{
  "id": "1812.01814",
  "version": "v1",
  "published": "2018-12-05T04:41:38.000Z",
  "updated": "2018-12-05T04:41:38.000Z",
  "title": "Zero-free intervals of chromatic polynomials of mixed hypergraphs",
  "authors": [
    "Ruixue Zhang",
    "Fengming Dong"
  ],
  "comment": "3 figures, 14 references",
  "categories": [
    "math.CO"
  ],
  "abstract": "A mixed hypergraph $\\mathcal{H}=(\\mathcal{V}, \\mathcal{C}, \\mathcal{D})$ consists of a vertex set $\\mathcal{V}$ and two subsets $\\mathcal{C}$ and $\\mathcal{D}$ of $\\{E\\subseteq \\mathcal{V}: |E|\\ge 1\\}$. For any positive integer $\\lambda$, a proper $\\lambda$-coloring of $\\mathcal{H}$ is an assignment of $\\lambda$ colors to vertices in $\\mathcal{H}$ such that each member in $\\mathcal{C}$ contains at least two vertices assigned the same color and each member in $\\mathcal{D}$ contains at least two vertices assigned different colors. The chromatic polynomial of $\\mathcal{H}$, denoted by $P(\\mathcal{H}, \\lambda)$, is the function which counts the number of proper $\\lambda$-colorings of $\\mathcal{H}$ whenever $\\lambda$ is a positive integer. In this paper, we show that $P(\\mathcal{H}, \\lambda)$ is zero-free in the intervals $(-\\infty, 0)$ and $(0, 1)$ under certain conditions. This result extends known results on zero-free intervals of the chromatic polynomials of graphs and hypergraphs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-12-05T04:41:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C15",
      "05C31",
      "05C65"
    ],
    "keywords": [
      "chromatic polynomial",
      "zero-free intervals",
      "mixed hypergraph",
      "positive integer",
      "vertex set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}