Generalized DP-colorings of digraphs

In this paper we consider the following three coloring concepts for digraphs. First of all, the generalized coloring concept, in which the same colored vertices of a digraph induce a subdigraph that satisfies a given digraph property. Second, the concept of variable degeneracy, introduced for graphs by Borodin, Kostochka and Toft in 2000; this allows to give a common generalization of the point partition number and the list dichromatic number. Finally, the DP-coloring concept as introduced for graphs by Dvo\v{r}\'ak and Postle in 2018, in which a list assignment of a graph is replaced by a cover. Combining these three coloring concepts leads to generalizations of several classical coloring results for graphs and digraphs, including the theorems of Brooks, of Gallai, of Erd\H{o}s, Rubin, and Taylor, and of Bernshteyn, Kostochka, and Pron for graphs, and the corresponding theorems for digraphs due to Harutyunyan and Mohar. Our main result combines the DP-coloring and variable degeneracy concepts for digraphs.