Complete minors in digraphs with given dichromatic number

TL;DR

This paper improves bounds on the existence of complete minors in digraphs with a given dichromatic number, connecting the problem to recent advances in Hadwiger's conjecture for directed graphs.

Contribution

It provides nearly linear bounds for complete minors in digraphs with specified dichromatic number, advancing the understanding of directed graph minors.

Findings

Improved bounds on complete minors in digraphs with given dichromatic number

Reduced the problem to a recent result on Hadwiger's conjecture for directed graphs

Established almost linear bounds for the problem

Abstract

The dichromatic number $\vec{\chi}(D)$ of a digraph $D$ is the smallest $k$ for which it admits a $k$-coloring where every color class induces an acyclic subgraph. Inspired by Hadwiger's conjecture for undirected graphs, several groups of authors have recently studied the containment of directed graph minors in digraphs with given dichromatic number. In this short note we improve several of the existing bounds and prove almost linear bounds by reducing the problem to a recent result of Postle on Hadwiger's conjecture.