  The $r$-edge bootstrap percolation on a graph is an activation process of the
edges. The process starts with some initially activated edges and then, in each
round, any inactive edge whose one of endpoints is incident to at least $r$
active edges becomes activated. A set of initially activated edges leading to
the activation of all edges is said to be a percolating set. Denote the minimum
size of a percolating set in the $r$-edge bootstrap percolation process on a
graph $G$ by $m_e(G, r)$. The importance of the $r$-edge bootstrap percolation
relies on the fact that $m_e(G, r)$ provides bounds on $m(G, r)$, that is, the
minimum size of a percolating set in the $r$-neighbor bootstrap percolation
process on $G$. In this paper, we explicitly determine $m_e(K_n^d, r)$, where
$K_n^d$ is the Cartesian product of $d$ copies of the complete graph on $n$
vertices which is referred as Hamming graph. Using this, we show that $m(K_n^d,
r)=(1+o(1))\frac{d^{r-1}}{r!}$ when $n, r$ are fixed and $d$ goes to infinity
which extends a known result on hypercubes.
