On the minimum number of arcs in 4-dicritical oriented graphs

The dichromatic number $\overrightarrow{\chi }\left(D\right)$ 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 $\overrightarrow{\chi }\left(D\right)=k$ and each proper subdigraph $H$ of $D$ satisfies $\overrightarrow{\chi }\left(H\right)<k$. For integers $k$ and $n$, we define $d_k\left(n\right)$ (resp., $o_k\left(n\right)$) as the minimum number of arcs possible in a $k$-dicritical digraph (resp., oriented graph). Kostochka and Stiebitz have shown that $d_4\left(n\right)⩾\frac{10}{3}n-\frac{4}{3}$. They also conjectured that there is a constant $c$ such that $o_k\left(n\right)⩾c{d}_k\left(n\right)$ for $k⩾3$ and $n$ large enough. This conjecture is known to be true for $k=3$. In this work, we prove that every 4-dicritical oriented graph on $n$ vertices has at least $\left(\frac{10}{3}+\frac{1}{51}\right)n-1$ arcs, showing the conjecture for $k=4$. We also characterise exactly the 4-dicritical digraphs on $n$ vertices with exactly $\frac{10}{3}n-\frac{4}{3}$ arcs.