# Coloring dense digraphs

## TL;DR

This paper extends the concept of heroes in tournaments to dense digraphs, introducing superheroes, and proves that the classes of heroes and superheroes coincide, providing a complete characterization.

## Contribution

It establishes that a digraph is a superhero if and only if it is a hero, thus fully characterizing superheroes in the context of dense digraphs.

## Key findings

- Superheroes and heroes are equivalent classes of digraphs.
- Complete characterization of superheroes in dense digraphs.
- Answers a previously open question in the field.

## Abstract

The chromatic number of a digraph $D$ is the minimum number of acyclic subgraphs covering the vertex set of $D$. A tournament $H$ is a hero if every $H$-free tournament $T$ has chromatic number bounded by a function of $H$. Inspired by the celebrated Erd\H{o}s--Hajnal conjecture, Berger et al. fully characterized the class of heroes in 2013. We extend this framework to dense digraphs: A digraph $H$ is a superhero if every $H$-free digraph $D$ has chromatic number bounded by a function of $H$ and $\alpha(D)$, the independence number of the underlying graph of $D$. We prove here that a digraph is a superhero if and only if it is a hero, and hence characterize all superheroes. This answers a question of Aboulker, Charbit and Naserasr.