Connectivity of random addable graphs

A non-empty class A of labelled graphs is weakly addable if for each graph G ∈ A and any two distinct components of G, any graph that can be obtain by adding an edge between the two components is also in A. For a weakly addable graph class A, we consider a random element Rn chosen uniformly from the set of all graph in A on the vertex set {1, . . . , n}. McDiarmid, Steger and Welsh conjecture [5] that the probability that Rn is connected is at least e −1/2 + o(1) as n → ∞, and showed that it is at least e−1 for all n. Balister, Bollobás and Gerke improved the result by showing that this probability is at least e−0.7983 for sufficiently large n. In this paper the results on the connectivity of random addable graphs are surveyed and some extensions of the conjecture are discussed.