Equitable chromatic threshold of Kronecker products of complete graphs

A proper vertex coloring of a graph is equitable if the sizes of color classes differ by at most 1. The equitable chromatic threshold of a graph $G$, denoted by ${\chi }_{=}^{\ast }(G)$, is the minimum $k$ such that $G$ is equitably $k^{\mathrm{\prime }}$-colorable for all $k^{\mathrm{\prime }}\ge k$. Let $G\times H$ denote the direct product of graphs $G$ and $H$. For $n\ge m\ge 2$ we prove that ${\chi }_{=}^{\ast }(K_m\times K_n)$ equals $\lceil \frac{mn}{m+1}\rceil$ if $n\equiv 2,…,m (\textup{mod} m+1)$,andequals$m\lceil\frac{n}{s^\star}\rceil$if$n\equiv 0,1 (\textup{mod} m+1)$,where$s^\star$ is the minimum positive integer such that $s^{\star }\nmid n$ and $s^{\star }\ge m+2.$