Aboulker et al. proved that a digraph with large enough dichromatic number contains any fixed digraph as a subdivision. The dichromatic number of a digraph is the smallest order of a partition of its vertex set into acyclic induced subdigraphs. A digraph is dicritical if the removal of any arc or vertex decreases its dichromatic number. In this paper we give sufficient conditions on a dicritical digraph of large order or large directed girth to contain a given digraph as a subdivision. In particular, we prove that (i) for every integers $k,\ell$, large enough dicritical digraphs with dichromatic number $k$ contain an orientation of a cycle with at least $\ell$ vertices; (ii) there are functions $f,g$ such that for every subdivision $F^*$ of a digraph $F$, digraphs with directed girth at least $f(F^*)$ and dichromatic number at least $g(F)$ contain a subdivision of $F^*$, and if $F$ is a tree, then $g(F)=|V(F)|$; (iii) there is a function $f$ such that for every subdivision $F^*$ of $TT_3$ (the transitive tournament on three vertices), digraphs with directed girth at least $f(F^*)$ and minimum out-degree at least $2$ contain $F^*$ as a subdivision.

翻译：Aboulker等人证明了，具有足够大二色数的有向图包含任何固定有向图作为其细分。有向图的二色数是指将其顶点集划分为无环诱导子有向图的最小颜色数。若删除任意一条弧或一个顶点都会降低其二色数，则该有向图称为临界有向图。本文给出了大阶数或大定向围长的临界有向图包含给定有向图作为细分的充分条件。特别地，我们证明了：(i) 对任意整数$k,\ell$，具有二色数$k$且阶数足够大的临界有向图包含一个至少含$\ell$个顶点的有向圈定向；(ii) 存在函数$f,g$，使得对任意有向图$F$的细分$F^*$，定向围长至少为$f(F^*)$且二色数至少为$g(F)$的有向图包含$F^*$的细分，且若$F$为树，则$g(F)=|V(F)|$；(iii) 存在函数$f$，使得对$TT_3$（三顶点传递竞赛图）的任意细分$F^*$，定向围长至少为$f(F^*)$且最小出度至少为$2$的有向图包含$F^*$作为细分。