Component behaviour and excess of random bipartite graphs near the critical point

TL;DR

This paper analyzes the component structure of random bipartite graphs near the critical point, revealing the emergence of a giant component and providing detailed asymptotic results and bounds.

Contribution

It introduces new bounds for bipartite graphs and characterizes the phase transition and component distribution near the critical point.

Findings

Existence of a unique giant component near the critical point

Asymptotic size and excess of the giant component

Distribution of the number of small components

Abstract

The binomial random bipartite graph $G(n,n,p)$ is the random graph formed by taking two partition classes of size $n$ and including each edge between them independently with probability $p$. It is known that this model exhibits a similar phase transition as that of the binomial random graph $G(n,p)$ as $p$ passes the critical point of $\frac{1}{n}$. We study the component structure of this model near to the critical point. We show that, as with $G(n,p)$, for an appropriate range of $p$ there is a unique `giant' component and we determine asymptotically its order and excess. We also give more precise results for the distribution of the number of components of a fixed order in this range of $p$. These results rely on new bounds for the number of bipartite graphs with a fixed number of vertices and edges, which we also derive.