{
  "id": "1105.4314",
  "version": "v1",
  "published": "2011-05-22T07:59:22.000Z",
  "updated": "2011-05-22T07:59:22.000Z",
  "title": "Asymptotic value of the minimal size of a graph with rainbow connection number 2",
  "authors": [
    "Hengzhe Li",
    "Xueliang Li",
    "Yuefang Sun"
  ],
  "comment": "8 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "A path in an edge (vertex)-colored graph $G$, where adjacent edges (vertices) may have the same color, is called a rainbow path if no pair of edges (internal vertices) of the path are colored the same. The rainbow (vertex) connection number $rc(G)$ ($rvc(G)$) of $G$ is the minimum integer $i$ for which there exists an $i$-edge (vertex)-coloring of $G$ such that every two distinct vertices of $G$ are connected by a rainbow path. Denote by $\\mathcal{G}_d(n)$ ($\\mathcal{G'}_d(n)$) the set of all graphs of order $n$ with rainbow (vertex) connection number $d$, and define $e_d(n)=\\min\\{e(G)\\,|\\, G\\in \\mathcal{G}_d(n)\\}$ ($e_d'(n)=\\min\\{e(G)\\,|\\, G\\in \\mathcal{G'}_d(n)\\}$), where $e(G)$ denotes the number of edges in $G$. In this paper, we investigate the bounds of $e_2(n)$ and get the exact asymptotic value. i.e., $\\displaystyle \\lim_{n\\rightarrow \\infty}\\frac{e_2(n)}{n\\log_2 n}=1$. Meanwhile, we obtain $e'_d(n)=n-1$ for $d\\geq 2$, and the equality holds if and only if $G$ is such a graph such that deleting all leaves of $G$ results in a tree of order $d$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-05-22T07:59:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C15",
      "05C35",
      "05C40"
    ],
    "keywords": [
      "rainbow connection number",
      "minimal size",
      "rainbow path",
      "exact asymptotic value",
      "adjacent edges"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1105.4314L"
    }
  }
}