{
  "id": "2010.00063",
  "version": "v1",
  "published": "2020-09-30T19:10:31.000Z",
  "updated": "2020-09-30T19:10:31.000Z",
  "title": "A Tight Bound for Conflict-free Coloring in terms of Distance to Cluster",
  "authors": [
    "Sriram Bhyravarapu",
    "Subrahmanyam Kalyanasundaram"
  ],
  "comment": "29 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Given an undirected graph $G = (V,E)$, a conflict-free coloring with respect to open neighborhoods (CFON coloring) is a vertex coloring such that every vertex has a uniquely colored vertex in its open neighborhood. The minimum number of colors required for such a coloring is the CFON chromatic number of $G$, denoted by $\\chi_{ON}(G)$. In previous work [WG 2020], we showed the upper bound $\\chi_{ON}(G) \\leq dc(G) + 3$, where $dc(G)$ denotes the distance to cluster parameter of $G$. In this paper, we obtain the improved upper bound of $\\chi_{ON}(G) \\leq dc(G) + 1$. We also exhibit a family of graphs for which $\\chi_{ON}(G) > dc(G)$, thereby demonstrating that our upper bound is tight.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-09-30T19:10:31.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "conflict-free coloring",
      "tight bound",
      "upper bound",
      "open neighborhood",
      "cfon chromatic number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}