Ball packings with high chromatic numbers from strongly regular graphs

TL;DR

This paper constructs high chromatic number ball packings using strongly regular graphs derived from generalized quadrangles, surpassing previous bounds and providing new insights into geometric graph coloring.

Contribution

It introduces a novel method to generate ball packings with high chromatic numbers using strongly regular graphs from generalized quadrangles, improving known bounds.

Findings

Ball packings in dimension q^3 - q^2 + q with chromatic number q^3 + 1.

Use of strongly regular graphs from generalized quadrangles.

Improved lower bounds for the chromatic number of ball packings.

Abstract

Inspired by Bondarenko's counter-example to Borsuk's conjecture, we notice some strongly regular graphs that provide examples of ball packings whose chromatic numbers are significantly higher than the dimensions. In particular, from generalized quadrangles we obtain unit ball packings in dimension $q^3-q^2+q$ with chromatic number $q^3+1$, where $q$ is a prime power. This improves the previous lower bound for the chromatic number of ball packings.