Suppose $\displaystyle \chi(G)=k$ and $\displaystyle c:V(G) \rightarrow \{1,\ldots,k\}$ is a proper $\displaystyle k$-colouring of $\displaystyle G$. Must there be a path $\displaystyle x_1 \ldots x_k$ in $\displaystyle G$ with $\displaystyle c(x_i)=i$ for each $\displaystyle i$?

I've been trying to find a counter-example without any luck (but on the other hand can't come up with a proof either...)