Asymptotic enumeration of sparse 2-connected graphs

Abstract

We determine an asymptotic formula for the number of labelled 2-connected (simple) graphs on $n$ vertices and $m$ edges, provided that $m-n\to \infty$ and $m=O(n\log n)$ as $n\to \infty$. This is the entire range of $m$ not covered by previous results. The proof involves determining properties of the core and kernel of random graphs with minimum degree at least 2. The case of 2-edge-connectedness is treated similarly. We also obtain formulae for the number of 2-connected graphs with given degree sequence for most (`typical') sequences. Our main result solves a problem of Wright from 1983.