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(n1, n2, p), with partition classes of size n1 and n2, was initiated by Mohar and Jing, who showed that when n1 and n2 are comparable in size and p = p(n1, n2) is significantly larger than (n1n2) 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 = (n1n2) 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 n1 and n2 are comparable, when with high probability (whp) the genus is linear in the number of edges, than in the case where n1 is asymptotically smaller than n2, when whp the genus behaves like the genus of a sparse random graph G(n1, q) for an appropriately chosen q = q(p, n1, n2). © 2022 Society for Industrial and Applied Mathematics.