{
  "id": "2003.05637",
  "version": "v1",
  "published": "2020-03-12T06:20:51.000Z",
  "updated": "2020-03-12T06:20:51.000Z",
  "title": "Conflict-free coloring on closed neighborhoods of bounded degree graphs",
  "authors": [
    "Sriram Bhyravarapu",
    "Subrahmanyam Kalyanasundaram",
    "Rogers Mathew"
  ],
  "comment": "3 pages",
  "categories": [
    "math.CO",
    "cs.DM"
  ],
  "abstract": "The closed neighborhood conflict-free chromatic number of a graph $G$, denoted by $\\chi_{CN}(G)$, is the minimum number of colors required to color the vertices of $G$ such that for every vertex, there is a color that appears exactly once in its closed neighborhood. Pach and Tardos [Combin. Probab. Comput. 2009] showed that $\\chi_{CN}(G) = O(\\log^{2+\\varepsilon} \\Delta)$, for any $\\varepsilon > 0$, where $\\Delta$ is the maximum degree. In [Combin. Probab. Comput. 2014], Glebov, Szab\\'o and Tardos showed existence of graphs $G$ with $\\chi_{CN}(G) = \\Omega(\\log^2\\Delta)$. In this paper, we bridge the gap between the two bounds by showing that $\\chi_{CN}(G) = O(\\log^2 \\Delta)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-03-12T06:20:51.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "bounded degree graphs",
      "conflict-free coloring",
      "closed neighborhood conflict-free chromatic number",
      "minimum number",
      "tardos showed existence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 3,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}