Asymptotic enumeration of sparse 2-connected graphs

TL;DR

This paper provides an asymptotic enumeration formula for sparse 2-connected graphs with specified edge and degree conditions, extending previous results and solving a problem posed by Wright in 1983.

Contribution

It introduces a comprehensive asymptotic formula for counting sparse 2-connected graphs, covering a broader range of parameters than prior work, and addresses the enumeration for typical degree sequences.

Findings

Derived asymptotic formulas for 2-connected graphs with given degree sequences.

Extended enumeration results to the entire range of sparse graphs not previously covered.

Solved a longstanding problem of Wright from 1983.

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.