Random graphs with few disjoint cycles

  • Authors:

  • Valentas Kurauskas;Colin Mcdiarmid

  • Affiliations:

  • Vilnius university, naugarduko 24, lt-03225 vilnius, lithuania (e-mail: valentas@gmail.com);Department of statistics, university of oxford, 1 south parks road, oxford ox1 3tg, uk (e-mail: cmcd@stats.ox.ac.uk)

  • Venue:
  • Combinatorics, Probability and Computing
  • Year:
  • 2011

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Abstract

The classical ErdÅ聭s-Pósa 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,.ï戮 .ï戮 .,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 ErdÅ聭s-Pósa theorem with the extra 'redundancy' property that B-v is still a blocker for all but at most k vertices v â聢聢 B.