{
  "abstract": "For $k\\ge2$, a graph is $C_{2k}$-connected if every two distinct edges lie together on an even cycle of length at most $2k$ in that graph. Let $f_k(n,m)$ be the largest number of edges in a $C_{2k}$-connected subgraph that every $n$-vertex graph with $m$ edges must contain. In this paper, we study two cycle-connected problems raised by Duke, Erdős and Rödl. First, Duke, Erdős and Rödl showed that $f_3(n,\\lfloor n^{2-β}\\rfloor)=Θ(n^{2-3β})$ for every fixed $0< β<1/2$ and ask for the case of $β\\ge 1/2$. We solve this problem in a stronger form by determining that the optimal range of $β$ is $(0,2/3]$. Second, we prove that $f_2(n,\\binom n2-s)=(1/(2e)+o(1))n^3/s$ whenever $\\frac{s}{n}\\to\\infty$ and $s=o(n^2)$, determining the leading constant in an estimate of Duke, Erdős, and Rödl. The first result uses a finite-field construction and a count of three-edge paths. The second uses the known structure of $C_4$-connected graphs and estimates for complete bipartite subgraphs in a random graph.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.34307",
  "author_total": 2,
  "authors": [
    {
      "name": "Guorong Gao"
    },
    {
      "name": "Run Zhao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4jfjsc7rs0t1g.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.34307",
  "fulltext_url": null,
  "id": "4jfjsc7rs0t1g",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.34307",
  "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": "W7223878287",
  "pdf_url": "https://arxiv.org/pdf/2610.34307",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4jfjsc7rs0t1g.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.34307",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.34307"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Sharp bounds for Duke--Erdős--Rödl problems on cycle-connected subgraphs",
  "title_text": "Sharp bounds for Duke--Erdős--Rödl problems on cycle-connected subgraphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/4jfjsc7rs0t1g",
  "venue": "arXiv",
  "versions": []
}