Grad and classes with bounded expansion III. Restricted graph homomorphism dualities

We study restricted homomorphism dualities in the context of classes with bounded expansion (which are defined by means of the greatest reduced average densities—grads). This presents a generalization of restricted dualities obtained earlier for bounded degree graphs and also for proper minor closed classes. This is related to distance coloring of graphs and to the “approximate version” of the Hadwiger conjecture.