Component Behaviour and Excess of Random Bipartite Graphs Near the Critical Point

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