Asymptotics for the number of bipartite graphs with fixed surplus

TL;DR

This paper derives asymptotic formulas for counting bipartite graphs with a fixed surplus, expanding on recent bounds by employing probabilistic encodings of bipartite random graphs.

Contribution

It introduces a probabilistic approach to obtain asymptotics for the number of bipartite graphs with fixed surplus, building on recent bounds.

Findings

Provides asymptotic formulas for large bipartite graphs with fixed surplus.

Uses probabilistic encodings to derive enumeration results.

Extends previous upper bounds to precise asymptotics.

Abstract

In a recent work on the bipartite Erd\H{o}s-R\'{e}nyi graph, Do et al. (2023) established upper bounds on the number of connected labeled bipartite graphs with a fixed surplus. We use some recent encodings of bipartite random graphs in order to provide a probabilistic formula for the number of bipartite graphs with fixed surplus. Using this, we obtain asymptotics as the number of vertices in each class tend to infinity.