{
  "id": "2003.01112",
  "version": "v1",
  "published": "2020-03-02T03:58:14.000Z",
  "updated": "2020-03-02T03:58:14.000Z",
  "title": "Combinatorial Nullstellensatz and DP-coloring of Graphs",
  "authors": [
    "Hemanshu Kaul",
    "Jeffrey A. Mudrock"
  ],
  "comment": "17 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "We initiate the study of applying the Combinatorial Nullstellensatz to the DP-coloring of graphs even though, as is well-known, the Alon-Tarsi theorem does not apply to DP-coloring. We define the notion of good covers of prime order which allows us to apply the Combinatorial Nullstellensatz to DP-coloring. We apply these tools to DP-coloring of the cones of certain bipartite graphs and uniquely 3-colorable graphs. We also extend a result of Akbari, Mirrokni, and Sadjad (2006) on unique list colorability to the context of DP-coloring. We establish a sufficient algebraic condition for a graph $G$ to satisfy $\\chi_{DP}(G) \\leq 3$, and we completely determine the DP-chromatic number of squares of all cycles.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-03-02T03:58:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C15",
      "05C25",
      "05C31",
      "05C69"
    ],
    "keywords": [
      "combinatorial nullstellensatz",
      "dp-coloring",
      "unique list colorability",
      "sufficient algebraic condition",
      "alon-tarsi theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}