  A graph $G$ is $r$-equitably $k$-colorable if its vertex set can be
partitioned into $k$ independent sets, any two of which differ in size by at
most $r$. The $r$-equitable chromatic threshold of a graph $G$, denoted by
$\chi_{r=}^*(G)$, is the minimum $k$ such that $G$ is $r$-equitably
$k'$-colorable for all $k'\ge k$. Let $G\times H$ denote the Kronecker product
of graphs $G$ and $H$. In this paper, we completely determine the exact value
of $\chi_{r=}^*(K_m\times K_n)$ for general $m,n$ and $r$. As a consequence, we
show that for $r\ge 2$, if $n\ge \frac{1}{r-1}(m+r)(m+2r-1)$ then $K_m\times
K_n$ and its spanning supergraph $K_{m(n)}$ have the same $r$-equitable
colorability, and in particular $\chi_{r=}^*(K_m\times
K_n)=\chi_{r=}^*(K_{m(n)})$, where $K_{m(n)}$ is the complete $m$-partite graph
with $n$ vertices in each part.
