Planarity and genus of sparse random bipartite graphs

The genus of the binomial random graph G(n, p) is well understood for a wide range of p = p(n). Recently, the study of the genus of the random bipartite graph G(n ₁, n ₂, p), with partition classes of size n ₁ and n ₂, was initiated by Mohar and Jing, who showed that when n ₁ and n ₂ are comparable in size and p = p(n ₁, n ₂) is significantly larger than (n ₁n ₂) ^1/2 the genus of the random bipartite graph has a similar behavior to that of the binomial random graph. In this paper we show that there is a threshold for planarity of the random bipartite graph at p = (n ₁n ₂) ^1/2 and investigate the genus close to this threshold, extending the results of Mohar and Jing. It turns out that there is qualitatively different behavior in the case where n ₁ and n ₂ are comparable, when with high probability (whp) the genus is linear in the number of edges, than in the case where n ₁ is asymptotically smaller than n ₂, when whp the genus behaves like the genus of a sparse random graph G(n ₁, q) for an appropriately chosen q = q(p, n ₁, n ₂).