{
  "id": "2201.01455",
  "version": "v1",
  "published": "2022-01-05T05:10:11.000Z",
  "updated": "2022-01-05T05:10:11.000Z",
  "title": "Odd Colorings of Sparse Graphs",
  "authors": [
    "Daniel W. Cranston"
  ],
  "comment": "8 pages",
  "journal": "Journal of Combinatorics. Vol. 15(4), 2024, pp. 439-450",
  "categories": [
    "math.CO"
  ],
  "abstract": "A proper coloring of a graph is called \\emph{odd} if every non-isolated vertex has some color that appears an odd number of times on its neighborhood. The smallest number of colors that admits an odd coloring of a graph $G$ is denoted $\\chi_o(G)$. This notion was introduced by Petru\\v{s}evski and \\v{S}krekovski, who proved that if $G$ is planar then $\\chi_o(G)\\le 9$; they also conjectured that $\\chi_o(G)\\le 5$. For a positive real number $\\alpha$, we consider the maximum value of $\\chi_o(G)$ over all graphs $G$ with maximum average degree less than $\\alpha$; we denote this value by $\\chi_o(\\mathcal{G}_{\\alpha})$. We note that $\\chi_o(\\mathcal{G}_{\\alpha})$ is undefined for all $\\alpha\\ge 4$. In contrast, for each $\\alpha\\in[0,4)$, we give a (nearly sharp) upper bound on $\\chi_o(\\mathcal{G}_{\\alpha})$. Finally, we prove $\\chi_o(\\mathcal{G}_{20/7})= 5$ and $\\chi_o(\\mathcal{G}_3)= 6$. Both of these results are sharp.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-01-05T05:10:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C15"
    ],
    "keywords": [
      "odd coloring",
      "sparse graphs",
      "maximum average degree",
      "upper bound",
      "smallest number"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}