Minimum saturated graphs without $4$-cycles and $5$-cycles

Abstract

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