Reed in 1998 conjectured that every graph $G$ satisfies $\chi(G) \leq \lceil \frac{\Delta(G)+1+\omega(G)}{2} \rceil$. As a partial result, he proved the existence of $\varepsilon > 0$ for which every graph $G$ satisfies $\chi(G) \leq \lceil (1-\varepsilon)(\Delta(G)+1)+\varepsilon\omega(G) \rceil$. We propose an analogue conjecture for digraphs. Given a digraph $D$, we denote by $\vec{\chi}(D)$ the dichromatic number of $D$, which is the minimum number of colours needed to partition $D$ into acyclic induced subdigraphs. We let $\overleftrightarrow{\omega}(D)$ denote the size of the largest biclique (a set of vertices inducing a complete digraph) of $D$ and $\tilde{\Delta}(D) = \max_{v\in V(D)} \sqrt{d^+(v) \cdot d^-(v)}$. We conjecture that every digraph $D$ satisfies $\vec{\chi}(D) \leq \lceil \frac{\tilde{\Delta}(D)+1+\overleftrightarrow{\omega}(D)}{2} \rceil$, which if true implies Reed's conjecture. As a partial result, we prove the existence of $\varepsilon >0$ for which every digraph $D$ satisfies $\vec{\chi}(D) \leq \lceil (1-\varepsilon)(\tilde{\Delta}(D)+1)+\varepsilon\overleftrightarrow{\omega}(D) \rceil$. This implies both Reed's result and an independent result of Harutyunyan and Mohar for oriented graphs. To obtain this upper bound on $\vec{\chi}$, we prove that every digraph $D$ with $\overleftrightarrow{\omega}(D) > \frac{2}{3}(\Delta_{\max}(D)+1)$, where $\Delta_{\max}(D) = \max_{v\in V(D)} \max(d^+(v),d^-(v))$, admits an acyclic set of vertices intersecting each biclique of $D$, which generalises a result of King.

翻译：Reed于1998年猜想：对于任意图$G$，满足$\chi(G) \leq \lceil \frac{\Delta(G)+1+\omega(G)}{2} \rceil$。作为部分结果，他证明了存在$\varepsilon > 0$，使得任意图$G$满足$\chi(G) \leq \lceil (1-\varepsilon)(\Delta(G)+1)+\varepsilon\omega(G) \rceil$。我们针对有向图提出了一个类比猜想。给定有向图$D$，记$\vec{\chi}(D)$为其双色数，即将$D$划分为无环诱导子有向图所需的最小颜色数。令$\overleftrightarrow{\omega}(D)$表示$D$中最大双团（诱导出完全有向图的顶点集）的大小，并定义$\tilde{\Delta}(D) = \max_{v\in V(D)} \sqrt{d^+(v) \cdot d^-(v)}$。我们猜想：任意有向图$D$满足$\vec{\chi}(D) \leq \lceil \frac{\tilde{\Delta}(D)+1+\overleftrightarrow{\omega}(D)}{2} \rceil$，若该猜想成立则可推出Reed猜想。作为部分结果，我们证明了存在$\varepsilon >0$，使得任意有向图$D$满足$\vec{\chi}(D) \leq \lceil (1-\varepsilon)(\tilde{\Delta}(D)+1)+\varepsilon\overleftrightarrow{\omega}(D) \rceil$。这一结果同时蕴含了Reed的结论以及Harutyunyan和Mohar针对定向图的独立结果。为得到该$\vec{\chi}$上界，我们证明了：对于任意满足$\overleftrightarrow{\omega}(D) > \frac{2}{3}(\Delta_{\max}(D)+1)$的有向图$D$（其中$\Delta_{\max}(D) = \max_{v\in V(D)} \max(d^+(v),d^-(v))$），总存在一个无环顶点集能与$D$的每个双团相交，该结论推广了King的一个结果。