Dichromatic number and forced subdivisions

TL;DR

This paper establishes bounds on the dichromatic number of digraphs avoiding certain subdivisions, proving key results for cycles, cactus graphs, bioriented forests, and tournaments, extending classical graph theory results to directed graphs.

Contribution

It proves that for orientations of cycles, the minimal dichromatic number is equal to the number of vertices, and extends this to cactus graphs and bioriented forests, also determining bounds for tournaments of order four.

Findings

For orientations of cycles, the minimal dichromatic number equals the number of vertices.

Every tournament of order four has a dichromatic number of at most 4.

Extended classical graph results to directed graphs with new bounds.

Abstract

We investigate bounds on the dichromatic number of digraphs which avoid a fixed digraph as a topological minor. For a digraph $F$, denote by $\text{mader}_{\vec{\chi}}(F)$ the smallest integer $k$ such that every $k$-dichromatic digraph contains a subdivision of $F$. As our first main result, we prove that if $F$ is an orientation of a cycle then $\text{mader}_{\vec{\chi}}(F)=v(F)$. This settles a conjecture of Aboulker, Cohen, Havet, Lochet, Moura and Thomass\'{e}. We also extend this result to the more general class of orientations of cactus graphs, and to bioriented forests. Our second main result is that $\text{mader}_{\vec{\chi}}(F)=4$ for every tournament $F$ of order $4$. This is an extension of the classical result by Dirac that $4$-chromatic graphs contain a $K_4$-subdivision to directed graphs.