Cartesian product graphs and $k$-tuple total domination

A $k$-tuple total dominating set ($k$TDS) 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 $k$TDS is denoted $\gamma_{\times k,t}(G)$. We give a Vizing-like inequality for Cartesian product graphs, namely $\gamma_{\times k,t}(G) \gamma_{\times k,t}(H) \leq 2k \gamma_{\times k,t}(G \Box H)$ provided $\gamma_{\times k,t}(G) \leq 2k\rho(G)$, where $\rho$ is the packing number. We also give bounds on $\gamma_{\times k,t}(G \Box H)$ in terms of (open) packing numbers, and consider the extremal case of $\gamma_{\times k,t}(K_n \Box K_m)$, i.e., the rook's graph, giving a constructive proof of a general formula for $\gamma_{\times 2, t}(K_n \Box K_m)$.