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Acyclic subgraphs of tournaments with high chromatic number

Jacob Fox, Matthew Kwan, Benny Sudakov

Abstract

We prove that every $n$-vertex tournament $G$ has an acyclic subgraph with chromatic number at least $n^{5/9-o(1)}$, while there exists an $n$-vertex tournament $G$ whose every acyclic subgraph has chromatic number at most $n^{3/4+o(1)}$. This establishes in a strong form a conjecture of Nassar and Yuster and improves on another result of theirs. Our proof combines probabilistic and spectral techniques together with some additional ideas. In particular, we prove a lemma showing that every tournament with many transitive subtournaments has a large subtournament that is almost transitive. This may be of independent interest.

Acyclic subgraphs of tournaments with high chromatic number

Abstract

We prove that every -vertex tournament has an acyclic subgraph with chromatic number at least , while there exists an -vertex tournament whose every acyclic subgraph has chromatic number at most . This establishes in a strong form a conjecture of Nassar and Yuster and improves on another result of theirs. Our proof combines probabilistic and spectral techniques together with some additional ideas. In particular, we prove a lemma showing that every tournament with many transitive subtournaments has a large subtournament that is almost transitive. This may be of independent interest.

Paper Structure

This paper contains 3 sections, 6 theorems, 6 equations.

Table of Contents

  1. Introduction
  2. Finding an acyclic subgraph with high chromatic number
  3. Iterative alignment and refinement

Key Result

Lemma 2.1

If an $n$-vertex tournament $G$ has fewer than $k!$ copies of $\mathrm{TT}_{k}$ then there is an ordering $\pi$ with $\alpha\mathopen{}\mathclose{\left(G_{\pi}\right)\le k$, so $\chi\mathopen{}\mathclose{\left(G_{\pi}}\right)\ge n/k$.

Theorems & Definitions (15)

  • Lemma 2.1
  • proof : Proof of \ref{['lem:few-Tk-alpha']}
  • Lemma 2.2
  • Lemma 2.3
  • proof : Proof of \ref{['thm:main']} given \ref{['lem:few-Tk-alpha', 'lem:structure-many-Tk', 'lem:almost-transitive-alpha']}
  • Lemma 2.4
  • proof : Proof of \ref{['lem:structure-many-Tk']}, given \ref{['lem:structure-many-Tk-aux']}
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • ...and 5 more