 Chromatic number from acyclic longest paths

For a finite graph, let $\lambda (D)$ count the vertices on a longest directed path in an acyclic orientation $D$. Its minimum over those orientations is the chromatic number. A proper $k$-colouring gives an orientation from smaller to larger colours, so every directed path has strictly increasing colours and at most $k$ vertices. Conversely, assign to each vertex the number of vertices in a longest directed path ending there. Along an edge $u\to v$ this number increases: an ending path at $u$ cannot contain $v$ without forming a directed cycle, so it can be extended. The resulting labels give a proper colouring using at most $\lambda (D)$ colours. These two constructions prove equality. Counting edges instead of vertices in path length subtracts one from the displayed minimum.