Asymptotic enumeration of graph factors by cumulant expansion

TL;DR

This paper develops a precise asymptotic enumeration method for subgraphs with a given degree sequence in dense, well-expanding graphs, extending to irregular degrees and providing explicit terms for regular graphs.

Contribution

It introduces a novel cumulant expansion approach for counting graph factors, applicable under weak conditions and handling large degree variations.

Findings

Derived an asymptotic formula for the number of subgraphs with specified degrees.

Established explicit terms in the expansion for regular graphs.

Developed a new Fourier inversion technique for complex integral bounds.

Abstract

Let $G$ be a dense graph with good expansion properties and not too close to being bipartite. Let $\boldsymbol d$ be a graphical degree sequence. Under very weak conditions, we find the number of subgraphs of $G$ with degree sequence $\boldsymbol d$ to arbitrary precision. The average degree can be any power of $n$ and the variation in degrees can be very large. The method uses an explicit bound on the tail of the cumulant generating function found by the first author. As a first application, we prove that there is an asymptotic expansion for the number of regular graphs and find several terms explicitly. We believe that this is the first combinatorial application of the Fourier inversion method for which the integral outside the dominant regions cannot be bounded by the integral of the absolute value, and we give a general method for dealing with that situation.