For any undirected connected graph $G$, let $\chi(G)$ be the minimum number of colours needed to colour all the vertices of $G$ in such a way that no two adjacent vertices have the same colour. Prove that for every vertex $v$ of $G$, there exists a path of length $\chi(G)-1$ starting from $v$. A path of length $k$ is a sequence of $k+1$ distinct vertices $v_{1}, v_{2}, \ldots, v_{k+1}$ such that $\left(v_{i}, v_{i+1}\right)$ is an edge for $1 \leq i \leq k$.
