{
  "id": "1110.1268",
  "version": "v1",
  "published": "2011-10-05T06:38:07.000Z",
  "updated": "2011-10-05T06:38:07.000Z",
  "title": "Note on rainbow connection number of dense graphs",
  "authors": [
    "Jiuying Dong",
    "Xueliang Li"
  ],
  "comment": "4 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "An edge-colored graph $G$ is rainbow connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection number of a connected graph $G$, denoted by $rc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow connected. Following an idea of Caro et al., in this paper we also investigate the rainbow connection number of dense graphs. We show that for $k\\geq 2$, if $G$ is a non-complete graph of order $n$ with minimum degree $\\delta (G)\\geq \\frac{n}{2}-1+log_{k}{n}$, or minimum degree-sum $\\sigma_{2}(G)\\geq n-2+2log_{k}{n}$, then $rc(G)\\leq k$; if $G$ is a graph of order $n$ with diameter 2 and $\\delta (G)\\geq 2(1+log_{\\frac{k^{2}}{3k-2}}{k})log_{k}{n}$, then $rc(G)\\leq k$. We also show that if $G$ is a non-complete bipartite graph of order $n$ and any two vertices in the same vertex class have at least $2log_{\\frac{k^{2}}{3k-2}}{k}log_{k}{n}$ common neighbors in the other class, then $rc(G)\\leq k$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-10-05T06:38:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C15",
      "05C40"
    ],
    "keywords": [
      "rainbow connection number",
      "dense graphs",
      "non-complete bipartite graph",
      "non-complete graph",
      "common neighbors"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 4,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1110.1268D"
    }
  }
}