  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.

Combinatorics;Discrete Mathematics