The Chromatic Number of Finite Group Cayley Tables

The chromatic number of a latin square L, denoted chi(L), is the minimum num- ber of partial transversals needed to cover all of its cells. It has been conjectured that every latin square satisfies chi(L) <= vertical bar L vertical bar+ 2. If true, this would resolve a long- standing conjecture commonly attributed to Brualdi that every latin square has a partial transversal of size vertical bar L vertical bar - 1. Restricting our attention to Cayley tables of finite groups, we prove two 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 vertical bar G vertical bar or vertical bar G vertical bar + 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 vertical bar G vertical bar >= 3, this improves the best-known general upper bound from 2 vertical bar G vertical bar to 3/2 vertical bar G vertical bar, while yielding an even stronger result in infinitely many cases.