{
  "id": "1903.10133",
  "version": "v1",
  "published": "2019-03-25T04:56:44.000Z",
  "updated": "2019-03-25T04:56:44.000Z",
  "title": "On Star 5-Colorings of Sparse Graphs",
  "authors": [
    "Ilkyoo Choi",
    "Boram Park"
  ],
  "comment": "22 pages, 16 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "Given a graph $G$, a star $k$-coloring of $G$ is a proper vertex $k$-coloring of $G$ such that the vertices on a path of length three receive at least three colors. The star chromatic number, denoted $\\chi_s(G)$, of a graph $G$ is the minimum integer $k$ for which $G$ admits a star $k$-coloring. Studying star coloring of sparse graphs is an active area of research, especially in terms of the maximum average degree of a graph; the maximum average degree, denoted mad$(G)$, of a graph $G$ is $\\max\\left\\{ \\frac{2|E(H)|}{|V(H)|}:{H \\subset G}\\right\\}$. It is known that for a graph $G$, if mad$(G)<{8\\over 3}$, then $\\chi_s(G)\\leq 6$, and if mad$(G)< \\frac{18}{7}$ and its girth is at least 6, then $\\chi_s(G)\\le 5$. For a graph $G$, let $\\Delta(G)$ denote the maximum degree of $G$. In this paper, we show that for a graph $G$, if mad$(G)\\le \\frac{8}{3}$ and $\\Delta(G)\\not\\in\\{4,5,6\\}$ then $\\chi_s(G)\\le 5$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-03-25T04:56:44.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "sparse graphs",
      "maximum average degree",
      "star chromatic number",
      "maximum degree",
      "proper vertex"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}