{
  "id": "1211.4340",
  "version": "v3",
  "published": "2012-11-19T09:10:25.000Z",
  "updated": "2013-08-07T22:49:52.000Z",
  "title": "On $r$-Equitable Coloring of Complete Multipartite Graphs",
  "authors": [
    "Chih-Hung Yen"
  ],
  "comment": "8 pages, 1 figure",
  "journal": "Taiwanese Journal of Mathematics 13(7), 991-998, 2013",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $r \\geqslant 0$ and $k \\geqslant 1$ be integers. We say that a graph $G$ has an $r$-equitable $k$-coloring if there exists a proper $k$-coloring of $G$ such that the sizes of any two color classes differ by at most $r$. The least $k$ such that a graph $G$ has an $r$-equitable $k$-coloring is denoted by $\\chi_{r=} (G)$, and the least $n$ such that a graph $G$ has an $r$-equitable $k$-coloring for all $k \\geqslant n$ is denoted by $\\chi^*_{r=} (G)$. In this paper, we propose a necessary and sufficient condition for a complete multipartite graph $G$ to have an $r$-equitable $k$-coloring, and also give exact values of $\\chi_{r=} (G)$ and $\\chi^*_{r=} (G)$.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2013-08-07T22:49:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C15"
    ],
    "keywords": [
      "complete multipartite graph",
      "equitable coloring",
      "color classes differ",
      "sufficient condition",
      "exact values"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1211.4340Y"
    }
  }
}