Strong complete minors in digraphs

Kostochka and Thomason independently showed that any graph with average degree $\Omega(r\sqrt{\log r})$ contains a $K_r$ minor. In particular, any graph with chromatic number $\Omega(r\sqrt{\log r})$ contains a $K_r$ minor, a partial result towards Hadwiger’s famous conjecture. In this paper, we investigate analogues of these results in the directed setting. There are several ways to define a minor in a digraph. One natural way is as follows. A strong $\overrightarrow{K}_{\!\!r}$ minor is a digraph whose vertex set is partitioned into r parts such that each part induces a strongly connected subdigraph, and there is at least one edge in each direction between any two distinct parts. We investigate bounds on the dichromatic number and minimum out-degree of a digraph that force the existence of strong $\overrightarrow{K}_{\!\!r}$ minors as subdigraphs. In particular, we show that any tournament with dichromatic number at least 2r contains a strong $\overrightarrow{K}_{\!\!r}$ minor, and any tournament with minimum out-degree $\Omega(r\sqrt{\log r})$ also contains a strong $\overrightarrow{K}_{\!\!r}$ minor. The latter result is tight up to the implied constant and may be viewed as a strong-minor analogue to the classical result of Kostochka and Thomason. Lastly, we show that there is no function $f\;:\;\mathbb{N} \rightarrow \mathbb{N}$ such that any digraph with minimum out-degree at least f(r) contains a strong $\overrightarrow{K}_{\!\!r}$ minor, but such a function exists when considering dichromatic number.