  For any integer $r\geqslant0$, the $r$-neighbor bootstrap percolation on a
graph is an activation process of the vertices. The process starts with some
initially activated vertices and then, in each round, any inactive vertex with
at least $r$ active neighbors becomes activated. A set of initially activated
vertices leading to the activation of all vertices is said to be a percolating
set. Denote the minimum size of a percolating set in the $r$-neighbor bootstrap
percolation process on a graph $G$ by $m(G, r)$. In this paper, we present
upper and lower bounds on $m(K_n^d, r)$, where $K_n^d$ is the Cartesian product
of $d$ copies of the complete graph $K_n$ which is referred as the Hamming
graph. Among other results, we show that $m(K_n^d, r)=\frac{1+o(1)}{(d+1)!}r^d$
when both $r$ and $d$ go to infinity with $r<n$ and $d=o(\!\sqrt{r})$.
