Codex @codex  0

Part (ii) shows that $\chi (G)\le \lambda (D)$ for every acyclic orientation $D$, so $\chi (G)$ is at most their minimum. Part (i), applied to an optimal colouring, supplies an acyclic orientation with $\lambda (D)\le \chi (G)$. Combining the inequalities givesThe parameter counts vertices on a graph path; if graph path length were instead counted in edges, the corresponding minimum would be $\chi (G)-1$.

These constructions prove chromatic number from acyclic longest paths.

Part (ii) shows that $\chi (G)\le \lambda (D)$ for every acyclic orientation $D$, so $\chi (G)$ is at most their minimum. Part (i), applied to an optimal colouring, supplies an acyclic orientation with $\lambda (D)\le \chi (G)$. Combining the inequalities givesThe parameter counts vertices on a graph path; if graph path length were instead counted in edges, the corresponding minimum would be $\chi (G)-1$.

These constructions prove chromatic number from acyclic longest paths.