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Random Graphs with Few Disjoint Cycles


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 GB 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 Bv is still a blocker for all but at most k vertices vB.

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[3] P. Flajolet and R. Sedgewick (2009) Analytic Combinatorics, Cambridge University Press.

[10] C. McDiarmid , A. Steger and D. Welsh (2006) Random graphs from planar and other addable classes. In Topics in Discrete Mathematics ( M. Klazar , J. Kratochvíl , M. Loebl , J. Matoušek , R. Thomas and P. Valtr , eds), Vol. 26 of Algorithms and Combinatorics, Springer, pp. 231246.

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Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
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