Triangle-free graphs with large chromatic number and no induced wheel

TL;DR

This paper proves that Burling graphs, which are triangle-free with arbitrarily large chromatic number, do not contain induced wheels, answering a longstanding question and disproving a previous conjecture.

Contribution

It establishes that Burling graphs lack induced wheels, providing new insights into the structure of triangle-free graphs with large chromatic number.

Findings

Burling graphs are triangle-free with large chromatic number.

No Burling graph contains an induced wheel.

The result disproves a conjecture by Scott and Seymour.

Abstract

A wheel is a graph consisting of an induced cycle of length at least four and a single additional vertex with at least three neighbours on the cycle. We prove that no Burling graph contains an induced wheel. Burling graphs are triangle-free and have arbitrarily large chromatic number, so this answers a question of Trotignon and disproves a conjecture of Scott and Seymour.