{
  "id": "1907.06630",
  "version": "v1",
  "published": "2019-07-13T01:18:40.000Z",
  "updated": "2019-07-13T01:18:40.000Z",
  "title": "Cover and variable degeneracy",
  "authors": [
    "Fangyao Lu",
    "Qianqian Wang",
    "Tao Wang"
  ],
  "comment": "14 pages, 3 figures",
  "categories": [
    "math.CO",
    "cs.DM"
  ],
  "abstract": "Let $f$ be a nonnegative integer valued function on the vertex-set of a graph. A graph is {\\bf strictly $f$-degenerate} if each nonempty subgraph $\\Gamma$ has a vertex $v$ such that $\\deg_{\\Gamma}(v) < f(v)$. In this paper, we define a new concept, strictly $f$-degenerate transversal, which generalizes list coloring, $(f_{1}, f_{2}, \\dots, f_{\\kappa})$-partition, signed coloring, DP-coloring and $L$-forested-coloring. A {\\bf cover} of a graph $G$ is a graph $H$ with vertex set $V(H) = \\bigcup_{v \\in V(G)} X_{v}$, where $X_{v} = \\{(v, 1), (v, 2), \\dots, (v, \\kappa)\\}$; the edge set $\\mathscr{M} = \\bigcup_{uv \\in E(G)}\\mathscr{M}_{uv}$, where $\\mathscr{M}_{uv}$ is a matching between $X_{u}$ and $X_{v}$. A vertex set $R \\subseteq V(H)$ is a {\\bf transversal} of $H$ if $|R \\cap X_{v}| = 1$ for each $v \\in V(G)$. A transversal $R$ is a {\\bf strictly $f$-degenerate transversal} if $H[R]$ is strictly $f$-degenerate. The main result of this paper is a degree type result, which generalizes Brooks' theorem, Gallai's theorem, degree-choosable, signed degree-colorable, DP-degree-colorable. Similar to Borodin, Kostochka and Toft's variable degeneracy, the degree type result is also self-strengthening. Using these results, we can uniformly prove many new and known results.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-13T01:18:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C15"
    ],
    "keywords": [
      "degree type result",
      "vertex set",
      "degenerate transversal",
      "generalizes brooks",
      "tofts variable degeneracy"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}