SERIES EXPANSIONS FOR THE Q-COLOR PROBLEM ON THE SQUARE AND CUBIC LATTICES

We apply the finite lattice and cluster expansion methods to evaluate the numbers of q-colourings omega(D, q) of the square and cubic lattices. In two dimensions, we develop an efficient transfer matrix algorithm to obtain the 1/(q - 1)-expansion of omega(D = 2, q) to order 36, and in three dimensions we construct expansions about the long-range ordered states. We show (for D = 2) that it is also efficient to use the formulae of the finite-lattice method substituting into them numerical values of the partition functions of finite lattices. Reliable estimates of omega are obtained in all cases.