{
  "abstract": "For integers $k\\ge3$ and $0\\le \\ell<k$, we call a graph $(\\ell\\bmod k)$-cycle-restricted if every cycle has length congruent to $\\ell$ modulo $k$. Akbari, Jamaali, Mahmoody, and Seyed Fakhari [On the size of graphs whose cycles have length divisible by a fixed integer, Australas. J. Combin. 45 (2009), 67--72] determined the maximum number of edges in a $2$-connected $(0\\bmod k)$-cycle-restricted graph and characterized the graphs attaining equality. In this paper, we determine the maximum number of edges in a $2$-connected $(\\ell\\bmod k)$-cycle-restricted graph for every $\\ell \\neq 0$, and completely characterize all graphs attaining equality. Combined with the result of Akbari et al., our result gives a complete solution for all residue classes modulo $k$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.15151",
  "author_total": 1,
  "authors": [
    {
      "name": "Takahiro Ueoro"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e8cnvmq6s89n6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.15151",
  "fulltext_url": null,
  "id": "e8cnvmq6s89n6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.15151",
  "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": "W7221784358",
  "pdf_url": "https://arxiv.org/pdf/2610.15151",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e8cnvmq6s89n6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.15151",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.15151"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Extremal graphs with cycle lengths in a fixed residue class",
  "title_text": "Extremal graphs with cycle lengths in a fixed residue class",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/e8cnvmq6s89n6",
  "venue": "arXiv",
  "versions": []
}