{
  "abstract": "In 1988, Jaeger conjectured that every bridgeless cubic graph $G$ admits a Petersen coloring; that is, a map $E(G) \\to E(P)$ mapping any two adjacent edges of $G$ to two adjacent edges of the Petersen graph $P$. A positive resolution of Jaeger's conjecture would have immediately resolved several other famous and long-standing problems in graph theory. In July 2026, a 68-vertex counterexample was announced on X. Shortly afterwards, Putman independently presented two non-isomorphic 112-vertex counterexamples, relying solely on computer-assisted verification. In this paper, we present two counterexamples of order $52$, currently the smallest known, and provide a purely theoretical proof. In the second part, we construct an infinite family of cyclically $4$-edge-connected cubic graphs without a Petersen coloring for every even order at least $60$. Additionally, through computational verification, we show that any counterexample must have order at least $40$. Moreover, we show that our counterexamples provide a negative answer to other related problems. Finally, we conclude the paper by discussing key open problems and highlighting avenues for future work.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.10028",
  "author_total": 6,
  "authors": [
    {
      "name": "Jan Goedgebeur"
    },
    {
      "name": "Jorik Jooken"
    },
    {
      "name": "Edita Máčajová"
    },
    {
      "name": "Davide Mattiolo"
    },
    {
      "name": "Giuseppe Mazzuoccolo"
    },
    {
      "name": "Nikolay Ulyanov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6engn2mnvphkn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.10028",
  "fulltext_url": null,
  "id": "6engn2mnvphkn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.10028",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7202207073",
  "pdf_url": "https://arxiv.org/pdf/2608.10028",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6engn2mnvphkn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2608.10028",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.10028"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Disproving the Petersen Coloring Conjecture: Theoretical Analysis and an Infinite Family of Counterexamples",
  "title_text": "Disproving the Petersen Coloring Conjecture: Theoretical Analysis and an Infinite Family of Counterexamples",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/6engn2mnvphkn",
  "venue": "arXiv",
  "versions": []
}