{
  "id": "1303.5171",
  "version": "v1",
  "published": "2013-03-21T05:40:07.000Z",
  "updated": "2013-03-21T05:40:07.000Z",
  "title": "The generalized 3-connectivity of random graphs",
  "authors": [
    "Ran Gu",
    "Xueliang Li",
    "Yongtang Shi"
  ],
  "comment": "14 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "The generalized connectivity of a graph $G$ was introduced by Chartrand et al. Let $S$ be a nonempty set of vertices of $G$, and $\\kappa(S)$ be defined as the largest number of internally disjoint trees $T_1, T_2, \\cdots, T_k$ connecting $S$ in $G$. Then for an integer $r$ with $2 \\leq r \\leq n$, the {\\it generalized $r$-connectivity} $\\kappa_r(G)$ of $G$ is the minimum $\\kappa(S)$ where $S$ runs over all the $r$-subsets of the vertex set of $G$. Obviously, $\\kappa_2(G)=\\kappa(G)$, is the vertex connectivity of $G$, and hence the generalized connectivity is a natural generalization of the vertex connectivity. Similarly, let $\\lambda(S)$ denote the largest number $k$ of pairwise edge-disjoint trees $T_1, T_2, \\ldots, T_k$ connecting $S$ in $G$. Then the {\\it generalized $r$-edge-connectivity} $\\lambda_r(G)$ of $G$ is defined as the minimum $\\lambda(S)$ where $S$ runs over all the $r$-subsets of the vertex set of $G$. Obviously, $\\lambda_2(G) = \\lambda(G)$. In this paper, we study the generalized 3-connectivity of random graphs and prove that for every fixed integer $k\\geq 1$, $$p=\\frac{{\\log n+(k+1)\\log \\log n -\\log \\log \\log n}}{n}$$ is a sharp threshold function for the property $\\kappa_3(G(n, p)) \\geq k$, which could be seen as a counterpart of Bollob\\'{a}s and Thomason's result for vertex connectivity. Moreover, we obtain that $\\delta (G(n,p)) - 1 = \\lambda (G(n,p)) - 1 = \\kappa (G(n,p)) - 1 \\le {\\kappa_3}(G(n,p)) \\le {\\lambda_3}(G(n,p)) \\le \\kappa (G(n,p)) = \\lambda (G(n,p)) = \\delta (G(n,p))$ almost surely holds, which could be seen as a counterpart of Ivchenko's result.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-03-21T05:40:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C40",
      "05C80",
      "05D40"
    ],
    "keywords": [
      "random graphs",
      "vertex connectivity",
      "vertex set",
      "largest number",
      "sharp threshold function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1303.5171G"
    }
  }
}