Planarity and genus of sparse random bipartite graphs

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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 languageEnglish
Pages (from-to)1394-1415
Number of pages22
JournalSIAM Journal on Discrete Mathematics
Volume36
Issue number2
DOIs
Publication statusPublished - 13 Jun 2022

Keywords

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

ASJC Scopus subject areas

  • Mathematics(all)

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