The dichromatic number $\vec{\chi}(D)$ of a digraph $D$ is the minimum number of colours needed to colour the vertices of a digraph such that each colour class induces an acyclic subdigraph. A digraph $D$ is $k$-dicritical if $\vec{\chi}(D) = k$ and each proper subdigraph $H$ of $D$ satisfies $\vec{\chi}(H) < k$. For integers $k$ and $n$, we define $d_k(n)$ (respectively $o_k(n)$) as the minimum number of arcs possible in a $k$-dicritical digraph (respectively oriented graph). Kostochka and Stiebitz have shown that $d_4(n) \geq \frac{10}{3}n -\frac{4}{3}$. They also conjectured that there is a constant $c$ such that $o_k(n) \geq cd_k(n)$ for $k\geq 3$ and $n$ large enough. This conjecture is known to be true for $k=3$ (Aboulker et al.). In this work, we prove that every $4$-dicritical oriented graph on $n$ vertices has at least $(\frac{10}{3}+\frac{1}{51})n-1$ arcs, showing the conjecture for $k=4$. We also characterise exactly the $k$-dicritical digraphs on $n$ vertices with exactly $\frac{10}{3}n -\frac{4}{3}$ arcs.

翻译：有向图$D$的二色数$\vec{\chi}(D)$是给有向图顶点着色所需的最小颜色数，使得每个颜色类诱导一个无圈子有向图。有向图$D$是$k$-临界的，如果$\vec{\chi}(D) = k$且$D$的每个真子有向图$H$满足$\vec{\chi}(H) < k$。对于整数$k$和$n$，我们定义$d_k(n)$（分别地$o_k(n)$）为$k$-临界有向图（分别地定向图）中可能的最小弧数。Kostochka和Stiebitz已证明$d_4(n) \geq \frac{10}{3}n -\frac{4}{3}$。他们还猜想存在常数$c$使得对于$k\geq 3$且足够大的$n$，有$o_k(n) \geq cd_k(n)$。该已知对$k=3$成立（Aboulker等人）。在本文中，我们证明每个在$n$个顶点上的$4$-临界定向图至少有$(\frac{10}{3}+\frac{1}{51})n-1$条弧，从而证明了$k=4$时的猜想。我们还精确刻画了恰好有$\frac{10}{3}n -\frac{4}{3}$条弧的$n$个顶点上的$k$-临界有向图。