{
  "id": "2408.09081",
  "version": "v1",
  "published": "2024-08-17T03:19:48.000Z",
  "updated": "2024-08-17T03:19:48.000Z",
  "title": "B-colorings of planar and outerplanar graphs",
  "authors": [
    "Ryan R. Martin",
    "Miklós Ruszinkó",
    "Gábor N. Sárközy"
  ],
  "comment": "15 pages, 4 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "A coloring of the edges of a graph $G$ in which every $K_{1,2}$ is totally multicolored is known as a proper coloring and a coloring of the edges of $G$ in which every $K_{1,2}$ and every $K_{2,2}$ is totally multicolored is called a B-coloring. In this paper, we establish that a planar graph with maximum degree $\\Delta$ can be B-colored with $\\max\\{2\\Delta,32\\}$ colors. This is best-possible for large $\\Delta$ because $K_{2,\\Delta}$ requires $2\\Delta$ colors. In addition, there is an example with $\\Delta=4$ that requires $12$ colors. We also establish that an outerplanar graph with maximum degree $\\Delta$ can be B-colored with $\\max\\{\\Delta,6\\}$ colors. This is almost best-possible because $\\Delta$ colors are necessary and there is an example with $\\Delta=4$ that requires $5$ colors.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-17T03:19:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C15",
      "05C10",
      "05C35"
    ],
    "keywords": [
      "outerplanar graph",
      "maximum degree",
      "b-coloring"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}