Connectivity in bridge-addable graph classes: The McDiarmid-Steger-Welsh conjecture

The study of typical properties of random graphs is of particular importance for the theoretical analysis of complex networks. In this field, many models of randomness (such as Erdös-Rényi or random planar graphs, preferential attachment models) have been successfully analysed thanks to the fact that their underlying structure enables one to perform explicit computations of some observables. Another approach, pioneered by McDiarmid, Steger and Welsh (2005) is to consider graphs taken uniformly from an abstract graph class, assuming only some global property of the class but without fully specifying it. Despite the fact that exact computations are no longer possible, results obtained in this setup are arguably very robust, since they apply universally for many different models of random graphs. The foundational and most studied problem in this topic is a conjecture of these authors on bridge-addable classes that we prove in this paper. A class of graphs is bridge-addable if any graph obtained by adding an edge between two connected components of a graph in the class, is also in the class. Examples of bridge-addable classes include forests, planar graphs, graphs with bounded tree-width, or graphs excluding any 2-connected minor. We prove that a random graph from a bridge-addable class is connected with probability at least e^-1/2 + o(l), when its number of vertices tends to infinity. This lower bound is tight since it is reached for forests. The best previously known constants where e^-1, e^-7983 and e^-2/3 proved respectively by McDiarmid, Steger and Welsh, by Balister, Bollobas and Gerke, and by Norin.