Chordal Directed Graphs Are Not $\chi$-Bounded

Pierre Aboulker (1) , Nicolas Bousquet (2) , Rémi de Verclos

1 DI-ENS - Département d'informatique - ENS Paris
2 GOAL - Graphes, AlgOrithmes et AppLications

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 in the negative a question of Carbonero, Hompe, Moore, and Spirkl (and strengthens one of their results).

Domaines: Mathématique discrète [cs.DM]

Pierre Aboulker, Nicolas Bousquet, Rémi de Verclos. Chordal Directed Graphs Are Not $\chi$-Bounded. The Electronic Journal of Combinatorics, 2022, 29 (2), ⟨10.37236/11050⟩. ⟨hal-03829690⟩