Chordal directed graphs are not $\chi$-bounded

TL;DR

The paper demonstrates that certain classes of directed graphs, specifically those avoiding transitive tournaments of size three and having only directed 3-cycles, can have arbitrarily large dichromatic number, disproving a previous conjecture.

Contribution

It provides a counterexample to the conjecture that such graphs are $ ext{chi}$-bounded, extending prior results and answering an open question.

Findings

Directed graphs with no transitive 3-vertex tournament can have unbounded dichromatic number.

The result extends previous work by disproving $ ext{chi}$-boundedness in this class.

Answers an open question posed by Carbonero, Hompe, Moore, and Spirkl.

Abstract

We show that digraphs with no transitive tournament on $3$ vertices and in which every induced directed cycle has length $3$ can have arbitrarily large dichromatic number. This answers to the negative a question of Carbonero, Hompe, Moore, and Spirkl (and extends some of their results).