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

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Abstract:

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^{...More

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