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Random Graphs from a Minor-Closed Class

Published online by Cambridge University Press:  01 July 2009

COLIN McDIARMID
COLIN McDIARMID*
Affiliation:
Department of Statistics, Oxford University (e-mail: cmcd@stats.ox.ac.uk)
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Abstract

A minor-closed class of graphs is addable if each excluded minor is 2-connected. We see that such a class of labelled graphs has smooth growth; and, for the random graph Rn sampled uniformly from the n-vertex graphs in , the fragment not in the giant component asymptotically has a simple ‘Boltzmann Poisson distribution’. In particular, as n → ∞ the probability that Rn is connected tends to 1/A(ρ), where A(x) is the exponential generating function for and ρ is its radius of convergence.

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Type
Paper
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Combinatorics, Probability and Computing , Volume 18 , Issue 4 , July 2009 , pp. 583 - 599
Copyright
Copyright © Cambridge University Press 2009

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References

[1]Addario-Berry, L., McDiarmid, C. and Reed, B. (2007) Connectivity of addable monotone graph classes. Manuscript.Google Scholar
[2]Balister, P., Bollobás, B. and Gerke, S. (2008) Connectivity of addable graph classes. J. Combin. Theory Ser. B 98 577–584.CrossRefGoogle Scholar
[3]Bell, J., Bender, E. A., Cameron, P. J. and Richmond, L. B. (2000) Asymptotics for the probability of connectedness and the distribution of number of components. Electron. J. Combin. 7 #33.CrossRefGoogle Scholar
[4]Bender, E. A., Canfield, E. R. and Richmond, L. B. (2008) Coefficients of functional compositions often grow smoothly. Electron. J. Combin. 15 #21.CrossRefGoogle Scholar
[5]Bender, E. A., Gao, Z. and Wormald, N. C. (2002) The number of labeled 2-connected planar graphs. Electron. J. Combin. 9 #43.CrossRefGoogle Scholar
[6]Bernardi, O., Noy, M. and Welsh, D. (2007) On the growth rate of minor-closed classes of graphs. Manuscript: arXiv:0710.2995.Google Scholar
[7]Bodirsky, M., Giménez, O., Kang, M. and Noy, M. (2005) On the number of series-parallel and outerplanar graphs. In Proc. European Conference on Combinatorics, Graph Theory, and Applications (EuroComb 2005), Discrete Math. Theor. Comput. Sci. Proc., Vol. AE, pp. 383–388.CrossRefGoogle Scholar
[8]Bodirsky, M., Giménez, O., Kang, M. and Noy, M. (2007) Enumeration and limit laws for series-parallel graphs. Europ. J. Combin. 28 2091–2105.CrossRefGoogle Scholar
[9]Bodirsky, M., Löffler, M. L., Kang, M. and McDiarmid, C. (2007) Random cubic planar graphs. Random Struct. Algebra 30 78–94.CrossRefGoogle Scholar
[10]Compton, K. J. (1987) Some methods for computing component distribution probabilities in relational structures. Discrete Math. 66 59–77.CrossRefGoogle Scholar
[11]Diestel, R. (2005) Graph Theory, Springer.Google Scholar
[12]Gerke, S., Giménez, O., Noy, M. and Weiβl, A. (2008) The number of graphs not containing K 3,3 as a minor. Electron. J. Combin. 15 R114.CrossRefGoogle Scholar
[13]Giménez, O. and Noy, M. (2009) Asymptotic enumeration and limit laws of planar graphs. J. Amer. Math. Soc. 22 309–329.CrossRefGoogle Scholar
[14]Giménez, O., Noy, M. and Rué, J. (2007) Graph classes with given 3-connected components: Asymptotic counting and critical phenomena. Manuscript.CrossRefGoogle Scholar
[15]Janson, S., Łuczak, T. and Ruciński, A. (2000) Random Graphs, Wiley Interscience.CrossRefGoogle Scholar
[16]Kolchin, V. F. (1986) Random Mappings, Optimization Software Inc., New York.Google Scholar
[17]Kolchin, V. F. (1999) Random Graphs, CUP.Google Scholar
[18]Kurauskas, V. and McDiarmid, C. (2008) Random graphs without disjoint cycles. In preparation.Google Scholar
[19]Mader, W. (1968) Homomorphiesätze für Graphen. Math. Ann. 178 154–168.CrossRefGoogle Scholar
[20]McDiarmid, C. (2008) Random graphs on surfaces. J. Combin. Theory Ser. B 98 778–797.CrossRefGoogle Scholar
[21]McDiarmid, C. (2008) Minor-closed unlabelled graph classes. In preparation.Google Scholar
[22]McDiarmid, C. (2008) Forests of paths and caterpillars. In preparation.Google Scholar
[23]McDiarmid, C. (2008) A combinatorial proof of smoothness. In preparation.Google Scholar
[24]McDiarmid, C. and Reed, B. (2008) On the maximum degree of a random planar graph. Combin. Probab. Comput. 17 591–601.CrossRefGoogle Scholar
[25]McDiarmid, C., Steger, A. and Welsh, D. (2005) Random planar graphs. J. Combin. Theory Ser. B 93 187–205.CrossRefGoogle Scholar
[26]McDiarmid, C., Steger, A. and Welsh, D. (2006) Random graphs from planar and other addable classes. In Topics in Discrete Mathematics (Klazar, M., Kratochvil, J., Loebl, M., Matousek, J., Thomas, R. and Valtr, P., eds), Vol. 26 of Algorithms and Combinatorics, Springer, pp. 231–246.CrossRefGoogle Scholar
[27]Moon, J. W. (1970) Counting Labelled Trees, Vol. 1 of Canadian Mathematical Monographs.Google Scholar
[28]Norine, S., Seymour, P., Thomas, R. and Wollan, P. (2006) Proper minor-closed families are small. J. Combin. Theory Ser. B 96 754–757.CrossRefGoogle Scholar
[29]Rényi, A. (1959) Some remarks on the theory of trees. Publications of the Mathematical Institute of the Hungarian Academy of Sciences 4 73–85.Google Scholar
[30]Robertson, N., Seymour, P. D. and Thomas, R. (1993) Hadwiger's conjecture for K 6-free graphs. Combinatorica 13 279–361.CrossRefGoogle Scholar
[31]Robertson, N. and Seymour, P. D. (1983–2004) Graph minors I–XX. J. Combin. Theory Ser. B.Google Scholar