Cartesian Product Graphs and k-Tuple Total Domination

A k-tuple total dominating set (kTDS) of a graph G is a set S of vertices in which every vertex in G is adjacent to at least k vertices in S; the minimum size of a kTDS is denoted gamma(xk,t)(G). We give a Vizing-like inequality for Cartesian product graphs, namely gamma(xk,t)(G)gamma(xk,t)(H) <= 2k gamma(xk,t)(G square H) provided gamma(xk,t)(G) <= 2k rho(G) holds, where rho denotes the packing number. We also give bounds on gamma(xk,t)(G square H) in terms of (open) packing numbers, and consider the extremal case of gamma(xk,t)(K-n square K-m), i.e., the rook's graph, giving a constructive proof of a general formula for gamma(x2,t)(K-n square K-m).