Partite saturation number of cycles

Abstract

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 $E\left(G\right)﹨E\left(H\right)$ 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 $C_{\ell }$ be the cycle of length ℓ. In this paper we give an asymptotically tight bound of $sat(K_k^n,C_{\ell })$ for all $\ell \ge 4,k\ge 2$ except $(\ell ,k)=(4,4)$. Moreover, we determine the exact value of $sat(K_k^n,C_{\ell })$ for $k>\ell =4$ and $5\ge \ell >k\ge 3$ and $(\ell ,k)=(6,2)$.