The genus of the Erdős-Rényi random graph and the fragile genus property

Chris Dowden, Mihyun Kang, Michael Krivelevich

Research output: Contribution to journalArticlepeer-review

Abstract

We investigate the genus g(n,m) of the Erdős-Rényi random graph G(n,m), providing a thorough description of how this relates to the function m = m(n), and finding that there is different behavior depending on which “region” m falls into. Results already exist for (Formula presented.) and (Formula presented.) for (Formula presented.), and so we focus on the intermediate cases. We establish that (Formula presented.) whp (with high probability) when n ≪ m = n1 + o(1), that g(n,m) = (1 + o(1))μ(λ)m whp for a given function μ(λ) when m∼λn for (Formula presented.), and that (Formula presented.) whp when (Formula presented.) for n2/3 ≪ s ≪ n. We then also show that the genus of a fixed graph can increase dramatically if a small number of random edges are added. Given any connected graph with bounded maximum degree, we find that the addition of ϵn edges will whp result in a graph with genus Ω(n), even when ϵ is an arbitrarily small constant! We thus call this the “fragile genus” property.

Original languageEnglish
Pages (from-to)97-121
Number of pages25
JournalRandom Structures and Algorithms
Volume56
Issue number1
DOIs
StatePublished - 1 Jan 2020

Keywords

  • fragile genus
  • genus
  • random graphs

All Science Journal Classification (ASJC) codes

  • Software
  • Applied Mathematics
  • General Mathematics
  • Computer Graphics and Computer-Aided Design

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