{
  "abstract": "A strong odd coloring of a graph is a proper vertex coloring such that for every vertex $v$, each color appearing on the neighborhood of $v$ appears an odd number of times. We investigate strong odd colorings of sparse graphs, where sparsity of a graph $G$ is measured in terms of maximum average degree, denoted $\\operatorname{mad}(G)$. We prove that the optimal threshold on the maximum average degree guaranteeing a strong odd $c$-coloring is $4-Θ(\\frac{1}{c})$ as $c\\to\\infty$. Moreover, for every $c\\geq 5$, this threshold is strictly smaller than the threshold for odd $c$-coloring. We also determine the sharp threshold on the maximum average degree guaranteeing a strong odd $5$-coloring: every graph $G$ with $\\operatorname{mad}(G)<\\frac{7}{3}$ has a strong odd $5$-coloring, and this bound is sharp. In addition, we show that every graph $G$ with $\\operatorname{mad}(G)<\\frac{11}{5}$ and no induced $5$-cycle has a strong odd $4$-coloring. Note that the $5$-cycle has maximum average degree 2 and no strong odd $4$-coloring.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.34874",
  "author_total": 2,
  "authors": [
    {
      "name": "Ilkyoo Choi"
    },
    {
      "name": "Hojin Chu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8bzs9jtfv4m1a.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.34874",
  "fulltext_url": null,
  "id": "8bzs9jtfv4m1a",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.34874",
  "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",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7223838598",
  "pdf_url": "https://arxiv.org/pdf/2610.34874",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8bzs9jtfv4m1a.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.34874",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.34874"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "On strong odd colorings of sparse graphs",
  "title_text": "On strong odd colorings of sparse graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/8bzs9jtfv4m1a",
  "venue": "arXiv",
  "versions": []
}