Past exam of the graph Theory and Probabilistic Combinatorics course of the University of Oxford / 2011 / A / 2 / b / ii

For each vertex $v$, let $c(v)$ be the maximum number of vertices in a directed path ending at $v$. These numbers lie between one and $\ell$.

If $u\to v$ is an edge, a longest graph path ending at $u$ cannot already contain $v$, since that would give a directed cycle on appending $u\to v$. Appending it therefore creates a graph path ending at $v$, so $c(v)\ge c(u)+1$. Adjacent vertices consequently receive different colours. HenceAcyclicity is exactly what justifies extending the graph path without repeating a vertex.