The genus of a random bipartite graph

Archdeacon and Grable (1995) proved that the genus of the random graph $G\in {\mathcal{G}}_{n,p}$ is almost surely close to $p{n}^2\mathrm{/}12$ if $p=p(n)\geq3(\ln n)^2n^{-1/2}$. In this paper we prove an analogous result for random bipartite graphs in ${\mathcal{G}}_{n_1,n_2,p}$. If $n_1\ge n_2\gg 1$, phase transitions occur for every positive integer $i$ when $p=\mathrm{\Theta }((n_1{n}_2{)}^{-\frac{i}{2i+1}})$. A different behaviour is exhibited when one of the bipartite parts has constant size, $n_1\gg 1$ and $n_2$ is a constant. In that case, phase transitions occur when $p=\mathrm{\Theta }(n_1^{-1\mathrm{/}2})$ and when $p=\mathrm{\Theta }(n_1^{-1\mathrm{/}3})$.