{
  "abstract": "We study graphs containing no cycle whose length is divisible by $3$ or congruent to $4$ modulo $6$. We prove that every such $n$-vertex graph $G$, where $n \\ge 2$, satisfies $e(G) \\le (11/8)n-7/4$. Moreover, equality holds if and only if $n=8k+2$ for some nonnegative integer $k$ and $G$ is isomorphic to the explicitly constructed graph $H_k$. We also construct, for every $n\\geq2$, an $n$-vertex graph with $\\left\\lfloor (11/8)n-7/4 \\right\\rfloor$ edges satisfying the same cycle restriction. Consequently, this is the exact maximum number of edges for every $n \\ge 2$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.06698",
  "author_total": 1,
  "authors": [
    {
      "name": "Takahiro Ueoro"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bjxbtdknew8y9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.06698",
  "fulltext_url": null,
  "id": "bjxbtdknew8y9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.06698",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7202112684",
  "pdf_url": "https://arxiv.org/pdf/2608.06698",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bjxbtdknew8y9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2608.06698",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.06698"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "On graphs without cycles of length $0$ modulo $3$ or $4$ modulo $6$",
  "title_text": "On graphs without cycles of length $0$ modulo $3$ or $4$ modulo $6$",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/bjxbtdknew8y9",
  "venue": "arXiv",
  "versions": []
}