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

Abstract A dynamic colouring of a graph is a proper colouring in which no neighbourhood of a non-leaf vertex is monochromatic. The 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 ) ≤ χ ( G ) + 2 for all regular graphs G , which would significantly improve the best current upper bound χ 2 ( G ) ≤ 2 χ ( G ) . In this note, however, we show that this last upper bound is sharp by constructing, for every integer n ≥ 2 , a regular graph G with χ ( G ) = n but χ 2 ( G ) = 2 n . In particular, this disproves Montgomery’s conjecture.