Connectivity of addable graph classes

Abstract

A non-empty class A of labeled graphs is weakly addable if for each graph G ∈ A and any two distinct components of G, any graph that can be obtained by adding an edge between the two components is also in A . For a weakly addable graph class A , we consider a random element R n chosen uniformly from the set of all graphs in A on the vertex set { 1 , … , n } . McDiarmid, Steger and Welsh conjecture that the probability that R n is connected is at least e − 1 / 2 + o ( 1 ) as n → ∞ , and showed that it is at least e −1 for all n [C. McDiarmid, A. Steger, D.J.A. Welsh, Random graphs from planar and other addable classes, in: M. Klazar, J. Kratochvil, M. Loebl, J. Matousek, R. Thomas, P. Valtr (Eds.), Topics in Discrete Mathematics, Dedicated to Jarik Nešetril on the occasion of his 60th birthday, Algorithms Combin., vol. 26, Springer-Verlag, Berlin, 2006, pp. 231–246]. We improve the result, and show that this probability is at least e −0.7983 for sufficiently large n. We also consider 2-addable graph classes B where for each graph G ∈ B and for any two distinct components of G, the graphs that can be obtained by adding at most 2 edges between the components are in B . We show that a random element of a 2-addable graph class on n vertices is connected with probability tending to 1 as n tends to infinity.