{
  "id": "1402.6466",
  "version": "v1",
  "published": "2014-02-26T09:21:33.000Z",
  "updated": "2014-02-26T09:21:33.000Z",
  "title": "Bipartite decomposition of random graphs",
  "authors": [
    "Noga Alon"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "For a graph $G=(V,E)$, let $\\tau(G)$ denote the minimum number of pairwise edge disjoint complete bipartite subgraphs of $G$ so that each edge of $G$ belongs to exactly one of them. It is easy to see that for every graph $G$, $\\tau(G) \\leq n -\\alpha(G)$, where $\\alpha(G)$ is the maximum size of an independent set of $G$. Erd\\H{o}s conjectured in the 80s that for almost every graph $G$ equality holds, i.e., that for the random graph $G(n,0.5)$, $\\tau(G)=n-\\alpha(G)$ with high probability, that is, with probability that tends to $1$ as $n$ tends to infinity. Here we show that this conjecture is (slightly) false, proving that for most values of $n$ tending to infinity and for $G=G(n,0.5)$, $\\tau(G) \\leq n-\\alpha(G)-1$ with high probability, and that for some sequences of values of $n$ tending to infinity $\\tau(G) \\leq n-\\alpha(G)-2$ with probability bounded away from $0$. We also study the typical value of $\\tau(G)$ for random graphs $G=G(n,p)$ with $p < 0.5$ and show that there is an absolute positive constant $c$ so that for all $p \\leq c$ and for $G=G(n,p)$, $\\tau(G)=n-\\Theta(\\alpha(G))$ with high probability.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-02-26T09:21:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C80"
    ],
    "keywords": [
      "random graph",
      "bipartite decomposition",
      "high probability",
      "edge disjoint complete bipartite subgraphs",
      "pairwise edge disjoint complete bipartite"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1402.6466A"
    }
  }
}