(Claw, C_3)-free digraphs with unbounded dichromatic number

TL;DR

This paper constructs specific orientations of claw-free graphs that lack directed 3-cycles but have unbounded dichromatic number, disproving a previous conjecture and advancing understanding of graph orientations.

Contribution

It provides a counterexample to a conjecture by constructing orientations of claw-free graphs with unbounded dichromatic number and no directed 3-cycle.

Findings

Disproved a conjecture on dichromatic number in claw-free graphs

Constructed orientations with unbounded dichromatic number and no directed C3

Improved upon previous results in graph orientation theory

Abstract

We construct orientations of rook graphs (whose underlying graphs are claw-free) that contain no directed $C_3$ but have unbounded dichromatic number. This disproves a conjecture of Aboulker, Charbit and Naserasr and improves a result of Carbonero, Koerts, Moore and Spirkl.