Bojan Mohar: The Genus of a Random Bipartite Graph

Povzetek. Archdeacon and Grable (1995) proved that the genus of the random graph $G\in G_{n,p}$ is almost surely close to $pn^2/12$ if $p=p(n) \geq 3(\ln n)^2n^{-1/2}$. We prove an analogous result for random bipartite graphs in $G_{n_1,n_2,p}$. If $n_1\ge n_2 \gg 1$, phase transitions occur for every positive integer $i$ when $p = \Theta((n_1n_2)^{-i/(2i+1)})$. 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 $p = \Theta(n_1^{-1/3})$. The last phase transition leads to and especially interesting problem about genus embeddings of complete 3-uniform hypergraphs.

This is a joint work with Yifan Jing.