Coloring cayley tables of finite groups
Kevin Halasz · Summit (Simon Fraser University) · 2017
The chromatic number of a latin square L, denoted χ(L), is defined as the minimum number of partial transversals needed to cover all of its cells.It has been conjectured that every latin square L satisfies χ(L) ≤ |L| + 2. If true, this would resolve a longstanding conjecture, commonly attributed to Brualdi, that every latin square has a partial transversal of length |L|-1.Restricting our attention to Cayley tables of finite groups, we prove two results.First, we constructively show that all finite Abelian groups G have Cayley tables with chromatic number |G| + 2. Second, we give an upper bound for the chromatic number of Cayley tables of arbitrary finite groups.For |G| ≥ 3, this improves the best-known general upper bound from 2|G| to 3 2 |G|, while yielding an even stronger result in infinitely many cases.