  Approximate random k-colouring of a graph G=(V,E) is a very well studied
problem in computer science and statistical physics. It amounts to constructing
a k-colouring of G which is distributed close to Gibbs distribution, i.e. the
uniform distribution over all the k-colourings of G. Here, we deal with the
problem when the underlying graph is an instance of Erdos-Renyi random graph
G(n,p), where p=d/n and d is fixed.
  We propose a novel efficient algorithm for approximate random k-colouring
with the following properties: given an instance of G(n,d/n) and for any
k>(2+\epsilon)d, it returns a k-colouring distributed within total variation
distance n^{-Omega(1)} from the Gibbs distribution, with probability
1-n^{-Omega(1)}.
  What we propose is neither a MCMC algorithm nor some algorithm inspired by
the message passing heuristics that were introduced by statistical physicist.
Our algorithm is of combinatorial nature. It is based on a rather simple
recursion which reduces the random k-colouring of G(n,d/n) to random
k-colouring simpler subgraphs first.
  The lower bound on the number of colours for our algorithm to run in
polynomial time is dramatically smaller than the corresponding bounds we have
for any previous algorithm.
