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    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Giménez, Omer Mitsche, Dieter and Noy, Marc 2016. Maximum degree in minor-closed classes of graphs. European Journal of Combinatorics, Vol. 55, p. 41.

    McDiarmid, Colin and Kurauskas, Valentas 2014. Random graphs containing few disjoint excluded minors. Random Structures & Algorithms, Vol. 44, Issue. 2, p. 240.

    McDiarmid, Colin 2011. On graphs with few disjoint <mml:math altimg="si1.gif" display="inline" overflow="scroll" xmlns:xocs="" xmlns:xs="" xmlns:xsi="" xmlns="" xmlns:ja="" xmlns:mml="" xmlns:tb="" xmlns:sb="" xmlns:ce="" xmlns:xlink="" xmlns:cals=""><mml:mi>t</mml:mi></mml:math>-star minors. European Journal of Combinatorics, Vol. 32, Issue. 8, p. 1394.

  • Combinatorics, Probability and Computing, Volume 20, Issue 5
  • September 2011, pp. 763-775

Random Graphs with Few Disjoint Cycles

  • DOI:
  • Published online: 09 June 2011

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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[2]P. Erdös and L. Pósa (1965) On independent circuits in a graph. Canad. J. Math. 17 347352.

[4]M. Kang and C. McDiarmid (2011) Random unlabelled graphs containing few disjoint cycles. Random Struct. Alg. 38 174204.

[7]L. Lovász (1993) Combinatorial Problems and Exercises, second edition, North-Holland.

[9]C. McDiarmid , A. Steger and D. Welsh (2005) Random planar graphs. J. Combin. Theory Ser. B 93 187205.

[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.

[11]B. A. Reed , N. Robertson , P. D. Seymour and R. Thomas (1996) Packing directed circuits. Combinatorica 16 535554.

[13]N. Robertson and P. Seymour (1986) Graph minors V: Excluding a planar graph. J. Combin. Theory Ser. B 41 92114.

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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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