{
  "abstract": "For integers $k\\ge 1$ and $n\\ge k+2$, let $g_k(n)$ be the maximum number of edges in an $n$-vertex graph containing no cycle with a vertex incident with at least $k$ chords. Erdős conjectured that $g_k(n)=(k+1)(n-k-1)$ for $n\\ge 2k+2$. Lewin found a counterexample. Bollobás later conjectured that there exists a function $n(k)$ such that $g_k(n)=(k+1)(n-k-1)$ for all $n\\ge n(k)$. Jiang confirmed this by proving the formula for all $n\\ge3k+3$ when $k\\ge1$. In this paper, we determine $g_k(n)$ completely. For all $k\\ge1$ and $n\\ge k+2$, we prove $g_k(n)=\\big\\{\\lfloor\\frac{(k+1)n}{2}\\rfloor,\\max\\{a(n-a)+\\lfloor\\frac{a(k+1-a)}{2}\\rfloor : a\\in\\mathbb Z,\\; \\lfloor {(k+1)}/{2}\\rfloor+1\\le a\\le k+1\\}\\big\\}$. For $k\\ge2$, we prove $g_k(n)=(k+1)(n-k-1)$ when $n\\ge \\lceil(5k+1)/2\\rceil$, and this threshold is sharp. Our proof builds on the method developed by Ma and the second author in [Ma and Ning, 2020].",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.15330",
  "author_total": 2,
  "authors": [
    {
      "name": "Xiaozheng Chen"
    },
    {
      "name": "Bo Ning"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bcwx55z5wrk9c.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.15330",
  "fulltext_url": null,
  "id": "bcwx55z5wrk9c",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.15330",
  "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": "W7213269871",
  "pdf_url": "https://arxiv.org/pdf/2609.15330",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bcwx55z5wrk9c.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.15330",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.15330"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "On Erdős Problem 767: Cycles with Chords",
  "title_text": "On Erdős Problem 767: Cycles with Chords",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/bcwx55z5wrk9c",
  "venue": "arXiv",
  "versions": []
}