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, χ k (G), is the smallest t so that there is a k-tuple coloring of G using t colors. It is well known that χ(GH) = max{χ(G), χ(H)}. In this paper, we show that there exist graphs G and H such that χ k (GH) > max{χ k (G), χ k (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