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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 nocycle 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 aresult 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. Weprove 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 knownlower bound. We prove that the number of connected components induced by thevertices of degree $2(k-1)$ of a $k$-dicritical digraph is at most the numberof connected components in the rest of the digraph, generalising a result ofStiebitz. Finally, we generalise a Theorem of Thomassen on list-chromaticnumber of undirected graphs to list-dichromatic number of digraphs.

Various bounds on the minimum number of arcs in a $k$-dicritical digraph