{
  "id": "1307.0079",
  "version": "v3",
  "published": "2013-06-29T09:02:28.000Z",
  "updated": "2013-07-03T04:06:53.000Z",
  "title": "The 3-rainbow index of a graph",
  "authors": [
    "Lily Chen",
    "Xueliang Li",
    "Kang Yang",
    "Yan Zhao"
  ],
  "comment": "13 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $G$ be a nontrivial connected graph with an edge-coloring $c: E(G)\\rightarrow \\{1,2,...,q\\},$ $q \\in \\mathbb{N}$, where adjacent edges may be colored the same. A tree $T$ in $G$ is a $rainbow tree$ if no two edges of $T$ receive the same color. For a vertex subset $S\\subseteq V(G)$, a tree that connects $S$ in $G$ is called an $S$-tree. The minimum number of colors that are needed in an edge-coloring of $G$ such that there is a rainbow $S$-tree for each $k$-subset $S$ of $V(G)$ is called $k$-rainbow index, denoted by $rx_k(G)$. In this paper, we first determine the graphs whose 3-rainbow index equals 2, $m,$ $m-1$, $m-2$, respectively. We also obtain the exact values of $rx_3(G)$ for regular complete bipartite and multipartite graphs and wheel graphs. Finally, we give a sharp upper bound for $rx_3(G)$ of 2-connected graphs and 2-edge connected graphs, and graphs whose $rx_3(G)$ attains the upper bound are characterized.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2013-07-03T04:06:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C05",
      "05C15",
      "05C75"
    ],
    "keywords": [
      "regular complete bipartite",
      "sharp upper bound",
      "minimum number",
      "rainbow index",
      "nontrivial connected graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1307.0079C"
    }
  }
}