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The Genus of the Erdös-Rényi Random Graph and the Fragile Genus Property

Authors Chris Dowden, Mihyun Kang, Michael Krivelevich



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Chris Dowden
  • Institute of Discrete Mathematics, Graz University of Technology, 8010 Graz, Austria
Mihyun Kang
  • Institute of Discrete Mathematics, Graz University of Technology, 8010 Graz, Austria
Michael Krivelevich
  • School of Mathematical Sciences, Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 6997801, Israel

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Chris Dowden, Mihyun Kang, and Michael Krivelevich. The Genus of the Erdös-Rényi Random Graph and the Fragile Genus Property. In 29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 110, pp. 17:1-17:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.AofA.2018.17

Abstract

We investigate the genus g(n,m) of the Erdös-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 n/(2) + O(n^{2/3}) and when m is at least omega (n^{1+1/(j)}) for j in N, and so we focus on intermediate cases. In particular, we show that g(n,m) = (1+o(1)) m/(2) whp (with high probability) when n << m = n^{1+o(1)}; that g(n,m) = (1+o(1)) mu (lambda) m whp for a given function mu (lambda) when m ~ lambda n for lambda > 1/2; and that g(n,m) = (1+o(1)) (8s^3)/(3n^2) whp when m = n/(2) + s for n^(2/3) << s << n. 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 epsilon n edges will whp result in a graph with genus Omega (n), even when epsilon is an arbitrarily small constant! We thus call this the `fragile genus' property.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Random graphs
  • Mathematics of computing → Graphs and surfaces
Keywords
  • Random graphs
  • Genus
  • Fragile genus

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