  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.
