On the Genus of Random Regular Graphs

The genus of a graph is a topological invariant that measures the minimum genus of a surface on which the graph can be embedded without any edges crossing. Graph genus plays a fundamental role in topological graph theory, used to classify and study different types of graphs and their properties. We show that, for any integer $d \geq 2$, the genus of a random $d$-regular graph on $n$ nodes is $\frac{(d - 2)}{4}n(1 - \varepsilon)$ with high probability for any $\varepsilon > 0$.