{
  "abstract": "We determine the asymptotic minimum-degree threshold for decompositions into cycles of a fixed odd length. We prove that, for each odd integer $\\ell\\ge3$, every $\\eps>0$ and sufficiently large $n$, every $n$-vertex $C_\\ell$-divisible graph $G$ with minimum degree $δ(G)\\ge (\\ell/(2\\ell-2)+\\eps)n$ has a $C_\\ell$-decomposition, where $C_{\\ell}$ denotes the cycle of length $\\ell$. This settles the remaining cases $\\ell=5,7,\\ldots,71$ and completes the determination of the thresholds for all odd cycles. Our main ingredient is a sharp fractional decomposition theorem for weighted triangles: for every real $a\\ge 1$,every $n$-vertex graph $G$ with $δ(G)\\ge \\left(\\frac12+\\frac{1}{2(a+1)}\\right)n$ admits a fractional decomposition into triangles with edge weights $a,1,1$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.23471",
  "author_total": 3,
  "authors": [
    {
      "name": "Jianfeng Hou"
    },
    {
      "name": "Caiyun Hu"
    },
    {
      "name": "Hezhi Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bjvbn89zsx1dn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.23471",
  "fulltext_url": null,
  "id": "bjvbn89zsx1dn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.23471",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7222966514",
  "pdf_url": "https://arxiv.org/pdf/2610.23471",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bjvbn89zsx1dn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:38Z",
      "native_id": "2610.23471",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.23471"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "On decomposition thresholds for odd cycles",
  "title_text": "On decomposition thresholds for odd cycles",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/bjvbn89zsx1dn",
  "venue": "arXiv",
  "versions": []
}