{
  "abstract": "The Erdős-Gyárfás conjecture posits that every graph with minimum degree at least three contains a cycle of length some power of two. We prove a few simple structural properties for any minimal counter-example to this conjecture. In particular, the fraction of its vertices of degree three must be greater than $2/3$, thus improving on the prior bound of $4/7$ (Carr, 2026). Furthermore, it is either biconnected or the $1$-clique-sum of two biconnected graphs. By exploiting some of these properties, we were able to verify the conjecture for every graph of order at most $40$, every bipartite graph of order at most $66$, and every cubic graph of order at most $48$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.28594",
  "author_total": 2,
  "authors": [
    {
      "name": "Guillaume Ducoffe"
    },
    {
      "name": "Bogdan Dumitru"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f68zcx1q9370t.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.28594",
  "fulltext_url": null,
  "id": "f68zcx1q9370t",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.28594",
  "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",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7214319329",
  "pdf_url": "https://arxiv.org/pdf/2609.28594",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f68zcx1q9370t.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.28594",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.28594"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Towards a more structured search for Erdős-Gyárfás counter-examples",
  "title_text": "Towards a more structured search for Erdős-Gyárfás counter-examples",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/f68zcx1q9370t",
  "venue": "arXiv",
  "versions": []
}