= Solution
Part (ii) shows that $\chi(G)\leq\lambda(D)$ 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 $\lambda(D)\leq\chi(G)$. Combining the <inequalities> gives
$$
\boxed{\chi(G)=\min_{D\text{ acyclic orientation of }G}\lambda(D).}
$$
The 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>.
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