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 $\text{sat}(n,\mathcal{F})$. Let ${\mathcal{C}}_{\ge r}$ be the family of cycles of length at least r. Ferrara et al. (2012) gave lower and upper bounds of $\text{sat}(n,C_{\ge r})$ and determined the exact values of $\text{sat}(n,C_{\ge r})$ for $3\le r\le 5$. In this paper, we determine the exact value of $\text{sat}(n,{\mathcal{C}}_{\ge r})$ for $r=6$ and $28\le \frac{n}{2}\le r\le n$ and give new upper and lower bounds for the other cases.

给定一个图族 $\mathcal{F}$, 一个图G被称为$\mathcal{F}$-saturated 如果G不包含F的副本作为任何子图$F\in \mathcal{F}$，但添加任何边 $\mathrm{电子}\notin 乙(G)$ 创建一些的至少一个副本 $F\in \mathcal{F}$G内。一个的最小尺寸$\mathcal{F}$-n个顶点上的饱和图称为饱和数，记为$\text{坐}(n,\mathcal{F})$. 让${\mathcal{C}}_{\ge r}$是长度至少为r的循环族。费拉拉等人。(2012) 给出了下限和上限$\text{坐}(n,C_{\ge r})$ 并确定了的确切值 $\text{坐}(n,C_{\ge r})$ 为了 $3\le r\le 5$. 在本文中，我们确定了$\text{坐}(n,{\mathcal{C}}_{\ge r})$ 为了 $r=6$ 和 $28\le \frac{n}{2}\le r\le n$ 并为其他情况提供新的上限和下限。