The saturation number of $C_6$

A graph $G$ is called $C_k$-saturated if $G$ is $C_k$-free but $G+e$ not for any $e\in E(\overline{G})$. The saturation number of $C_k$, denoted $sat(n,C_k)$, is the minimum number of edges in a $C_k$-saturated graph on $n$ vertices. Finding the exact values of $sat(n,C_k)$ has been one of the most intriguing open problems in extremal graph theory. In this paper, we study the saturation number of $C_6$. We prove that ${4n}/{3}-2 \le sat(n,C_6) \le {(4n+1)}/{3}$ for $n\ge9$, which significantly improves the existing lower and upper bounds for $sat(n,C_6)$.