Counterexample to Babai's lonely colour conjecture

TL;DR

This paper disproves Babai's conjecture by constructing graphs with arbitrarily large girth and chromatic number that still admit a proper edge-colouring where each cycle avoids having a uniquely coloured edge.

Contribution

It provides a counterexample to Babai's lonely colour conjecture, showing such graphs can have unbounded chromatic number despite the colouring constraints.

Findings

Constructed graphs with large girth and chromatic number

Graphs admit proper edge-colouring with no cycle having a uniquely coloured edge

Disproved Babai's conjecture on no-lonely-colour graphs

Abstract

Motivated by colouring minimal Cayley graphs, in 1978, Babai conjectured that no-lonely-colour graphs have bounded chromatic number. We disprove this in a strong sense by constructing graphs of arbitrarily large girth and chromatic number that have a proper edge-colouring in which each cycle contains no colour exactly once.