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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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