{
  "id": "2205.09856",
  "version": "v1",
  "published": "2022-05-19T21:00:08.000Z",
  "updated": "2022-05-19T21:00:08.000Z",
  "title": "List Multicoloring of Planar Graphs and Related Classes",
  "authors": [
    "Glenn G. Chappell"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "For positive integers $a$ and $b$, a graph $G$ is $(a:b)$-choosable if, for each assignment of lists of $a$ colors to the vertices of $G,$ each vertex can be colored with a set of $b$ colors from its list so that adjacent vertices are colored with disjoint sets. We show that for positive integers $a$ and $b$, every bipartite planar graph is $(a:b)$-choosable iff $\\frac{a}{b} \\ge 3$. For general planar graphs, we show that if $\\frac{a}{b} < 4\\frac{2}{5}$, then there exists a planar graph that is not $(a:b)$-choosable, thus improving on a result of X. Zhu, which had $4\\frac{2}{9}$. Lastly, we show that every $K_5$-minor-free graph is $(a:b)$-choosable iff $\\frac{a}{b} \\ge 5$. Along the way, we mention some open problems.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-05-19T21:00:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C15",
      "05C10",
      "05C83"
    ],
    "keywords": [
      "related classes",
      "list multicoloring",
      "positive integers",
      "general planar graphs",
      "bipartite planar graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}