{
  "id": "0903.3048",
  "version": "v1",
  "published": "2009-03-17T21:03:03.000Z",
  "updated": "2009-03-17T21:03:03.000Z",
  "title": "Biclique Coverings and the Chromatic Number",
  "authors": [
    "Dhruv Mubayi",
    "Sundar Vishwanathan"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Consider a graph $G$ with chromatic number $k$ and a collection of complete bipartite graphs, or bicliques, that cover the edges of $G$. We prove the following two results: \\medskip \\noindent $\\bullet$ If the bicliques partition the edges of $G$, then their number is at least $2^{\\sqrt{\\log_2 k}}$. This is the first improvement of the easy lower bound of $\\log_2 k$, while the Alon-Saks-Seymour conjecture states that this can be improved to $k-1$. \\medskip \\noindent $\\bullet$ The sum of the orders of the bicliques is at least $(1-o(1))k\\log_2 k$. This generalizes, in asymptotic form, a result of Katona and Szemer\\'edi who proved that the minimum is $k\\log_2 k$ when $G$ is a clique.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-03-17T21:03:03.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "chromatic number",
      "biclique coverings",
      "complete bipartite graphs",
      "alon-saks-seymour conjecture states",
      "bicliques partition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0903.3048M"
    }
  }
}