  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 $\Delta=O(1)$) when the number of
available colours is greater than $\Delta$, 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.
