All minimum $C_4$-saturated multipartite graphs

A subgraph $H$ of $G$ is said to be $F$-saturated relative to $G$, if $H$ does not contain any copy of $F$, but the addition of any edge $e$ in $E(G)\backslash E(H)$ would create a copy of $F$. The minimum size of an $F$-saturated graph relative to $G$ is denoted by $sat(G,F)$. Let $K_k^n$ be the complete $k$-partite graph with $n$ vertices in each part. In this paper, we determine $sat(K_4^n,C_4)$ for all $n \geq 2$. Moreover, we determine all extremal configurations of $sat(K_k^n,C_4)$ for all $n\ge 2$ and $k\ge 4$.