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，使得每个颜色类导出一个无圈子有向图。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 $o_k(n) \geq cd_k(n)$。该猜想已知对 $k=3$ 成立（Aboulker 等人）。在本工作中，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，从而验证了 $k=4$ 时的猜想。我们还精确刻画了在 $n$ 个顶点上恰好有 $\frac{10}{3}n -\frac{4}{3}$ 条弧的 $k$-临界有向图。