# Coloring tournaments: from local to global

## TL;DR

This paper demonstrates that tournaments with locally simple neighborhoods have bounded chromatic number, showing a significant simplification compared to general graphs, and answers a question posed by Berger et al.

## Contribution

It establishes a bound on the chromatic number of tournaments based on local neighborhood properties, revealing a fundamental difference from undirected graphs.

## Key findings

- Locally simple tournaments have bounded chromatic number
- There exists a function f such that local chromatic bounds imply global bounds
- Answers a question of Berger et al.

## Abstract

The \emph{chromatic number} of a directed graph $D$ is the minimum number of colors needed to color the vertices of $D$ such that each color class of $D$ induces an acyclic subdigraph. Thus, the chromatic number of a tournament $T$ is the minimum number of transitive subtournaments which cover the vertex set of $T$. We show in this paper that tournaments are significantly simpler than graphs with respect to coloring. Indeed, while undirected graphs can be altogether "locally simple" (every neighborhood is a stable set) and have large chromatic number, we show that locally simple tournaments are indeed simple. In particular, there is a function $f$ such that if the out-neighborhood of every vertex in a tournament $T$ has chromatic number at most $c$, then $T$ has chromatic number at most $f(c)$. This answers a question of Berger et al.