Partite saturation number of cycles

A graph $H$ 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 $E(G)\mathrm{\backslash }E(H)$ 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 containing $n$ vertices in each part and $C_{\mathrm{\ell }}$ be the cycle of length $\mathrm{\ell }$. In this paper we give an asymptotically tight bound of $sat(K_k^n,C_{\mathrm{\ell }})$ for all $\mathrm{\ell }\ge 4,k\ge 2$ except $(\mathrm{\ell },k)=(4,4)$. Moreover, we determined the exact value of $sat(K_k^n,C_{\mathrm{\ell }})$ for $k>\mathrm{\ell }=4$ and $5\ge \mathrm{\ell }>k\ge 3$ and $(\mathrm{\ell },k)=(6,2)$.