  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, viz.
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 for 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 to numerical transfer matrix calculations and a
previous Bethe ansatz study. We also show that for the square, honeycomb, 3-12,
and one-dimensional lattice, known exact results for the asymptotic scaling of
Z with the number of vertices can be expressed in a unified way as different
values of one and the same function.
