{
  "abstract": "Erdős and Gallai in 1959 proved the seminal result that every $n$-vertex graph with no cycle of length at least $2t+2$ has at most $\\tfrac{2t+1}{2}(n-1)$ edges. We prove the extension that, for every sufficiently large $t$, the same quantity is also the sharp extremal bound for graphs with no $t$ consecutive even cycle lengths, resolving a conjecture of Verstraëte. Thus, at the Erdős-Gallai threshold, forcing an entire interval of even cycle lengths costs no more than forcing its longest member. More precisely, every $n$-vertex graph $G$ with $e(G)\\ge \\tfrac{(2t+1)(n-1)}2$ $\\bullet$ either contains $t$ consecutive even cycle lengths, $\\bullet$ or equality holds and $G$ is connected with every block isomorphic to $K_{2t+1}$. As consequences, for every sufficiently large even $k$ we determine the sharp edge thresholds forcing a cycle of length $0\\pmod k$ or $2\\pmod k$, answering questions of Bai, Grzesik, Li, and Prorok and of Gao, Li, Ma and Xie, respectively, for sufficiently large even $k$. The proof develops a stability-enhanced sublinear expander method. Its main new ingredient is a dense-case decomposition that recovers the lengths lost in the expander extraction by combining a flexible dense core with rooted cycle families in the vertices outside the core.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.27404",
  "author_total": 5,
  "authors": [
    {
      "name": "Yaobin Chen"
    },
    {
      "name": "Hong Liu"
    },
    {
      "name": "Xia Wang"
    },
    {
      "name": "Xin Wei"
    },
    {
      "name": "Fan Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4d9z48vmd1th0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.27404",
  "fulltext_url": null,
  "id": "4d9z48vmd1th0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.27404",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7204521977",
  "pdf_url": "https://arxiv.org/pdf/2608.27404",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4d9z48vmd1th0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2608.27404",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.27404"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The Erdős-Gallai bound for consecutive even cycle lengths",
  "title_text": "The Erdős-Gallai bound for consecutive even cycle lengths",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/4d9z48vmd1th0",
  "venue": "arXiv",
  "versions": []
}