KURSCHAK 2003 Problem 2

Prove that if a graph $\mathcal{G}$ on $n\ge 3$ vertices has a unique $3$ -coloring, then $\mathcal{G}$ has at least $2n-3$ edges.

(A graph is $3$ -colorable when there exists a coloring of its vertices with $3$ colors such that no two vertices of the same color are connected by an edge. The graph can be $3$ -colored uniquely if there do not exist vertices $u$ and $v$ of the graph that are painted different colors in one $3$ -coloring, yet are colored the same in another.)