{
  "id": "2501.07129",
  "version": "v1",
  "published": "2025-01-13T08:33:18.000Z",
  "updated": "2025-01-13T08:33:18.000Z",
  "title": "$(2,4)$-Colorability of Planar Graphs Excluding $3$-, $4$-, and $6$-Cycles",
  "authors": [
    "Pongpat Sittitrai",
    "Wannapol Pimpasalee",
    "Kittikorn Nakprasit"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "A defective $k$-coloring is a coloring on the vertices of a graph using colors $1,2, \\dots, k$ such that adjacent vertices may share the same color. A $(d_1,d_2)$-\\emph{coloring} of a graph $G$ is a defective $2$-coloring of $G$ such that any vertex colored by color $i$ has at most $d_i$ adjacent vertices of the same color, where $i\\in\\{1,2\\}$. A graph $G$ is said to be $(d_1,d_2)$-\\emph{colorable} if it admits a $(d_1,d_2)$-coloring. Defective $2$-coloring in planar graphs without $3$-cycles, $4$-cycles, and $6$-cycles has been investigated by Dross and Ochem, as well as Sittitrai and Pimpasalee. They showed that such graphs are $(0,6)$-colorable and $(3,3)$-colorable, respectively. In this paper, we proved that these graphs are also $(2,4)$-colorable.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-01-13T08:33:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C15",
      "05C10"
    ],
    "keywords": [
      "planar graphs excluding",
      "adjacent vertices",
      "colorability",
      "pimpasalee"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}