A $k$-coloring of a graph $G$ is a coloring of its vertices using $k$ possible colors such that the end points of any edge have different colors. We say a graph $G$ is uniquely $k$-colorable if one hand it has a $k$-coloring, on the other hand there do not exist vertices $u$ and $v$ such that $u$ and $v$ have the same color in one $k$-coloring and $u$ and $v$ have different colors in another $k$-coloring. Prove that if a graph $G$ with $n$ vertices $(n \ge 3)$ is uniquely $3$-colorable, then it has at least $2n-3$ edges.