Counting strongly-connected, sparsely edged directed graphs

TL;DR

This paper derives a precise asymptotic formula for counting strongly connected directed graphs with given vertices and edges, solving a longstanding problem and extending classic undirected graph enumeration results.

Contribution

It provides the first asymptotic enumeration formula for strongly connected digraphs with specified parameters, including an explicit error term, addressing a problem posed in 1977.

Findings

Asymptotic formula for strongly connected digraphs with $m-n oinite$

Explicit error term included in the enumeration

Extension of classic undirected graph enumeration to directed graphs

Abstract

A sharp asymptotic formula for the number of strongly connected digraphs on $n$ labelled vertices with $m$ arcs, under a condition $m-n\to\infty$, $m=O(n)$, is obtained; this solves a problem posed by Wright back in $1977$. Our formula is a counterpart of a classic asymptotic formula, due to Bender, Canfield and McKay, for the total number of connected undirected graphs on $n$ vertices with $m$ edges. A key ingredient of their proof was a recurrence equation for the connected graphs count due to Wright. No analogue of Wright's recurrence seems to exist for digraphs. In a previous paper with Nick Wormald we rederived the BCM formula via counting two-connected graphs among the graphs of minimum degree $2$, at least. In this paper, using a similar embedding for directed graphs, we find an asymptotic formula, which includes an explicit error term, for the fraction of strongly-connected digraphs with parameters $m$ and $n$ among all such digraphs with positive in/out-degrees.