On the minimum number of arcs in $k$-dicritical oriented graphs

The dichromatic number $\text{dic}(D)$ of a digraph $D$ is the least integer $k$ such that $D$ can be partitioned into $k$ directed acyclic digraphs. A digraph is $k$-dicritical if $\text{dic}(D)=k$ and each proper subgraph $D^{\mathrm{\prime }}$ of $D$ satisfies $\text{dic}(D^{\mathrm{\prime }})\le k-1$. An oriented graph is a digraph with no directed cycle of length $2$. For integers $k$ and $n$, we denote by $o_k(n)$ the minimum number of edges of a $k$-critical oriented graph on $n$ vertices (with the convention $o_k(n)=+\mathrm{\infty }$ if there is no $k$-dicritical oriented graph of order $n$). The main result of this paper is a proof that $o_3(n)\ge \frac{7n+2}{3}$ together with a construction witnessing that $o_3(n)\le \lceil \frac{5n}{2}\rceil$ for all $n\ge 12$. We also give a construction showing that for all sufficiently large $n$ and all $k\ge 3$, $o_k(n)<(2k-3)n$, disproving a conjecture of Hoshino and Kawarabayashi. Finally, we prove that, for all $k\ge 2$, $o_k(n)\ge \text{pth}k-\frac{3}{4}-\frac{1}{4k-6}n+\frac{3}{4(2k-3)}$, improving the previous best known lower bound of Bang-Jensen, Bellitto, Schweser and Stiebitz.