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, when $d$ goes to infinity, we show that $m(K_n^d,r)=\frac{1+o\left(1\right)}{(d+1)!}{r}^d$ if $r\gg d^2$ and $n⩾r+1$. Furthermore, we explicitly determine $m(L\left(K_n\right),r)$, where $L\left(K_n\right)$ is the line graph of $K_n$ also known as triangular graph.

的 $\mathrm{[R}$图上的-Neighbor Bootstrap渗透是顶点的激活过程。该过程从一些初始激活的顶点开始，然后在每一轮中，至少包含至少一个不活动的顶点$\mathrm{[R}$活动邻居被激活。导致所有顶点激活的一组初始激活顶点被称为渗滤集。表示渗滤集中的最小尺寸$\mathrm{[R}$图上的邻居引导渗透过程 $G$ 通过 $米（G，\mathrm{[R}）$。在本文中，我们给出了$米（{ķ}_{ñ}^d，\mathrm{[R}）$，在哪里 ${ķ}_{ñ}^d$ 是的笛卡尔积 $d$ 完整图形的副本 ${ķ}_{ñ}$这就是汉明图。除其他结果外，何时$d$ 达到无穷大，我们表明 $米（{ķ}_{ñ}^d，\mathrm{[R}）=\frac{\mathrm{1个}+Ø（\mathrm{1个}）}{（d+\mathrm{1个}）！}{\mathrm{[R}}^d$ 如果 $\mathrm{[R}\gg d^2$ 和 $ñ⩾\mathrm{[R}+\mathrm{1个}$。此外，我们明确确定$米（\mathrm{大号}（{ķ}_{ñ}），\mathrm{[R}）$，在哪里 $\mathrm{大号}（{ķ}_{ñ}）$ 是的线图 ${ķ}_{ñ}$ 也称为三角图。