Coloring tournaments with forbidden substructures

TL;DR

This paper introduces a quasi-polynomial algorithm for coloring tournaments with forbidden substructures, providing explicit bounds and a constructive proof of the Erdős-Hajnal conjecture for certain classes.

Contribution

It presents the first quasi-polynomial time algorithm for coloring H-free tournaments with explicit bounds, avoiding the regularity lemma, and offers a constructive proof of the Erdős-Hajnal conjecture for specific classes.

Findings

Algorithm runs in e^{O(log(n)^2)} time

Provides explicit lower bounds on epsilon(H)

Constructive proof of Erdős-Hajnal conjecture for prime tournaments

Abstract

Coloring graphs is an important algorithmic problem in combinatorics with many applications in computer science. In this paper we study coloring tournaments. A chromatic number of a random tournament is of order $\Omega(\frac{n}{\log(n)})$. The question arises whether the chromatic number can be proven to be smaller for more structured nontrivial classes of tournaments. We analyze the class of tournaments defined by a forbidden subtournament $H$. This paper gives a first quasi-polynomial algorithm running in time $e^{O(\log(n)^{2})}$ that constructs colorings of $H$-free tournaments using only $O(n^{1-\epsilon(H)}\log(n))$ colors, where $\epsilon(H) \geq 2^{-2^{50|H|^{2}+1}}$ for many forbidden tournaments $H$. To the best of our knowledge all previously known related results required at least sub-exponential time and relied on the regularity lemma. Since we do not use the regularity lemma, we obtain the first known lower bounds on $\epsilon(H)$ that can be given by a closed-form expression. As a corollary, we give a constructive proof of the celebrated open Erd\H{o}s-Hajnal conjecture with explicitly given lower bounds on the EH coefficients for all classes of prime tournaments for which the conjecture is known. Such a constractive proof was not known before. Thus we significantly reduce the gap between best lower and upper bounds on the EH coefficients from the conjecture for all known prime tournaments that satisfy it. We also briefly explain how our methods may be used for coloring $H$-free tournaments under the following conditions: $H$ is any tournament with $\leq 5$ vertices or: $H$ is any but one tournament of six vertices.