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

Chris Dowden, Mihyun Kang, Michael Krivelevich, Marc Herbstritt
2018 International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms
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 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 ω n 1+ 1 j for j ∈ 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
more » ... (n, m) = (1 + o(1))µ(λ)m whp for a given function µ(λ) when m ∼ λn for λ > 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 n edges will whp result in a graph with genus Ω(n), even when is an arbitrarily small constant! We thus call this the 'fragile genus' property.