{
  "abstract": "The Erdos-Gyarfas conjecture states that every graph with minimum degree at least 3 contains a cycle whose length is a power of two. We prove by a SAT-based exhaustive search, certified by DRAT proofs, that every graph with minimum degree at least 3 on at most 23 vertices contains a cycle of length 4 or a cycle of length 8; consequently any counterexample has at least 24 vertices, improving the previously published bound of 16, and the smallest graph of minimum degree 3 with no 4-cycle and no 8-cycle has exactly 24 vertices. We show that the lemma underlying Exoo's 450-vertex construction for the bound f(5) at most 450 is false: the Tutte-Coxeter graph contains 8-cycles alternating between outer and chord edges, and the graph as specified contains 32-cycles. We repair the construction and verify the corrected graph, so the bound stands. We also give an exact window calculus for vertex-replacement constructions, prove that f(k) is at most 15 times the order of the smallest known cubic graph of girth 2 to the power (k-2) plus 1 for all k at least 4 (in particular f(6) is at most 32640, the first bound for f(6)), and show that Exoo's 78-vertex witness for f(4) at most 78 is optimal among gadget designs on bases with at most 12 vertices. All graphs, scripts and certificates are archived at doi:10.5281/zenodo.22180583.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.04686",
  "author_total": 1,
  "authors": [
    {
      "name": "Daniel Garcia"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/98s8g3va0jnwy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.04686",
  "fulltext_url": null,
  "id": "98s8g3va0jnwy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.04686",
  "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": "W7223731086",
  "pdf_url": "https://arxiv.org/pdf/2609.04686",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/98s8g3va0jnwy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:18Z",
      "native_id": "2609.04686",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.04686"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Small graphs without power-of-two cycles: a lower bound of 24, a correction to a construction of Exoo, and explicit bounds for f(k)",
  "title_text": "Small graphs without power-of-two cycles: a lower bound of 24, a correction to a construction of Exoo, and explicit bounds for f(k)",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:18Z",
  "url": "https://research.draconitic.com/paper/98s8g3va0jnwy",
  "venue": "arXiv",
  "versions": []
}