{
  "id": "2202.02586",
  "version": "v1",
  "published": "2022-02-05T15:58:11.000Z",
  "updated": "2022-02-05T15:58:11.000Z",
  "title": "A note on odd-coloring 1-planar graphs",
  "authors": [
    "Michael Lafferty",
    "Zi-Xia Song"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "A proper coloring of a graph is odd if every non-isolated vertex has some color that appears an odd number of times on its neighborhood. This notion was recently introduced by Petru\\v{s}evski and \\v{S}krekovski, who proved that every planar graph admits an odd $9$-coloring; they also conjectured that every planar graph admits an odd $5$-coloring. Shortly after, this conjecture was confirmed for planar graphs of girth at least seven by Cranston; outerplanar graphs by Caro, Petru\\v{s}evski and \\v{S}krekovski. Building on the work of Caro, Petru\\v{s}evski and \\v{S}krekovski, Petr and Portier then further proved that every planar graph admits an odd $8$-coloring. In this note we prove that every 1-planar graph admits an odd $47$-coloring, where a graph is 1-planar if it can be drawn in the plane so that each edge is crossed by at most one other edge.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-02-05T15:58:11.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "planar graph admits",
      "odd number",
      "odd-coloring",
      "outerplanar graphs",
      "non-isolated vertex"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}