Subdivisions in Digraphs of Large Out-Degree or Large Dichromatic Number

In 1985, Mader conjectured the existence of a function f such that every digraph with minimum out-degree at least f (k) contains a subdivision of the transitive tournament of order k. This conjecture is still completely open, as the existence of f (5) remains unknown. In this paper, we show that if D is an oriented path, or an in-arborescence (i.e., a tree with all edges oriented towards the root) or the union of two directed paths from x to y and a directed path from y to x, then every digraph with minimum out-degree large enough contains a subdivision of D. Additionally, we study Mader's conjecture considering another graph parameter. The dichromatic number of a digraph D is the smallest integer k such that D can be partitioned into k acyclic subdigraphs. We show that any digraph with dichromatic number greater than 4 m (n − 1) contains every digraph with n vertices and m arcs as a subdivision.