not yet peer-reviewed.",
      "sources": [
        {
          "label": "VibeMathed: Kotzig's Perfect 1-Factorisation Conjecture, Asymptotically",
          "url": "https://vibemathed.com/problem/kotzig-perfect-1-factorisation-asymptotic",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "arXiv:2607.09459 - The perfect 1-factorisation conjecture holds asymptotically",
          "url": "https://arxiv.org/abs/2607.09459",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "arXiv:2607.09459 - The perfect 1-factorisation conjecture holds asymptotically",
      "url": "https://arxiv.org/abs/2607.09459",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Kotzig's Perfect 1-Factorisation Conjecture, Asymptotically",
      "url": "https://vibemathed.com/problem/kotzig-perfect-1-factorisation-asymptotic",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "expand",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:kotzig-perfect-1-factorisation-asymptotic:signal:recent_partial_result",
      "problem_id": "vibemathed:kotzig-perfect-1-factorisation-asymptotic",
      "kind": "recent_partial_result",
      "reason": "A source-reported partial result may support bounded expansion.",
      "generated_at": "2026-09-13T16:28:40.178Z"
    }
  ],
  "uncertainties": [
    "VibeMath has not independently verified the mathematical claim.",
    "AI-attempt independence and training-data exposure are unknown unless explicitly documented.",
    "VibeMath has not independently audited the mathematical statement, proof, or novelty claim."
  ],
  "generated_at": "2026-09-13T16:28:40.178Z"
}