{
  "abstract": "For a positive rational number $x=p/q$, where $p$ and $q$ are coprime positive integers, let $\\operatorname{den}(x)=q$. For a positive integer $n$, let $\\operatorname{sd}(n)=\\max\\{\\operatorname{den}(χ_f(G)): |V(G)|=n\\}$. Let $D(n)=2^{-n}(n+1)^{(n+1)/2}$. It was proved by Alon, Tuza and Voigt that $\\operatorname{sd}(n)\\le D(n)$. In this paper, we prove that $\\operatorname{sd}(n)\\le D(n)n^{-\\frac12\\log_2 n+\\frac14\\log_2\\log_2 n+O(1)}$ for all sufficiently large $n$. For the lower bound on $\\operatorname{sd}(n)$, it was recently proved by Bonamy, Hylasová, Kaiser and Sereni that $\\limsup_{n\\to\\infty}\\operatorname{sd}(n)^{1/n}\\ge\\sqrt{2}$. We prove that $\\limsup_{n\\to\\infty}\\operatorname{sd}(n)^{1/n}\\ge 3^{1/3}>\\sqrt{2}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.41771",
  "author_total": 2,
  "authors": [
    {
      "name": "Xinbo Xu"
    },
    {
      "name": "Xuding Zhu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5ve16kxyvc9ze.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.41771",
  "fulltext_url": null,
  "id": "5ve16kxyvc9ze",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.41771",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2610.41771",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5ve16kxyvc9ze.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2610.41771",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.41771"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Bounds on the maximum denominator of fractional chromatic numbers",
  "title_text": "Bounds on the maximum denominator of fractional chromatic numbers",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:46Z",
  "url": "https://research.draconitic.com/paper/5ve16kxyvc9ze",
  "venue": "arXiv",
  "versions": []
}