Relatively Bridge-Addable Classes of Graphs

Abstract

AbstractIn recent years there has been a growing interest in random graphs sampled uniformly from a suitable structured class of (labelled) graphs, such as planar graphs. In particular, bridge-addable classes have received considerable attention. 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.In this extended abstract, we generalise this result to relatively bridge-addable classes of graphs, which are classes of graphs where some but not necessarily all of the possible bridges are allowed to be introduced. We also give a bound on the expected number of vertices not in the largest component. These results are related to the theory of expander graphs. Furthermore, we investigate whether these bounds are tight, and in particular give detailed results about random forests in the bipartite graph K n/2,n/2.Keywordsrandom graphslabelled graphsbridge-addableexpander-graphsforests in K n,n