{
  "id": "2108.12880",
  "version": "v1",
  "published": "2021-08-29T17:16:46.000Z",
  "updated": "2021-08-29T17:16:46.000Z",
  "title": "Five-List-Coloring Graphs on Surfaces: The Many Faces Far-Apart Generalization of Thomassen's Theorem",
  "authors": [
    "Luke Postle",
    "Robin Thomas"
  ],
  "comment": "14 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $G$ be a plane graph with $C$ the boundary of the outer face and let $(L(v):v\\in V(G))$ be a family of non-empty sets. By an $L$-coloring of a subgraph $J$ of $G$ we mean a (proper) coloring $\\phi$ of $J$ such that $\\phi(v)\\in L(v)$ for every vertex $v$ of $J$. Thomassen proved that if $v_1,v_2\\in V(C)$ are adjacent, $L(v_1)\\ne L(v_2)$, $|L(v)|\\ge3$ for every $v\\in V(C)\\setminus \\{v_1,v_2\\}$ and $|L(v)|\\ge5$ for every $v\\in V(G)\\setminus V(C)$, then $G$ has an $L$-coloring. As one final application in this last part of our series on $5$-list-coloring, we derive from all of our theory a far-reaching generalization of Thomassen's theorem, namely the generalization of Thomassen's theorem to arbitrarily many such faces provided that the faces are pairwise distance $D$ apart for some universal constant $D>0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-08-29T17:16:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C15",
      "05C10"
    ],
    "keywords": [
      "thomassens theorem",
      "faces far-apart generalization",
      "five-list-coloring graphs",
      "plane graph",
      "non-empty sets"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}