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

Chris Dowden, Mihyun Kang, Michael Krivelevich

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

We investigate the genus g(n,m) of the Erdos-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 behaviour depending on which 'region' m falls into. Existing results are known for when m is at most {equation Presented} and when m is at least {equation Presented} and so we focus on intermediate cases. In particular, we show that {equation Presented} whp (with high probability) when {equation Presented} whp for a given function μ(λ) when m ~ λn for {equation Presented}. We then also show that the genus of fixed graphs 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 n edges will whp result in a graph with genus (n), even when is an arbitrarily small constant! We thus call this the 'fragile genus' property.

Original languageEnglish
Title of host publication29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms, AofA 2018
EditorsMark Daniel Ward, James Allen Fill
ISBN (Electronic)9783959770781
DOIs
StatePublished - 1 Jun 2018
Event29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms, AofA 2018 - Uppsala, Sweden
Duration: 25 Jun 201829 Jun 2018

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume110

Conference

Conference29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms, AofA 2018
Country/TerritorySweden
CityUppsala
Period25/06/1829/06/18

Keywords

  • Fragile genus
  • Genus
  • Random graphs

All Science Journal Classification (ASJC) codes

  • Software

Cite this