{
  "id": "1311.6423",
  "version": "v3",
  "published": "2013-11-25T19:51:50.000Z",
  "updated": "2014-01-28T01:55:16.000Z",
  "title": "Rainbow Matchings and Hamilton Cycles in Random Graphs",
  "authors": [
    "Deepak Bal",
    "Alan Frieze"
  ],
  "comment": "We replaced graphs by k-uniform hypergraphs",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $HP_{n,m,k}$ be drawn uniformly from all $k$-uniform, $k$-partite hypergraphs where each part of the partition is a disjoint copy of $[n]$. We let $HP^{(\\k)}_{n,m,k}$ be an edge colored version, where we color each edge randomly from one of $\\k$ colors. We show that if $\\k=n$ and $m=Kn\\log n$ where $K$ is sufficiently large then w.h.p. there is a rainbow colored perfect matching. I.e. a perfect matching in which every edge has a different color. We also show that if $n$ is even and $m=Kn\\log n$ where $K$ is sufficiently large then w.h.p. there is a rainbow colored Hamilton cycle in $G^{(n)}_{n,m}$. Here $G^{(n)}_{n,m}$ denotes a random edge coloring of $G_{n,m}$ with $n$ colors. When $n$ is odd, our proof requires $m=\\om(n\\log n)$ for there to be a rainbow Hamilton cycle.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2014-01-28T01:55:16.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random graphs",
      "rainbow matchings",
      "rainbow hamilton cycle",
      "rainbow colored hamilton cycle",
      "sufficiently large"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1311.6423B"
    }
  }
}