Component Behaviour of Random Bipartite Graphs

Abstract

We study the component behaviour of the binomial random bipartite graph G(n, n, p) near the critical point. We show that, as is the case in the binomial random graph G(n, p), for an appropriate range of p there is a unique ‘giant’ component of order at least \(n^{\frac{2}{3}}\) and determine asymptotically its order and excess. Our proofs rely on good enumerative estimates for the number of bipartite graphs of a fixed order, as well as probabilistic techniques such as the sprinkling method.