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.

翻译：有向图$D$的双色数$\vec{\chi}(D)$是使得$D$存在一种用$k$种颜色染色且不含单色有向圈的最小整数$k$。本文考虑的有向图是有限的，允许存在反向平行弧，但不允许平行弧。若$D$的每个真子有向图$D'$均满足$\vec{\chi}(D')<\vec{\chi}(D)=k$，则称$D$为$k$-临界有向图。对于整数$k$和$n$，令$\overrightarrow{\mathrm{ext}}(k,n)$表示阶为$n$的$k$-临界有向图可能的最小弧数。容易证明，对所有$n\geq 2$有$\overrightarrow{\mathrm{ext}}(2,n)=n$，且对所有可能的$n$有$\overrightarrow{\mathrm{ext}}(3,n)\geq 2n$，其中等号成立当且仅当$n$为奇数且$n\geq 3$。作为主要结果，我们证明：若$n,k,p$为整数且满足$n=k+p$及$2\leq p \leq k-1$，则$\overrightarrow{\mathrm{ext}}(k,n)=2({\binom{n}{2}} - (p^2+1))$，并给出使等号成立的$k$-临界有向图的精确刻画。这一结果推广了Tibor Gallai于1963年获得的关于临界图的一个结论。