The genus of a random bipartite graph

TL;DR

This paper extends the understanding of the genus of random bipartite graphs, identifying phase transitions in genus growth depending on the size of the parts and the probability parameter.

Contribution

It provides the first analysis of genus phase transitions in bipartite graphs, generalizing previous results for non-bipartite graphs and exploring different size regimes.

Findings

Phase transitions occur at specific probabilities related to the sizes of bipartite parts.

Different behaviors are observed when one part of the bipartite graph is of constant size.

Genus growth patterns depend on the relation between $p$, $n_1$, and $n_2$.

Abstract

Archdeacon 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 $p=p(n)\geq3(\ln n)^2n^{-1/2}$. 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 $p=\Theta((n_1n_2)^{-\frac{i}{2i+1}})$. A different behaviour is exhibited when one of the bipartite parts has constant size, $n_1\gg1$ and $n_2$ is a constant. In that case, phase transitions occur when $p=\Theta(n_1^{-1/2})$ and when $p=\Theta(n_1^{-1/3})$.