On the Genus of Random Regular Graphs

The orientable genus of a graph $G$ is the minimum number of handles required to embed that graph on a surface. Determining graph genus is a fundamental yet often difficult task. 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 $\varepsilon > 0$Comment: 3 page