The genus of a random bipartite graph

Abstract

Archdeacon and Grable (1995) proved that the genus of the random graph $G\in {\mathcal{G}}_{n,p}$ is almost surely close to $p{n}^2/12$ if $p=p(n)\ge 3(\ln n{)}^2{n}^{-1/2}$. 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 $p=\Theta ((n_1{n}_2{)}^{-\frac{i}{2i+1}})$. 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 $p=\Theta (n_1^{-1/2})$ and when $p=\Theta (n_1^{-1/3})$.

arXiv comment19 pages