The Genus of a Random Bipartite Graph

Abstract

AbstractArchdeacon and Grable (1995) proved that the genus of the random graph$G\in {\mathcal{G}}_{n,p}$is almost surely close to$pn^{2}/12$if$p=p(n)\geqslant 3(\ln n)^{2}n^{-1/2}$. In this paper we prove an analogous result for random bipartite graphs in${\mathcal{G}}_{n_{1},n_{2},p}$. If$n_{1}\geqslant n_{2}\gg 1$, phase transitions occur for every positive integer$i$when$p=\unicode[STIX]{x1D6E9}((n_{1}n_{2})^{-i/(2i+1)})$. A different behaviour is exhibited when one of the bipartite parts has constant size,i.e.,$n_{1}\gg 1$and$n_{2}$is a constant. In that case, phase transitions occur when$p=\unicode[STIX]{x1D6E9}(n_{1}^{-1/2})$and when$p=\unicode[STIX]{x1D6E9}(n_{1}^{-1/3})$.