Bridge-Addability, Edge-Expansion and Connectivity

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.

Pendant appearances and components in random graphs from structured classes

“…In this section we present and prove two lemmas, Lemmas 4.1 and 4.2, which extend some earlier results on bridge-addable sets of graphs, see [1,11,14,16,19,23,24]. We shall use these new results in the proof of Lemma 5.2, which is used in the proof of Theorem 1.4, and elsewhere in the proof of Theorem 1.4.…”