On acyclic colorings of direct produts

A coloring of a graph G is an acyclic coloring if the union of any two color classes induces a forest. It is proved that the acyclic chromatic number of direct product of two trees T1 and T2 equals min{∆(T1)+1, ∆(T2)+1}. We also prove that the acyclic chromatic number of direct product of two complete graphs Km and Kn is mn − m − 2, where m ≥ n ≥ 4. Several bounds for the acyclic chromatic number of direct products are given and in connection to this some questions are raised.