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Bridge-Addability, Edge-Expansion and Connectivity

Part of: Combinatorial probability Graph theory

Published online by Cambridge University Press:  11 May 2017

COLIN McDIARMID  and
KERSTIN WELLER
COLIN McDIARMID
Affiliation:
Department of Statistics, University of Oxford, 24-29 St Giles', Oxford OX1 3LB, UK (e-mail: cmcd@stats.ox.ac.uk)
KERSTIN WELLER
Affiliation:
Department of Statistics, University of Oxford, 24-29 St Giles', Oxford OX1 3LB, UK (e-mail: cmcd@stats.ox.ac.uk) Institut für Theoretische Informatik, Eigenössische Technische Hochschule Zürich, 8092 Zürich, Switzerland
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Abstract

A class of graphs is called bridge-addable if, for each graph in the class and each pair u and v of vertices in different components, the graph obtained by adding an edge joining u and v must also be in the class. The concept was introduced in 2005 by McDiarmid, Steger and Welsh, who showed that, for a random graph sampled uniformly from such a class, the probability that it is connected is at least 1/e.

We generalize this and related results to bridge-addable classes with edge-weights which have an edge-expansion property. Here, a graph is sampled with probability proportional to the product of its edge-weights. We obtain for example lower bounds for the probability of connectedness of a graph sampled uniformly from a relatively bridge-addable class of graphs, where some but not necessarily all of the possible bridges are allowed to be introduced. Furthermore, we investigate whether these bounds are tight, and in particular give detailed results about random forests in complete balanced multipartite graphs.

MSC classification

Primary: 05C80: Random graphs
Secondary: 60C05: Combinatorial probability
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 26 , Issue 5 , September 2017 , pp. 697 - 719
Copyright
Copyright © Cambridge University Press 2017 

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References

[1] Addario-Berry, L., McDiarmid, C. and Reed, B. (2012) Connectivity for bridge-addable monotone graph classes. Combin. Probab. Comput. 21 803–815.CrossRefGoogle Scholar
[2] Alon, N. and Spencer, J. H. (2008) The Probabilistic Method, Wiley-Interscience Series in Discrete Mathematics and Optimization, Wiley.CrossRefGoogle Scholar
[3] Austin, T. L. (1960) The enumeration of point labelled chromatic graphs and trees. Canad. J. Math. 12 535–545.CrossRefGoogle Scholar
[4] Balister, P., Bollobás, B. and Gerke, S. (2008) Connectivity of addable graph classes. J. Combin. Theory Ser. B 98 577–584.CrossRefGoogle Scholar
[5] Balister, P., Bollobás, B. and Gerke, S. (2010) Connectivity of random addable graphs. In Proc. ICDM 2008, number 13, pp. 127–134.Google Scholar
[6] Bollobás, B. (2001) Random Graphs, Cambridge Studies in Advanced Mathematics, Cambridge University Press.CrossRefGoogle Scholar
[7] Chapuy, G. and Perarnau, G. (2015) Connectivity in bridge-addable graph classes: The McDiarmid–Steger–Welsh conjecture. arXiv:1504.06344 Google Scholar
[8] Flajolet, P. and Sedgewick, R. (2009) Analytic Combinatorics, Cambridge University Press.CrossRefGoogle Scholar
[9] Kang, M. and Panagiotou, K. (2013) On the connectivity of random graphs from addable classes. J. Combin. Theory Ser. B 103 306–312.CrossRefGoogle Scholar
[10] McDiarmid, C. (2009) Random graphs from a minor-closed class. Combin. Probab. Comput. 18 583–599.CrossRefGoogle Scholar
[11] McDiarmid, C. (2012) Connectivity for random graphs from a weighted bridge-addable class. Electron. J. Combin. 19 P53.CrossRefGoogle Scholar
[12] McDiarmid, C. (2013) Random graphs from a weighted minor-closed class. Electron. J. Combin. 20 P52.CrossRefGoogle Scholar
[13] McDiarmid, C., Steger, A. and Welsh, D. J. A. (2005) Random planar graphs. J. Combin. Theory Ser. B 93 187–205.CrossRefGoogle Scholar
[14] McDiarmid, C., Steger, A. and Welsh, D. J. A. (2006) Random graphs from planar and other addable classes. In Topics in Discrete Mathematics, Vol. 26 of Algorithms and Combinatorics, Springer, pp. 231–246.CrossRefGoogle Scholar
[15] McDiarmid, C. and Weller, K. (2014) Relatively bridge-addable classes of graphs (extended abstract). In LATIN 2014, Theoretical Informatics: Proc. 11th Latin American Symposium (Pardo, A. and Viola, A., eds), Vol. 8392 of Lecture Notes in Computer Science, Springer, pp. 391–398.CrossRefGoogle Scholar
[16] Moon, J. W. (1970) Counting Labelled Trees. From lectures delivered to the Twelfth Biennial Seminar of the Canadian Mathematical Congress (Vancouver) 1969: Canadian Mathematical Congress, Montreal, Quebec.Google Scholar
[17] Norine, S. (2013) Connectivity of addable classes of forests (draft). http://www.math.mcgill.ca/snorin/papers/Forests.pdf Google Scholar
[18] Rényi, A. (1959) Some remarks on the theory of trees. Magyar Tud. Akad. Mat. Kutató Int. Közl. 4 73–85.Google Scholar
[19] Scoins, H. I. (1962) The number of trees with nodes of alternate parity. Proc. Cambridge Philos. Soc. 58 12–16.CrossRefGoogle Scholar