Bounding the chromatic number of dense digraphs by arc neighborhoods

TL;DR

This paper proves that if every arc neighborhood in a dense directed graph has bounded chromatic number, then the entire graph also has bounded chromatic number, extending to oriented graphs with bounded independence number.

Contribution

It establishes a link between local arc neighborhoods and the global chromatic number in dense digraphs, extending previous results from tournaments to broader classes.

Findings

Bounded arc neighborhood chromatic number implies bounded overall chromatic number.

Extends results from tournaments to oriented graphs with bounded independence number.

Proves equivalence of two conjectures relating graph structure and chromatic number.

Abstract

The chromatic number of a directed graph is the minimum number of induced acyclic subdigraphs that cover its vertex set, and accordingly, the chromatic number of a tournament is the minimum number of transitive subtournaments that cover its vertex set. The neighborhood of an arc $uv$ in a tournament $T$ is the set of vertices that form a directed triangle with arc $uv$. We show that if the neighborhood of every arc in a tournament has bounded chromatic number, then the whole tournament has bounded chromatic number. This holds more generally for oriented graphs with bounded independence number, and we extend our proof from tournaments to this class of dense digraphs. As an application, we prove the equivalence of a conjecture of El-Zahar and Erd\H{o}s and a recent conjecture of Nguyen, Scott and Seymour relating the structure of graphs and tournaments with high chromatic number.