A counterexample to Montgomery's conjecture on dynamic colourings of regular graphs

Abstract

A \emph{dynamic colouring} of a graph is a proper colouring in which no neighbourhood of a non-leaf vertex is monochromatic. The \emph{dynamic colouring number} $χ_2(G)$ of a graph $G$ is the least number of colours needed for a dynamic colouring of $G$. Montgomery conjectured that $χ_2(G) \leq χ(G) + 2$ for all regular graphs $G$, which would significantly improve the best current upper bound $χ_2(G) \leq 2χ(G)$. In this note, however, we show that this last upper bound is sharp by constructing, for every integer $n \geq 2$, a regular graph $G$ with $χ(G) = n$ but $χ_2(G) = 2n$. In particular, this disproves Montgomery's conjecture.

4 pages, 1 colour figure