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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: [email protected])
COLIN McDIARMID
Affiliation:
Department of Statistics, University of Oxford, 1 South Parks Road, Oxford OX1 3TG, UK (e-mail: [email protected])
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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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References

[1]Dirac, G. (1965) Some results concerning the structure of graphs. Canad. Math. Bull. 8 459–463.Google Scholar
[2]Erdös, P. and Pósa, L. (1965) On independent circuits in a graph. Canad. J. Math. 17 347–352.CrossRefGoogle Scholar
[3]Flajolet, P. and Sedgewick, R. (2009) Analytic Combinatorics, Cambridge University Press.CrossRefGoogle Scholar
[4]Kang, M. and McDiarmid, C. (2011) Random unlabelled graphs containing few disjoint cycles. Random Struct. Alg. 38 174–204.Google Scholar
[5]Kurauskas, V. and McDiarmid, C. (2011) Random graphs containing few disjoint excluded minors. Manuscript.CrossRefGoogle Scholar
[6]Lovász, L. (1965) On graphs not containing independent circuits (Hungarian). Mat. Lapok 16 289–299.Google Scholar
[7]Lovász, L. (1993) Combinatorial Problems and Exercises, second edition, North-Holland.Google Scholar
[8]McDiarmid, C. (2009) Random graphs from a minor-closed class. Combin. Probab. Comput. 18 583–599.Google Scholar
[9]McDiarmid, C., Steger, A. and Welsh, D. (2005) Random planar graphs. J. Combin. Theory Ser. B 93 187–205.CrossRefGoogle Scholar
[10]McDiarmid, C., Steger, A. and Welsh, D. (2006) Random graphs from planar and other addable classes. In Topics in Discrete Mathematics (Klazar, M., Kratochvíl, J., Loebl, M., Matoušek, J., Thomas, R. and Valtr, P., eds), Vol. 26 of Algorithms and Combinatorics, Springer, pp. 231–246.CrossRefGoogle Scholar
[11]Reed, B. A., Robertson, N., Seymour, P. D. and Thomas, R. (1996) Packing directed circuits. Combinatorica 16 535–554.Google Scholar
[12]Rényi, A. (1959) Some remarks on the theory of trees. Publ. Math. Inst. Hung. Acad. Sci. 4 73–85.Google Scholar
[13]Robertson, N. and Seymour, P. (1986) Graph minors V: Excluding a planar graph. J. Combin. Theory Ser. B 41 92–114.Google Scholar