Strong complete minors in digraphs

TL;DR

This paper explores conditions under which directed graphs contain strong complete minors, establishing bounds related to out-degree and dichromatic number, and extending classical undirected minor results to the directed setting.

Contribution

It introduces the concept of strong $ ightarrow K_r$ minors in digraphs and provides bounds on out-degree and dichromatic number that guarantee their existence, extending classical graph minor theorems.

Findings

Any tournament with dichromatic number at least 2r contains a strong $ ightarrow K_r$ minor.

Any tournament with minimum out-degree $ ext{Omega}(r\sqrt{ ext{log} ext{r}})$ contains a strong $ ightarrow K_r$ minor.

No universal function bounds minimum out-degree to guarantee strong $ ightarrow K_r$ minors in all digraphs.

Abstract

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.