Subdivisions in dicritical digraphs with large order or digirth

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 $F^{\ast }$ of a digraph $F$, digraphs with directed girth at least $f\left(F^{\ast }\right)$ and dichromatic number at least $g\left(F\right)$ contain a subdivision of $F^{\ast }$, and if $F$ is a tree, then $g\left(F\right)=\left|V\left(F\right)\right|$; (iii) there is a function $f$ such that for every subdivision $F^{\ast }$ of $T{T}_3$ (the transitive tournament on three vertices), digraphs with directed girth at least $f\left(F^{\ast }\right)$ and minimum out-degree at least 2 contain $F^{\ast }$ as a subdivision.