The dichromatic number χ​(D) of a digraph D is the minimum size of
a partition of its vertices into acyclic induced subgraphs. We denote by
λ(D) the maximum local edge connectivity of a digraph D. Neumann-Lara
proved that for every digraph D, χ​(D)≤λ(D)+1. In this
paper, we characterize the digraphs D for which χ​(D)=λ(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.Comment: 34 pages, 11 figure