The dichromatic number $\vec\chi(D)$ of a digraph $D$ is the minimum size of a partition of its vertices into acyclic induced subgraphs. We denote by $\lambda(D)$ the maximum local edge connectivity of a digraph $D$. Neumann-Lara proved that for every digraph $D$, $\vec\chi(D) \leq \lambda(D) + 1$. In this paper, we characterize the digraphs $D$ for which $\vec\chi(D) = \lambda(D) + 1$. This generalizes an analogue result for undirected graph proved by Stiebitz and Toft as well as the directed version of Brooks' Theorem proved by Mohar. Along the way, we introduce a generalization of Haj\'os join that gives a new way to construct families of dicritical digraphs that is of independent interest.

翻译：有向图 $D$ 的二色数 $\vec\chi(D)$ 是其顶点可划分为无环诱导子图的最小数目。记 $\lambda(D)$ 为有向图 $D$ 的最大局部边连通度。Neumann-Lara 证明：对任意有向图 $D$，有 $\vec\chi(D) \leq \lambda(D) + 1$。本文刻画了满足 $\vec\chi(D) = \lambda(D) + 1$ 的有向图 $D$ 的结构。这推广了 Stiebitz 与 Toft 关于无向图的相应结论，以及 Mohar 证明的有向图 Brooks 定理。在研究过程中，我们引入了 Hajós 并的推广形式，为构造临界有向图族提供了新方法，该结果本身具有独立的研究价值。