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Acyclic subgraphs of digraphs with high chromatic number

Raphael Yuster

Abstract

For a digraph $G$, let $f(G)$ be the maximum chromatic number of an acyclic subgraph of $G$. For an $n$-vertex digraph $G$ it is proved that $f(G) \ge n^{5/9-o(1)}s^{-14/9}$ where $s$ is the bipartite independence number of $G$, i.e., the largest $s$ for which there are two disjoint $s$-sets of vertices with no edge between them. This generalizes a result of Fox, Kwan and Sudakov, who proved this for the case $s=0$ (i.e., tournaments and semicomplete digraphs). Consequently, if $s=n^{o(1)}$, then $f(G) \ge n^{5/9-o(1)}$ which polynomially improves the folklore bound $f(G) \ge n^{1/2-o(1)}$. As a corollary, with high probability, all orientations of the random $n$-vertex graph with edge probability $p=n^{-o(1)}$ (in particular, constant $p$, hence almost all $n$-vertex graphs) satisfy $f(G) \ge n^{5/9-o(1)}$. Our proof uses a theorem of Gallai and Milgram that together with several additional ideas, essentially reduces to the proof of Fox, Kwan and Sudakov.

Acyclic subgraphs of digraphs with high chromatic number

Abstract

For a digraph , let be the maximum chromatic number of an acyclic subgraph of . For an -vertex digraph it is proved that where is the bipartite independence number of , i.e., the largest for which there are two disjoint -sets of vertices with no edge between them. This generalizes a result of Fox, Kwan and Sudakov, who proved this for the case (i.e., tournaments and semicomplete digraphs). Consequently, if , then which polynomially improves the folklore bound . As a corollary, with high probability, all orientations of the random -vertex graph with edge probability (in particular, constant , hence almost all -vertex graphs) satisfy . Our proof uses a theorem of Gallai and Milgram that together with several additional ideas, essentially reduces to the proof of Fox, Kwan and Sudakov.
Paper Structure (4 sections, 10 theorems, 14 equations)

This paper contains 4 sections, 10 theorems, 14 equations.

Table of Contents

  1. Introduction
  2. A few helpful lemmas
  3. Generalization of the Fox-Kwan-Sudakov Theorem
  4. Proof of Lemma \ref{['l:l22']}

Key Result

Theorem 1.1

Let $G$ be an $n$-vertex digraph with $\alpha^*(G) \le s$. Then, $f(G) \ge n^{5/9-o(1)}s^{-14/9}$. In particular, if $\alpha^*(G)=n^{o(1)}$, then $f(G) \ge n^{5/9-o(1)}$.

Theorems & Definitions (22)

  • Theorem 1.1
  • Corollary 1.2
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 3.1
  • ...and 12 more