The dichromatic number $\vec{\chi}(G)$ of a digraph $G$ is the least integer $k$ such that $G$ can be partitioned into $k$ acyclic digraphs. A digraph is $k$-dicritical if $\vec{\chi}(G) = k$ and each proper subgraph $H$ of $G$ satisfies $\vec{\chi}(H) \leq k-1$. %An oriented graph is a digraph with no cycle of length $2$. We prove various bounds on the minimum number of arcs in a $k$-dicritical digraph, a structural result on $k$-dicritical digraphs and a result on list-dicolouring. We characterise $3$-dicritical digraphs $G$ with $(k-1)|V(G)| + 1$ arcs. For $k \geq 4$, we characterise $k$-dicritical digraphs $G$ on at least $k+1$ vertices and with $(k-1)|V(G)| + k-3$ arcs, generalising a result of Dirac. We prove that, for $k \geq 5$, every $k$-dicritical digraph $G$ has at least $(k-1/2 - 1/(k-1)) |V(G)| - k(1/2 - 1/(k-1))$ arcs, which is the best known lower bound. We prove that the number of connected components induced by the vertices of degree $2(k-1)$ of a $k$-dicritical digraph is at most the number of connected components in the rest of the digraph, generalising a result of Stiebitz. Finally, we generalise a Theorem of Thomassen on list-chromatic number of undirected graphs to list-dichromatic number of digraphs.

翻译：有向图 $G$ 的二分色数 $\vec{\chi}(G)$ 是使得 $G$ 可划分为 $k$ 个无环有向图的最小整数 $k$。若有向图 $G$ 满足 $\vec{\chi}(G) = k$ 且其每个真子图 $H$ 都有 $\vec{\chi}(H) \leq k-1$，则称 $G$ 是 $k$-迪临界的。%有向图是不含长度为2的环的有向图。我们证明了 $k$-迪临界有向图中最小弧数的若干界、$k$-迪临界有向图的一个结构结果以及关于列表-二分着色的一项结果。我们刻画了具有 $(k-1)|V(G)| + 1$ 条弧的 $3$-迪临界有向图 $G$。对于 $k \geq 4$，我们推广了Dirac的一个结果，刻画了顶点数至少为 $k+1$ 且具有 $(k-1)|V(G)| + k-3$ 条弧的 $k$-迪临界有向图 $G$。我们证明，对于 $k \geq 5$，每个 $k$-迪临界有向图 $G$ 至少有 $(k-1/2 - 1/(k-1)) |V(G)| - k(1/2 - 1/(k-1))$ 条弧，这是目前已知的最佳下界。我们推广了Stiebitz的一个结果，证明 $k$-迪临界有向图中由度数为 $2(k-1)$ 的顶点诱导的连通分支数不超过该有向图其余部分中的连通分支数。最后，我们将Thomassen关于无向图列表色数的一个定理推广到有向图的列表-二分色数。