DP-coloring is a relatively new coloring concept by Dvořak 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.

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关于有向图的 DP 着色

DP 着色是 Dvořak 和 Postle 提出的一个相对较新的着色概念，并作为（无向）图的列表着色的扩展引入。它将寻找具有列表赋值 $L$ 的给定图 $G$ 的列表着色问题转化为在具有顶点集 $\{(v,c) ~|~ v \ 的辅助图中寻找独立横向的问题在 V(G), c \in L(v)\}$。在本文中，我们使用 Neumann-Lara 的方法将 DP 着色的定义扩展到有向图，其中有向图的着色是顶点的着色，使得有向图不包含任何单色有向环。此外，我们证明了关于 DP 色数的布鲁克斯类型定理，它扩展了有向图的（列表）色数的各种结果。