A dualistic approach to bounding the chromatic number of a graph

We give a new and more direct proof of the characterization theorem for finitary homomorphism dualities of directed graphs. This result may be viewed as a characterization of Gallai–Hasse–Roy–Vitaver type theorems. We exhibit infinitely many examples where this general setting improves the bounds for chromatic number of graphs and we relate this to extremal problems for oriented graphs.