{
  "id": "2207.12129",
  "version": "v2",
  "published": "2022-07-21T22:49:36.000Z",
  "updated": "2022-12-20T17:34:34.000Z",
  "title": "Extensions of Thomassen's Theorem to Paths of Length At Most Four: Part II",
  "authors": [
    "Joshua Nevin"
  ],
  "comment": "70 pages, 8 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $G$ be a planar embedding with list-assignment $L$ and outer cycle $C$, and let $P$ be a path of length at most four on $C$, where each vertex of $G\\setminus C$ has a list of size at least five and each vertex of $C\\setminus P$ has a list of size at least three. This is the second paper in a sequence of three papers in which we prove some results about partial $L$-colorings $\\phi$ of $C$ with the property that any extension of $\\phi$ to an $L$-coloring of $\\textrm{dom}(\\phi)\\cup V(P)$ extends to $L$-color all of $G$, and, in particular, some useful results about the special case in which $\\textrm{dom}(\\phi)$ consists only of the endpoints of $P$. We also prove some results about the other special case in which $\\phi$ is allowed to color some vertices of $C\\setminus\\mathring{P}$ but we avoid taking too many colors away from the leftover vertices of $\\mathring{P}\\setminus\\textrm{dom}(\\phi)$. We use these results in a later sequence of papers to prove some results about list-colorings of high-representativity embeddings on surfaces.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-12-20T17:34:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C15",
      "G.2.2"
    ],
    "keywords": [
      "thomassens theorem",
      "special case",
      "second paper",
      "high-representativity embeddings",
      "outer cycle"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 70,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}