$3$-tuple total domination number of rook's graphs

Abstract

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 called the $k$-tuple total dominating number and it is denoted by ${\gamma }_{\times k,t}(G)$. We give a constructive proof of a general formula for ${\gamma }_{\times 3,t}(K_n\square K_m)$.