In this paper we propose a deterministic algorithm for approximate counting
the $k$-colourings of a graph. We consider two categories of graphs. The first
one is the sparse random graphs $G_{n,d/n}$, where each edge is chosen
independently with probability $d/n$ and $d$ is fixed. The second one is a
family we call {\em locally $\alpha$-dense graphs} of bounded maximum degree
\Delta. A graph $G=(V, E)$ in this family has following property: For all
$\{u,v\}\in E$ the number of vertices which are adjacent to $v$ but not
adjacent to $u$ are at most $(1-\alpha)\Delta$, where $\alpha\in [0,1]$ is a
parameter of the model.

W.h.p. our algorithm computes in polynomial time a $(1\pm
poly(n))$-approximation of the logarithm of the $k$-colourings of $G_{n,d/n}$
for $k\geq 2.1 d$. Also, for $G$ a locally $\alpha$-dense graph of bounded
$\Delta$ it computes in polynomial time a $(1 \pm poly(log n))$-approximation
of the logarithm of the k-colourings, for $k>(2-a)\Delta$. Restricting the
treewidth of the neighbourhoods in $G$ we improve the approximation.

Our algorithms is related to the algorithms of A. Bandyopadhyay et al. in
SODA '06, and A. Montanari et al. in SODA '06 However, here, we establish
correlation decay properties for the {\em Gibbs distribution} in a completely
different context. I.e. given the graph $G=(V,E)$, we alter the graph structure
in some specific region $\Lambda \subseteq V$ (by deleting edges between
vertices of $\Lambda$) and then we show that the effect of this change on the
marginals of Gibbs distribution, diminishes as we move away from $\Lambda$. Our
approach is novel and suggests a new context for the study of deterministic
counting algorithms.