Dimer and fermionic formulations of a class of colouring problems

Abstract

We show that the number Z of q-edge-colourings of a simple regular graph of degree q is deducible from functions describing dimers on the same graph, namely the dimer generating function or equivalently the set of connected dimer correlation functions. Using this relationship to the dimer problem, we derive fermionic representations for Z in terms of Grassmann integrals with quartic actions. Expressions are given for planar graphs and nonplanar graphs embeddable (without edge crossings) on a torus. We discuss exact numerical evaluations of the Grassmann integrals using an algorithm by Creutz and present an application to the 4-edge-colouring problem on toroidal square lattices, comparing the results with numerical transfer matrix calculations and a previous Bethe ansatz study.