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    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="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ja="http://www.elsevier.com/xml/ja/dtd" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:cals="http://www.elsevier.com/xml/common/cals/dtd"><mml:mi>t</mml:mi></mml:math>-star minors. European Journal of Combinatorics, Vol. 32, Issue. 8, p. 1394.


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

Random Graphs with Few Disjoint Cycles

  • VALENTAS KURAUSKAS (a1) and COLIN McDIARMID (a2)
  • DOI: http://dx.doi.org/10.1017/S0963548311000186
  • Published online: 09 June 2011
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 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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This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[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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