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(G2H) = max{χ(G), χ(H)}. In this paper, we show that there exist graphs G and H such that χk(G2H) > 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.