Dimer and fermionic formulations of a class of colouring problems

TL;DR

This paper establishes a connection between q-edge-colourings of regular graphs and dimer models, deriving fermionic Grassmann integral representations, and applies these to evaluate colouring problems on various lattices.

Contribution

It introduces a novel fermionic formulation for graph colouring problems based on dimer model relationships, enabling new analytical and numerical approaches.

Findings

Derived Grassmann integral representations for colouring counts.

Applied methods to toroidal and planar graphs, including square lattices.

Unified asymptotic scaling expressions for different lattice types.

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, 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.