The Chromatic Number of Finite Group Cayley Tables

TL;DR

This paper investigates the chromatic number of Cayley tables of finite groups, proving exact values for Abelian groups and improving bounds for general groups, advancing understanding of partial transversals in Latin squares.

Contribution

It provides a complete characterization of the chromatic number for Abelian group Cayley tables and improves the upper bound for all finite groups.

Findings

Cayley table of an Abelian group has chromatic number |G| or |G|+2.

The chromatic number is |G|+2 if and only if the group has nontrivial cyclic Sylow 2-subgroups.

The upper bound for the chromatic number of any finite group Cayley table is improved to (3/2)|G|.

Abstract

The chromatic number of a latin square $L$, denoted $\chi(L)$, is the minimum number of partial transversals needed to cover all of its cells. It has been conjectured that every latin square satisfies $\chi(L) \leq |L|+2$. If true, this would resolve a longstanding conjecture---commonly attributed to Brualdi---that every latin square has a partial transversal of size $|L|-1$. Restricting our attention to Cayley tables of finite groups, we prove two main results. First, we resolve the chromatic number question for Cayley tables of finite Abelian groups: the Cayley table of an Abelian group $G$ has chromatic number $|G|$ or $|G|+2$, with the latter case occurring if and only if $G$ has nontrivial cyclic Sylow 2-subgroups. Second, we give an upper bound for the chromatic number of Cayley tables of arbitrary finite groups. For $|G|\geq 3$, this improves the best-known general upper bound from $2|G|$ to $\frac{3}{2}|G|$, while yielding an even stronger result in infinitely many cases.