  We show that if $\gS=(V,E)$ is a regular bipartite graph for which the
expansion of subsets of a single parity of $V$ is reasonably good and which
satisfies a certain local condition (that the union of the neighbourhoods of
adjacent vertices does not contain too many pairwise non-adjacent vertices),
and if $\cM$ is a Markov chain on the set of proper 3-colourings of $\gS$ which
updates the colour of at most $\rho|V|$ vertices at each step and whose
stationary distribution is uniform, then for $\rho \approx .22$ and $d$
sufficiently large the convergence to stationarity of $\cM$ is (essentially)
exponential in $|V|$. In particular, if $\gS$ is the $d$-dimensional hypercube
$Q_d$ (the graph on vertex set $\{0,1\}^d$ in which two strings are adjacent if
they differ on exactly one coordinate) then the convergence to stationarity of
the well-known Glauber (single-site update) dynamics is exponentially slow in
$2^d/(\sqrt{d}\log d)$. A combinatorial corollary of our main result is that in
a uniform 3-colouring of $Q_d$ there is an exponentially small probability (in
$2^d$) that there is a colour $i$ such the proportion of vertices of the even
subcube coloured $i$ differs from the proportion of the odd subcube coloured
$i$ by at most $.22$. Our proof combines a conductance argument with
combinatorial enumeration methods.
