Coloring digraphs with $Δ-b$ colors

The dichromatic number of a digraph is the minimum number of colors needed to partition its vertex set into acyclic subdigraphs. A biclique is a set of vertices inducing all possible pairs of opposite arcs. For a digraph $D$, define $\Delta(D) = \max_{v\in V(D)} \sqrt{d^+(v) \cdot d^-(v)}$. We prove that, for every fixed integer $b\in\mathbb{N}$, every digraph $D$ with $\Delta(D) = \Delta$ being sufficiently large with respect to $b$ either contains a biclique whose size exceeds $\Delta-2b$ or has dichromatic number at most $\Delta-b$. This extends a classical result of Reed to the directed setting and supports a conjecture of the present authors. Furthermore, the theorem is tight, as for all integers $b$ and $\Delta\geq 3b$ there exists a digraph $D$ with $\Delta(D)= \Delta$, dichromatic number $\Delta-b+1$, and whose largest biclique has size $\Delta-2b+1$.