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 1n 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 1n 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 |
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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)