Random graphs from structured classes

Abstract

Given a class $\mathcal G$ of graphs, let ${\mathcal G}_n$ denote the set of graphs in $\mathcal G$ on vertex set $[n]$. For certain classes $\mathcal G$, we are interested in the asymptotic behaviour of a random graph $R_n$ sampled uniformly from ${\mathcal G}_n$. Call $\mathcal G$ smooth if $ n |{\mathcal G}_{n-1}| / |{\mathcal G}_n|$ tends to a limit as $n \to \infty$. Showing that a graph class is smooth is a key step in an approach to investigating properties of $R_n$, in particular the asymptotic probability that $R_n$ is connected, and more generally the asymptotic behaviour of the fragment of $R_n$ outside the largest component. The composition method of Bender, Canfield and Richmond shows that the class of graphs embeddable in a given surface is smooth; and similarly we have smoothness for any minor-closed class of graphs with 2-connected excluded minors. Here we develop the approach further, and give results encompassing both these cases and much more. We see that, under quite general conditions, our graph classes are smooth and we can describe for example the limiting distribution of the fragment of $R_n$ and the size of the core; and we obtain similar results for the graphs in the class with minimum degree at least 2.