# On DP-Coloring of Digraphs

## TL;DR

This paper extends the concept of DP-coloring to directed graphs (digraphs), introduces a new theoretical framework, and proves a Brooks' type theorem for the DP-chromatic number of digraphs.

## Contribution

It generalizes DP-coloring to digraphs and establishes a Brooks' type theorem, advancing the understanding of coloring properties in directed graphs.

## Key findings

- Extended DP-coloring definition to digraphs
- Proved a Brooks' type theorem for DP-chromatic number of digraphs
- Connected DP-coloring with classical chromatic number results

## Abstract

DP-coloring is a relatively new coloring concept by Dvo\v{r}\'ak and Postle and was introduced as an extension of list-colorings of (undirected) graphs. It transforms the problem of finding a list-coloring of a given graph $G$ with a list-assignment $L$ to finding an independent transversal in an auxiliary graph with vertex set $\{(v,c) ~|~ v \in V(G), c \in L(v)\}$. In this paper, we extend the definition of DP-colorings to digraphs using the approach from Neumann-Lara where a coloring of a digraph is a coloring of the vertices such that the digraph does not contain any monochromatic directed cycle. Furthermore, we prove a Brooks' type theorem regarding the DP-chromatic number, which extends various results on the (list-)chromatic number of digraphs.