@inproceedings{4ce0e1ed0bc6462f92eabfde8795335b,
title = "The genus of the Erdos-R{\'e}nyi random graph and the fragile genus property",
abstract = "We investigate the genus g(n,m) of the Erdos-R{\'e}nyi 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.",
keywords = "Fragile genus, Genus, Random graphs",
author = "Chris Dowden and Mihyun Kang and Michael Krivelevich",
note = "Publisher Copyright: {\textcopyright} Chris Dowden, Mihyun Kang, and Michael Krivelevich; licensed under Creative Commons License CC-BY.; 29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms, AofA 2018 ; Conference date: 25-06-2018 Through 29-06-2018",
year = "2018",
month = jun,
day = "1",
doi = "https://doi.org/10.4230/LIPIcs.AofA.2018.17",
language = "الإنجليزيّة",
series = "Leibniz International Proceedings in Informatics, LIPIcs",
editor = "Ward, {Mark Daniel} and Fill, {James Allen}",
booktitle = "29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms, AofA 2018",
}