{
  "id": "1310.2783",
  "version": "v2",
  "published": "2013-10-10T12:10:45.000Z",
  "updated": "2013-10-18T09:42:28.000Z",
  "title": "The $(k,\\ell)$-rainbow index for complete bipartite and multipartite graphs",
  "authors": [
    "Qingqiong Cai",
    "Xueliang Li",
    "Jiangli Song"
  ],
  "comment": "13 pages. arXiv admin note: substantial text overlap with arXiv:1212.6845",
  "categories": [
    "math.CO"
  ],
  "abstract": "A tree in an edge-colored graph $G$ is said to be a rainbow tree if no two edges on the tree share the same color. Given two positive integers $k$, $\\ell$ with $k\\geq 3$, the \\emph{$(k,\\ell)$-rainbow index} $rx_{k,\\ell}(G)$ of $G$ is the minimum number of colors needed in an edge-coloring of $G$ such that for any set $S$ of $k$ vertices of $G$, there exist $\\ell$ internally disjoint rainbow trees connecting $S$. This concept was introduced by Chartrand et al., and there have been very few results about it. In this paper, we investigate the $(k,\\ell)$-rainbow index for complete bipartite graphs and complete multipartite graphs. Some asymptotic values of their $(k,\\ell)$-rainbow index are obtained.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-10-18T09:42:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C05",
      "05C15",
      "05C80",
      "05D40"
    ],
    "keywords": [
      "rainbow index",
      "complete multipartite graphs",
      "complete bipartite graphs",
      "minimum number",
      "asymptotic values"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1310.2783C"
    }
  }
}