A counterexample to a conjecture about triangle-free induced subgraphs of graphs with large chromatic number

TL;DR

This paper constructs a graph with large chromatic number and small clique number that serves as a counterexample to a longstanding conjecture about triangle-free induced subgraphs.

Contribution

It provides the first known counterexample to the conjecture, demonstrating that such graphs can have high chromatic number despite restrictions on induced subgraphs.

Findings

Constructed a graph with chromatic number at least n and clique number at most 3.

Showed every induced subgraph with clique number at most 2 has chromatic number at most 4.

Disproved the previously held conjecture about triangle-free induced subgraphs in graphs with large chromatic number.

Abstract

We prove that for every $n$, there is a graph $G$ with $\chi(G) \geq n$ and $\omega(G) \leq 3$ such that every induced subgraph $H$ of $G$ with $\omega(H) \leq 2$ satisfies $\chi(H) \leq 4$. This disproves a well-known conjecture. Our construction is a digraph with bounded clique number, large dichromatic number, and no induced directed cycles of odd length at least 5.