The 3-edge-colouring problem on the 4-8 and 3-12 lattices

TL;DR

This paper calculates the number of 3-edge-colourings on the 4-8 and 3-12 lattices by mapping these problems to known models, revealing their equivalence to certain vertex-colouring and Potts models on related lattices.

Contribution

It provides exact solutions for the 3-edge-colouring problem on complex Archimedean lattices by mapping to well-studied models, extending understanding of edge-colouring in these structures.

Findings

Mapped the 4-8 lattice colouring problem to a six-vertex model.

Mapped the 3-12 lattice colouring problem to the honeycomb lattice problem.

Established the equivalence to zero-temperature 3-state antiferromagnetic Potts models.

Abstract

We consider the problem of counting the number of 3-colourings of the edges (bonds) of the 4-8 lattice and the 3-12 lattice. These lattices are Archimedean with coordination number 3, and can be regarded as decorated versions of the square and honeycomb lattice, respectively. We solve these edge-colouring problems in the infinite-lattice limit by mapping them to other models whose solution is known. The colouring problem on the 4-8 lattice is mapped to a completely packed loop model with loop fugacity n=3 on the square lattice, which in turn can be mapped to a six-vertex model. The colouring problem on the 3-12 lattice is mapped to the same problem on the honeycomb lattice. The 3-edge-colouring problems on the 4-8 and 3-12 lattices are equivalent to the 3-vertex-colouring problems (and thus to the zero-temperature 3-state antiferromagnetic Potts model) on the "square kagome" ("squagome") and "triangular kagome" lattices, respectively.