  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.
