Minimum saturated graphs without 4-cycles and 5-cycles

Given a family of graphs $F$, a graph G is said to be $F$-saturated if G does not contain a copy of F as a subgraph for any $F\in F$, but the addition of any edge $e\notin E\left(G\right)$ creates at least one copy of some $F\in F$ within G. The minimum size of an $F$-saturated graph on n vertices is called the saturation number, denoted by $\text{sat}(n,F)$. Let $C_r$ be the cycle of length r. In this paper, we study on $\text{sat}(n,F)$ when $F$ is a family of cycles. In particular, we determine that $\text{sat}(n,\{C_4,C_5\left\}\right)=\lceil \frac{5n}{4}-\frac{3}{2}\rceil$ for any positive integer n.