{
  "id": "1510.04814",
  "version": "v1",
  "published": "2015-10-16T08:20:31.000Z",
  "updated": "2015-10-16T08:20:31.000Z",
  "title": "On the decomposition of random hypergraphs",
  "authors": [
    "Xing Peng"
  ],
  "comment": "15 pages, 3 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "For an $r$-uniform hypergraph $H$, let $f(H)$ be the minimum number of complete $r$-partite $r$-uniform subhypergraphs of $H$ whose edge sets partition the edge set of $H$. For a graph $G$, $f(G)$ is the bipartition number of $G$ which was introduced by Graham and Pollak in 1971. In 1988, Erd\\H{o}s conjectured that if $G \\in G(n,1/2)$, then with high probability $f(G)=n-\\alpha(G)$, where $\\alpha(G)$ is the independence number of $G$. This conjecture and related problems have received a lot of attention recently. In this paper, we study the value of $f(H)$ for a typical $r$-uniform hypergraph $H$. More precisely, we prove that if $(\\log n)^{2.001}/n \\leq p \\leq 1/2$ and $H \\in H^{(r)}(n,p)$, then with high probability $f(H)=(1-\\pi(K^{(r-1)}_r)+o(1))\\binom{n}{r-1}$, where $\\pi(K^{(r-1)}_r)$ is the Tur\\'an density of $K^{(r-1)}_r$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-10-16T08:20:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C80"
    ],
    "keywords": [
      "random hypergraphs",
      "decomposition",
      "uniform hypergraph",
      "high probability",
      "edge sets partition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151004814P"
    }
  }
}