We solve, in a fully decentralised way (\ie with no message passing), the classic problem of colouring a graph. We propose a novel algorithm that is automatically responsive to topology changes, and we prove that it converges quickly to a proper colouring in $O(N\log{N})$ time with high probability for generic graphs (and in $O(\log{N})$ time if $Δ=O(1)$) when the number of available colours is greater than $Δ$, the maximum degree of the graph. We believe the proof techniques used in this work are of independent interest and provide new insight into the properties required to ensure fast convergence of decentralised algorithms.