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

## TL;DR

This paper constructs specific regular graphs demonstrating that Montgomery's conjecture, which proposed a tight bound on the dynamic colouring number relative to the chromatic number, is false.

## Contribution

It provides a counterexample for all integers n ≥ 2, disproving Montgomery's conjecture on the upper bound of dynamic colouring numbers.

## Key findings

- Counterexamples for all n ≥ 2 with χ(G)=n and χ₂(G)=2n
- Disproves Montgomery's conjecture on dynamic colourings
- Shows the upper bound χ₂(G) ≤ 2χ(G) is sharp

## 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} $\chi_2(G)$ of a graph $G$ is the least number of colours needed for a dynamic colouring of $G$. Montgomery conjectured that $\chi_2(G) \leq \chi(G) + 2$ for all regular graphs $G$, which would significantly improve the best current upper bound $\chi_2(G) \leq 2\chi(G)$. 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 $\chi(G) = n$ but $\chi_2(G) = 2n$. In particular, this disproves Montgomery's conjecture.