  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 $\varepsilon > 0$.
