  The dichromatic number $\vec{\chi}(D)$ of a digraph $D$ is the least integer
$k$ for which $D$ has a coloring with $k$ colors such that there is no
monochromatic directed cycle in $D$. The digraphs considered here are finite
and may have antiparallel arcs, but no parallel arcs. A digraph $D$ is called
$k$-critical if each proper subdigraph $D'$ of $D$ satisfies
$\vec{\chi}(D')<\vec{\chi}(D)=k$. For integers $k$ and $n$, let
$\overrightarrow{\mathrm{ext}}(k,n)$ denote the minimum number of arcs possible
in a $k$-critical digraph of order $n$. It is easy to show that
$\overrightarrow{\mathrm{ext}}(2,n)=n$ for all $n\geq 2$, and
$\overrightarrow{\mathrm{ext}}(3,n)\geq 2n$ for all possible $n$, where
equality holds if and only if $n$ is odd and $n\geq 3$. As a main result we
prove that if $n, k$ and $p$ are integers with $n=k+p$ and $2\leq p \leq k-1$,
then $\overrightarrow{\mathrm{ext}}(k,n)=2({\binom{n}{2}} - (p^2+1))$, and we
give an exact characterisation of $k$-critical digraphs for which equality
holds. This generalizes a result about critical graphs obtained in 1963 by
Tibor Gallai.
