On the Genus of Random Regular Graphs

TL;DR

This paper establishes that the genus of a random d-regular graph on n nodes is approximately rac{(d - 2)}{4}n with high probability, providing a key topological invariant for such graphs.

Contribution

It provides a probabilistic formula for the genus of random regular graphs, linking graph degree and size to topological complexity.

Findings

Genus of random d-regular graphs scales linearly with n.

High probability bounds for the genus are established.

The genus depends on the degree d of the regular graph.

Abstract

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 $\varepsilon > 0$.