Asymptotic enumeration of sparse 2-connected graphs

الأصل · EN

We determine an asymptotic formula for the number of labelled 2-connected (simple) graphs on n vertices and m edges, provided that m-n→∞ and m=O(n n) as n→∞. 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.