{
  "id": "1504.01975",
  "version": "v1",
  "published": "2015-04-08T14:09:19.000Z",
  "updated": "2015-04-08T14:09:19.000Z",
  "title": "On the b-chromatic number of the Cartesian product of two complete graphs",
  "authors": [
    "Frédéric Maffray",
    "Artur Mesquita Barbosa"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "A b-coloring of a graph $G$ is a coloring of its vertices such that every color class contains a vertex that has neighbors in all other classes. The b-chromatic number of $G$ is the largest integer $k$ such that $G$ has a b-coloring with $k$ colors. Javadi and Omoomi (\"On b-coloring of cartesian product of graphs\", Ars Combinatoria 107 (2012) 521-536) proved that the b-chromatic number of $K_n \\times K_n$ (the Cartesian product of two complete graphs on $n$ vertices) is in the set $\\{2n-3, 2n-2\\}$ and conjectured that the exact value is $2n-3$ for all $n \\ge 5$. We give counterexamples to this conjecture for $n=5$, $n=6$ and $n=7$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-04-08T14:09:19.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "cartesian product",
      "b-chromatic number",
      "complete graphs",
      "color class contains",
      "conjecture"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150401975M"
    }
  }
}