On the connectivity of random graphs from addable classes

Abstract

A class A of graphs is called weakly addable (or bridge-addable) if for any G∈A and any two distinct components C1 and C2 in G, any graph that can be obtained by adding an edge between C1 and C2 is also in A. McDiarmid, Steger and Welsh (2006) conjectured in [6] that a graph chosen uniformly at random among all graphs with n vertices in a weakly addable A is connected with probability at least e−1/2+o(1), as n→∞. In this paper we show that the conjecture is true under a stronger assumption. A class G of graphs is called bridge-alterable, if for any G∈G and any bridge e in G, G∈G if and only if G−e∈G. We prove that a graph chosen uniformly at random among all graphs with n vertices in a bridge-alterable G is connected with probability at least e−1/2+o(1), as n→∞.The main tool in our analysis is a tight enumeration result that addresses the number of ways in which a given forest can be complemented to a forest with fewer components.