English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

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 onthe 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 $m \sim \lambdan$ 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 $n^{2/3} \ll s \lln$.  We then also show that the genus of a fixed graph can increase dramaticallyif a small number of random edges are added. Given any connected graph withbounded maximum degree, we find that the addition of $\epsilon n$ edges willwhp result in a graph with genus $\Omega (n)$, even when $\epsilon$ is anarbitrarily small constant! We thus call this the `fragile genus' property.

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