The genus of the ErdH{o}s-R\'enyi random graph and the fragile genus property

We investigate the genus $g(n,m)$ of the Erd\H{o}s-R\'enyi 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. Results already exist for $m \le \frac{n}{2} + O(n^{2/3})$ and $m = \omega \left( n^{1+\frac{1}{j}} \right)$ for $j \in \mathbb{N}$, and so we focus on the intermediate cases. We establish that $g(n,m) = (1+o(1)) \frac{m}{2}$ whp (with high probability) when $n \ll m = n^{1+o(1)}$, that $g(n,m) = (1+o(1)) \mu (\lambda) m$ whp for a given function $\mu (\lambda)$ when $m \sim \lambda n$ for $\lambda > \frac{1}{2}$, and that $g(n,m) = (1+o(1)) \frac{8s^{3}}{3n^{2}}$ whp when $m = \frac{n}{2} + s$ for $n^{2/3} \ll s \ll n$. 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 $\epsilon n$ 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.