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双色数的有向图包含任意固定有向图作为细分。有向图的双色数是指将其顶点集划分为无环诱导子有向图的最小划分数目。若删除任意弧或顶点均会降低其双色数，则该有向图被称为双临界的。本文针对具有大阶数或有向大围长的双临界有向图，给出了其包含给定有向图作为细分的充分条件。具体而言，我们证明了：(i) 对于任意整数$k,\ell$，具有双色数$k$且足够大的双临界有向图包含至少具有$\ell$个顶点的某个有向环定向；(ii) 存在函数$f,g$，使得对于有向图$F$的任意细分$F^*$，若有向图具有至少$f(F^*)$的有向围长和至少$g(F)$的双色数，则包含$F^*$的细分，且当$F$为树时，$g(F)=|V(F)|$；(iii) 存在函数$f$，使得对于$TT_3$（三个顶点上的传递竞赛图）的任意细分$F^*$，若有向图具有至少$f(F^*)$的有向围长和至少$2$的最小出度，则包含$F^*$作为细分。