{
  "id": "1212.6845",
  "version": "v1",
  "published": "2012-12-31T09:24:43.000Z",
  "updated": "2012-12-31T09:24:43.000Z",
  "title": "Solutions to conjectures on the $(k,\\ell)$-rainbow index of complete graphs",
  "authors": [
    "Qingqiong Cai",
    "Xueliang Li",
    "Jiangli Song"
  ],
  "comment": "9 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "The $(k,\\ell)$-rainbow index $rx_{k, \\ell}(G)$ of a graph $G$ was introduced by Chartrand et. al. For the complete graph $K_n$ of order $n\\geq 6$, they showed that $rx_{3,\\ell}(K_n)=3$ for $\\ell=1,2$. Furthermore, they conjectured that for every positive integer $\\ell$, there exists a positive integer $N$ such that $rx_{3,\\ell}(K_{n})=3$ for every integer $n \\geq N$. More generally, they conjectured that for every pair of positive integers $k$ and $\\ell$ with $k\\geq 3$, there exists a positive integer $N$ such that $rx_{k,\\ell}(K_{n})=k$ for every integer $n \\geq N$. This paper is to give solutions to these conjectures.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-12-31T09:24:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C40",
      "05C05",
      "05C15",
      "05D40"
    ],
    "keywords": [
      "rainbow index",
      "complete graph",
      "positive integer"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1212.6845C"
    }
  }
}