{
  "id": "1010.6278",
  "version": "v1",
  "published": "2010-10-29T17:17:42.000Z",
  "updated": "2010-10-29T17:17:42.000Z",
  "title": "Random graphs with few disjoint cycles",
  "authors": [
    "Valentas Kurauskas",
    "Colin McDiarmid"
  ],
  "journal": "Combinatorics, Probability and Computing 20 (2011) 763 -- 775",
  "categories": [
    "math.CO"
  ],
  "abstract": "The classical Erd\\H{o}s-P\\'{o}sa theorem states that for each positive integer k there is an f(k) such that, in each graph G which does not have k+1 disjoint cycles, there is a blocker of size at most f(k); that is, a set B of at most f(k) vertices such that G-B has no cycles. We show that, amongst all such graphs on vertex set {1,..,n}, all but an exponentially small proportion have a blocker of size k. We also give further properties of a random graph sampled uniformly from this class; concerning uniqueness of the blocker, connectivity, chromatic number and clique number. A key step in the proof of the main theorem is to show that there must be a blocker as in the Erd\\H{o}s-P\\'{o}sa theorem with the extra `redundancy' property that B-v is still a blocker for all but at most k vertices v in B.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-10-29T17:17:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C80"
    ],
    "keywords": [
      "disjoint cycles",
      "exponentially small proportion",
      "theorem states",
      "chromatic number",
      "clique number"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1010.6278K"
    }
  }
}