On the chromatic number of structured Cayley graphs

TL;DR

This paper investigates the chromatic number of Cayley graphs derived from algebraic groups, establishing lower bounds using advanced number theory and representation theory, with applications to matrices over finite fields and rings.

Contribution

It introduces new lower bounds for the chromatic number of Cayley graphs of algebraic groups using Lang-Weil and Weil bounds, extending to matrices over finite fields and rings.

Findings

Established lower bounds for chromatic numbers of Cayley graphs from algebraic groups.

Applied Lang-Weil bound and representation theory to derive bounds.

Proved analogous results for $ ext{SL}_2$ over finite rings using Weil's bound.

Abstract

In this paper, we will study the chromatic number of Cayley graphs of algebraic groups that arise from algebraic constructions. Using Lang-Weil bound and representation theory of finite simple groups of Lie type, we will establish lower bounds on the chromatic number of these graphs. This provides a lower bound for the chromatic number of Cayley graphs of the regular graphs associated to the ring of $n\times n$ matrices over finite fields. Using Weil's bound for Kloosterman sums we will also prove an analogous result for $\mathrm{SL}_2$ over finite rings.