Topology and chromatic number of random $\epsilon$-distance graphs on spheres

TL;DR

This paper investigates the topological bounds on the chromatic number of random spherical graphs, revealing limitations of certain invariants but providing useful bounds in low dimensions.

Contribution

It analyzes the effectiveness of the neighborhood complex connectivity as a bound for the chromatic number in random spherical graphs, extending understanding of topological methods.

Findings

Topological bounds perform poorly in general for these graphs.

The bounds are more effective in dimensions 1 and 2.

The study highlights limitations of topological invariants in high-dimensional random graphs.

Abstract

Given $0<\alpha\leq\pi$, ${\epsilon}>0$ and $n$, we define random graphs on the $d$-dimensional sphere by drawing $n$ i.i.d. uniform random points for the vertices, and edges $u {\sim} v$ whenever the geodesic distance between $u$ and $v$ is ${\epsilon}$-close to ${\alpha}$. This model generalizes distance graphs on spheres, and also random Borsuk graphs. Topological tools are known to give tight bounds for the chromatic number of Borsuk graphs. We now study the efficiency of one of these topological invariants, namely the connectivity of L\'ovasz's neighborhood complex, to bound the chromatic number of this model of random graphs. We show that, in general, this bound performs badly, however, it still produces some useful bounds in dimensions $d=1$ and 2.