{
  "id": "1907.07141",
  "version": "v1",
  "published": "2019-07-13T01:46:20.000Z",
  "updated": "2019-07-13T01:46:20.000Z",
  "title": "Variable degeneracy on toroidal graphs",
  "authors": [
    "Rui Li",
    "Tao Wang"
  ],
  "comment": "10 pages, 5 figures. arXiv admin note: text overlap with arXiv:1803.01197",
  "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 $\\deg_{\\Gamma}(v) < f(v)$. A {\\bf cover} of a graph $G$ is a graph $H$ with vertex set $V(H) = \\bigcup_{v \\in V(G)} L_{v}$, where $L_{v} = \\{\\,(v, 1), (v, 2), \\dots, (v, \\kappa)\\,\\}$; the edge set $\\mathscr{M} = \\bigcup_{uv \\in E(G)}\\mathscr{M}_{uv}$, where $\\mathscr{M}_{uv}$ is a matching between $L_{u}$ and $L_{v}$. A vertex set $R \\subseteq V(H)$ is a {\\bf transversal} of $H$ if $|R \\cap L_{v}| = 1$ for each $v \\in V(G)$. A transversal $R$ is a {\\bf strictly $f$-degenerate transversal} if $H[R]$ is strictly $f$-degenerate. In this paper, we give some structural results on planar and toroidal graphs with forbidden configurations, and give some sufficient conditions for the existence of strictly $f$-degenerate transversal by using these structural results.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-13T01:46:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C15"
    ],
    "keywords": [
      "toroidal graphs",
      "variable degeneracy",
      "degenerate transversal",
      "structural results",
      "vertex set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}