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Random Graphs with Few Disjoint Cycles

Published online by Cambridge University Press:  09 June 2011

VALENTAS KURAUSKAS  and
COLIN McDIARMID
VALENTAS KURAUSKAS
Affiliation:
Vilnius University, Naugarduko 24, LT-03225 Vilnius, Lithuania (e-mail: valentas@gmail.com)
COLIN McDIARMID
Affiliation:
Department of Statistics, University of Oxford, 1 South Parks Road, Oxford OX1 3TG, UK (e-mail: cmcd@stats.ox.ac.uk)
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Abstract

The classical Erdős–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,. . .,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 Erdős–Pósa theorem with the extra ‘redundancy’ property that B–v is still a blocker for all but at most k vertices v ∈ B.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 20 , Issue 5 , September 2011 , pp. 763 - 775
Copyright
Copyright © Cambridge University Press 2011

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