TY - UNPB

T1 - Planarity and genus of sparse random bipartite graphs

AU - Do, Tuan Anh

AU - Erde, Joshua

AU - Kang, Mihyun

PY - 2020/5/8

Y1 - 2020/5/8

N2 - 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 $G(n_1,n_2,p)$, with partition classes of size $n_1$ and $n_2$, was initiated by Mohar and Ying, who showed that when $n_1$ and $n_2$ are comparable in size and $p=p(n_1,n_2)$ is significantly larger than $(n_1n_2)^{-\frac{1}{2}}$ the genus of the random bipartite graph has a similar behaviour 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 $p=(n_1n_2)^{-\frac{1}{2}}$ and investigate the genus close to this threshold, extending the results of Mohar and Ying. It turns out that there is qualitatively different behaviour in the case where $n_1$ and $n_2$ are comparable, when whp the genus is linear in the number of edges, than in the case where $n_1$ is asymptotically smaller than $n_2$, when whp the genus behaves like the genus of a sparse random graph $G(n_1,q)$ for an appropriately chosen $q=q(p,n_1,n_2)$.

AB - 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 $G(n_1,n_2,p)$, with partition classes of size $n_1$ and $n_2$, was initiated by Mohar and Ying, who showed that when $n_1$ and $n_2$ are comparable in size and $p=p(n_1,n_2)$ is significantly larger than $(n_1n_2)^{-\frac{1}{2}}$ the genus of the random bipartite graph has a similar behaviour 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 $p=(n_1n_2)^{-\frac{1}{2}}$ and investigate the genus close to this threshold, extending the results of Mohar and Ying. It turns out that there is qualitatively different behaviour in the case where $n_1$ and $n_2$ are comparable, when whp the genus is linear in the number of edges, than in the case where $n_1$ is asymptotically smaller than $n_2$, when whp the genus behaves like the genus of a sparse random graph $G(n_1,q)$ for an appropriately chosen $q=q(p,n_1,n_2)$.

KW - Random graph

KW - genus

M3 - Working paper

BT - Planarity and genus of sparse random bipartite graphs

ER -