{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:petersen-coloring-conjecture",
  "title": "Petersen Coloring Conjecture",
  "canonical_statement": "Jaeger conjectured that every bridgeless cubic graph $G$ admits a Petersen coloring: a map $\\varphi\\colon E(G)\\to E(P)$ into the edges of the Petersen graph $P$ such that, for every vertex $v$ of $G$, the three edges at $v$ are sent to three edges meeting at a common vertex of $P$. Equivalently, by Jaeger's theorem, every bridgeless cubic graph has a normal 5-edge-coloring. The conjecture implies both the Berge-Fulkerson conjecture and the 5-cycle-double-cover conjecture. False: there is an explicit simple connected bridgeless cubic graph on $112$ vertices, of girth five and edge- and vertex-connectivity three, with no Petersen coloring.",
  "plain_summary": "VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
  "current_status": "resolved",
  "solution_events": [
    {
      "id": "vibemathed:petersen-coloring-conjecture:event",
      "problem_id": "vibemathed:petersen-coloring-conjecture",
      "type": "disproved",
      "status": "resolved",
      "occurred_at": "2026-07-23T00:00:00.000Z",
      "title": "arXiv",
      "summary": "This preprint was not the first disproof. A 68-vertex counterexample was posted to X on 23 July 2026 by @NeuralReformist, credited to GPT-5.6 Sol Ultra, sixteen days earlier. This site decoded that sparse6 string and checked it independently: 68 vertices, 102 edges, simple, cubic, connected, bridgeless, girth five, and no Petersen coloring under the same encoder used for the 112-vertex graph. Whether the two are independent is unknown - the preprint does not cite the post. The headline axes still record the preprint, the only complete writeup with certificates.\n\nThe implication runs one way: the Petersen coloring conjecture implies Berge-Fulkerson and the 5-cycle-double-cover conjecture, so refuting it leaves both of those open. The paper does not claim 112 is minimum, and it supplies a second, nonisomorphic $D_3$-symmetric 112-vertex counterexample. Combined with a theorem of Ma, Mattiolo, Steffen and Wolf, one counterexample yields infinitely many.",
      "ai_contribution": "ai_co_developed",
      "attempt_ids": [
        "vibemathed:petersen-coloring-conjecture:attempt"
      ],
      "method_family_ids": [
        "vibemathed:petersen-coloring-conjecture:method"
      ],
      "source_assertion_ids": [
        "vibemathed:petersen-coloring-conjecture:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:petersen-coloring-conjecture:attempt",
      "problem_id": "vibemathed:petersen-coloring-conjecture",
      "model": "ChatGPT",
      "provider": "OpenAI",
      "attempted_at": "2026-07-23T00:00:00.000Z",
      "human_collaborators": [
        "Bryce Putman"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The paper's \"Computational provenance and responsibility\" section states in full: \"OpenAI language-model systems were used extensively in the discovery, computational search, verification, and preparation of this work. The author reviewed the final claims and artifacts and accepts responsibility for the contents.\" No product name, model version or division of labour is given, so which of discovery, search, verification and write-up the model actually carried is not recoverable from the paper. The catalog records the model as ChatGPT because that is this catalog's convention for an unnamed OpenAI system; the paper itself names none.",
      "failed_routes": [],
      "artifacts": [
        {
          "label": "First known counterexample: 68 vertices, posted to X, 23 July 2026",
          "url": "https://x.com/NeuralReformist/status/2080153035045839069",
          "kind": "announcement",
          "license": null
        },
        {
          "label": "Open Problem Garden: Petersen coloring conjecture",
          "url": "http://www.openproblemgarden.org/op/petersen_coloring_conjecture",
          "kind": "problem-record",
          "license": null
        },
        {
          "label": "Zenodo artifact record v1.1.0 (CNFs, DRAT certificates, hashes)",
          "url": "https://doi.org/10.5281/zenodo.21845291",
          "kind": "code",
          "license": null
        },
        {
          "label": "Ma, Mattiolo, Steffen, Wolf - Sets of r-graphs that color all r-graphs",
          "url": "https://doi.org/10.1007/s00493-025-00144-4",
          "kind": "paper",
          "license": null
        }
      ],
      "sources": [
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2608.10012",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Petersen Coloring Conjecture",
          "url": "https://vibemathed.com/problem/petersen-coloring-conjecture",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:petersen-coloring-conjecture:verification",
      "problem_id": "vibemathed:petersen-coloring-conjecture",
      "solution_event_id": "vibemathed:petersen-coloring-conjecture:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "Reproduced here on 12 August 2026, independently of the paper's certificates. The 112-vertex graph was rebuilt from the appendix edge table, and the SHA-256 of its normalized sorted edge list reproduces the digest in Theorem 1.1 exactly, pinning the object under review to the one claimed. Every property in that theorem was rederived: 112 vertices, 168 edges, simple, cubic, connected, bridgeless, girth five, connectivity three. Non-existence of a Petersen coloring was then re-proved with a CNF encoding written here from the definition - each edge carries one of the 15 edges of $KG(5,2)$, each vertex selects one of the 10 target stars, the three edges at a vertex land in that star and are pairwise distinct - and solved with CaDiCaL via PySAT. UNSAT. That re-derives the unsatisfiability rather than replaying the shipped DRAT certificates, and the encoder was written without reference to the paper's: the same 3640 variables, forced by the problem shape, but 31,360 clauses against their 68,324. It ran twice in separate processes with identical results. Six controls - $K_4$, $K_{3,3}$, the 3-cube, the prism, Desargues and the Petersen graph itself - all came back satisfiable through the same encoder. Petersen is the important one, being a snark: a coloring for it rules out the encoder having quietly tested 3-edge-colorability. Not checked: the second $D_3$-symmetric counterexample, the normal-5-edge-coloring formulation, and the DRAT proofs. Four-day-old arXiv preprint, unrefereed.",
      "sources": [
        {
          "label": "VibeMathed: Petersen Coloring Conjecture",
          "url": "https://vibemathed.com/problem/petersen-coloring-conjecture",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2608.10012",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "arXiv",
      "url": "https://arxiv.org/abs/2608.10012",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Petersen Coloring Conjecture",
      "url": "https://vibemathed.com/problem/petersen-coloring-conjecture",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:petersen-coloring-conjecture:signal:watch_only",
      "problem_id": "vibemathed:petersen-coloring-conjecture",
      "kind": "watch_only",
      "reason": "Current public evidence does not meet the default replay threshold.",
      "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"
}