{
  "id": "1908.04902",
  "version": "v1",
  "published": "2019-08-14T00:57:30.000Z",
  "updated": "2019-08-14T00:57:30.000Z",
  "title": "3-choosable planar graphs with some precolored vertices and no $5^{-}$-cycles normally adjacent to $8^{-}$-cycles",
  "authors": [
    "Fangyao Lu",
    "Qianqian Wang",
    "Tao Wang"
  ],
  "comment": "9 pages, 2 figures",
  "categories": [
    "math.CO",
    "cs.DM"
  ],
  "abstract": "DP-coloring was introduced by Dvo\\v{r}\\'{a}k and Postle [J. Combin. Theory Ser. B 129 (2018) 38--54] as a generalization of list coloring. They used a \"weak\" version of DP-coloring to solve a longstanding conjecture by Borodin, stating that every planar graph without cycles of length $4$ to $8$ is $3$-choosable. Liu and Li improved the result by showing that every planar graph without adjacent cycles of length at most $8$ is $3$-choosable. In this paper, it is showed that every planar graph without $5^{-}$-cycles normally adjacent to $8^{-}$-cycles is $3$-choosable. Actually, all these three papers give more stronger results by stating them in the form of \"weakly\" DP-$3$-coloring and color extension.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-08-14T00:57:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C15"
    ],
    "keywords": [
      "planar graph",
      "cycles normally adjacent",
      "precolored vertices",
      "theory ser",
      "adjacent cycles"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}