{
  "id": "1101.2765",
  "version": "v2",
  "published": "2011-01-14T10:51:21.000Z",
  "updated": "2011-03-08T02:38:08.000Z",
  "title": "Rainbow connection of graphs with diameter 2",
  "authors": [
    "Hengzhe Li",
    "Xueliang Li",
    "Sujuan Liu"
  ],
  "comment": "10 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "A path in an edge-colored graph $G$, where adjacent edges may have the same color, is called a rainbow path if no two edges of the path are colored the same. The rainbow connection number $rc(G)$ of $G$ is the minimum integer $i$ for which there exists an $i$-edge-coloring of $G$ such that every two distinct vertices of $G$ are connected by a rainbow path. It is known that for a graph $G$ with diameter 2, to determine $rc(G)$ is NP-hard. So, it is interesting to know the best upper bound of $rc(G)$ for such a graph $G$. In this paper, we show that $rc(G)\\leq 5$ if $G$ is a bridgeless graph with diameter 2, and that $rc(G)\\leq k+2$ if $G$ is a connected graph of diameter 2 with $k$ bridges, where $k\\geq 1$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-03-08T02:38:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C15",
      "05C40"
    ],
    "keywords": [
      "rainbow path",
      "best upper bound",
      "rainbow connection number",
      "minimum integer",
      "distinct vertices"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1101.2765L"
    }
  }
}