## Abstract

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 Jing, 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
_{1}n
_{2})
^{1/2} the genus of the random bipartite graph has a similar behavior 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
_{1}n
_{2})
^{1/2} and investigate the genus close to this threshold, extending the results of Mohar and Jing. It turns out that there is qualitatively different behavior in the case where n
_{1} and n
_{2} are comparable, when with high probability (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}).

Original language | English |
---|---|

Pages (from-to) | 1394-1415 |

Number of pages | 22 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 36 |

Issue number | 2 |

DOIs | |

Publication status | Published - 13 Jun 2022 |

## Keywords

- Random graph
- genus
- random bipartite graphs
- components
- cycles
- random graphs
- faces

## ASJC Scopus subject areas

- Mathematics(all)