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- W3023190968 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 $(n_1n_2)^{-frac{1}{2}}$ 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 $p=(n_1n_2)^{-frac{1}{2}}$ 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 $q=q(p,n_1,n_2)$." @default.
- W3023190968 created "2020-05-13" @default.
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- W3023190968 date "2022-06-01" @default.
- W3023190968 modified "2023-10-14" @default.
- W3023190968 title "Planarity and Genus of Sparse Random Bipartite Graphs" @default.
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- W3023190968 doi "https://doi.org/10.1137/20m1341817" @default.
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