k-tuple chromatic number of the cartesian product of graphs

A k-tuple coloring of a graph G assigns a set of k colors to each vertex of G such that if two vertices are adjacent, the corresponding sets of colors are disjoint. The k-tuple chromatic number of G,χk(G), is the smallest t so that there is a k-tuple coloring of G using t colors. It is well known that χ(G□H)=max⁡{χ(G),χ(H)}. In this paper, we show that there exist graphs G and H such that χk(G□H)>max⁡{χk(G),χk(H)} for k≥2. Moreover, we also show that there exist graph families such that, for any k≥1, the k-tuple chromatic number of their cartesian product is equal to the maximum k-tuple chromatic number of its factors.