{
  "abstract": "Bernshteyn, Kostochka, and Zhu (2020) introduced the notion of fractional DP-coloring, which generalizes both fractional coloring and fractional list coloring. Among several foundational results, they proved that every $d$-degenerate bipartite graph $G$ satisfies $χ_f^{\\mathrm{DP}} \\le (1 + o(1))\\frac{d}{\\log d}$, and that this bound is optimal---a stark contrast to ordinary fractional coloring. In this paper, we extend this upper bound to all $d$-degenerate triangle-free graphs, proving that $χ_f^{\\mathrm{DP}} \\le (4 + o(1))\\frac{d}{\\log d}$. This generalizes a recent result of Martinsson and Steiner (2025) for ordinary fractional coloring. We derive this result as a corollary of a more general upper bound concerning locally sparse graph orderings. Specifically, a $d$-degenerate graph $G$ is left $k$-locally-sparse if it admits a degeneracy ordering in which, for every vertex $v$, the subgraph induced by its back-neighbors contains at most $k$ edges. We show that if a $d$-degenerate graph $G$ is left $\\frac{d^2}{f}$-locally-sparse, then \\[ χ_f^{\\mathrm{DP}}(G) \\le (8 + o(1))\\frac{d}{\\log f}. \\] This immediately yields an identical upper bound on the ordinary fractional chromatic number $χ_f(G)$, improving upon the leading constants of previously known bounds. Additionally, we establish the asymptotic sharpness of this result up to the leading constant. For any $1 \\ll f \\le d^2$, we construct $d$-degenerate graphs that are left $\\frac{d^2}{f}$-locally-sparse and satisfy $χ_f(G) \\ge (1 - o(1))\\frac{d}{\\log f}$. Finally, as applications of our main theorem, we obtain improved upper bounds on the fractional DP-chromatic number of $d$-degenerate $K_{1,t,t}$-free graphs, as well as $K_{t,t,t}$-free graphs with maximum degree $Δ$. Notably, these bounds improve upon existing results even in the setting of ordinary fractional coloring.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.10978",
  "author_total": 3,
  "authors": [
    {
      "name": "Abhishek Dhawan"
    },
    {
      "name": "Huy Nguyen"
    },
    {
      "name": "Rohan A. Rathi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fa0n6yprtwgb0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.10978",
  "fulltext_url": null,
  "id": "fa0n6yprtwgb0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.10978",
  "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",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7212299348",
  "pdf_url": "https://arxiv.org/pdf/2609.10978",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fa0n6yprtwgb0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:11Z",
      "native_id": "2609.10978",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.10978"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Fractional DP-colorings of $d$-degenerate locally sparse graphs",
  "title_text": "Fractional DP-colorings of d-degenerate locally sparse graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:11Z",
  "url": "https://research.draconitic.com/paper/fa0n6yprtwgb0",
  "venue": "arXiv",
  "versions": []
}