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

Guillaume Aubian, Luis Kuffner

TL;DR

This paper addresses the problem of finding digraphs with unbounded dichromatic number that avoid a directed triangle while being free of a claw and other obstacles. It provides an explicit construction: orienting the claw-free rook graphs $R_N$ into digraphs $D_N$ that contain no directed $C_3$. The key approach combines structural analysis showing $D_N$ is $ ext{vec}{C}_3$-free with a Ramsey-type argument via the Gallai–Witt theorem to prove unbounded dichromatic number, thereby disproving a conjecture and advancing understanding of dichromatic behavior in restricted digraph classes. The results specifically yield a counterexample for $H= ext{vec}{C}_3$ and $S$ any orientation of the claw $K_{1,3}$, improving prior work by CKMS25 and ACN21. This contributes a precise, constructive method for separating local subdigraph constraints from global coloring properties in oriented graphs.

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.

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

TL;DR

This paper addresses the problem of finding digraphs with unbounded dichromatic number that avoid a directed triangle while being free of a claw and other obstacles. It provides an explicit construction: orienting the claw-free rook graphs into digraphs that contain no directed . The key approach combines structural analysis showing is -free with a Ramsey-type argument via the Gallai–Witt theorem to prove unbounded dichromatic number, thereby disproving a conjecture and advancing understanding of dichromatic behavior in restricted digraph classes. The results specifically yield a counterexample for and any orientation of the claw , improving prior work by CKMS25 and ACN21. This contributes a precise, constructive method for separating local subdigraph constraints from global coloring properties in oriented graphs.

Abstract

We construct orientations of rook graphs (whose underlying graphs are claw-free) that contain no directed but have unbounded dichromatic number. This disproves a conjecture of Aboulker, Charbit and Naserasr and improves a result of Carbonero, Koerts, Moore and Spirkl.
Paper Structure (2 sections, 3 theorems)

This paper contains 2 sections, 3 theorems.

Table of Contents

  1. Introduction and notation
  2. Construction and proof

Key Result

Proposition 1

The digraph $D_N$ contains no $\vec{C}_3$.

Theorems & Definitions (6)

  • Proposition 1
  • proof
  • Lemma 1
  • proof
  • Theorem 1
  • proof