Mittagsseminar (in cooperation with J. Lengler, A. Steger, and D. Steurer)

Date and Time: Thursday, November 07, 2013, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Kerstin Weller (Univesity of Oxford)

Connectivity of relatively bridge-addable classes of graphs

In 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.
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 start with a bridge-addable class A and a host graph H, and consider the set of subgraphs of H in A. Our connectivity bound involves the edge-expansion properties of the host graph H. We also give a bound on the expected number of vertices not in the largest component. Furthermore, we investigate whether these bounds are tight, and in particular give detailed results about random forests in balanced complete multipartite graphs.

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