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

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

2009-09-08

arxiv.org/abs/0909.1396v2

J. Stat. Mech. (2010) P01004

Physics

Condensed Matter

Statistical Mechanics

10 pages, 4 figures (2 in colour). Added discussion, 2 refs. in Sec. 4

Scientific paper

10.1088/1742-5468/2010/01/P01004

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.

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