4-8和3-12格上的3-边着色问题
The 3-edge-colouring problem on the 4-8 and 3-12 lattices
摘要：
本文考虑了4-8格和3-12格的边（键）的3-着色数的计数问题。这些格子是阿基米德的，配位数为3，可以分别看作是正方形和蜂窝格子的装饰版本。我们解决这些边着色问题的无限格的限制，将它们映射到其他模型的解决方案是已知的。将4-8格上的着色问题映射到正方形格上一个圈逸度n=3的完全填充圈模型上，再将其映射到六顶点模型上。将3-12格子上的着色问题映射到蜂窝格子上的着色问题。4-8和3-12晶格上的3-边着色问题分别等价于“正方形kagome”（“squagome”）和“三角形kagome”晶格上的3-顶点着色问题（从而等价于零温3态反铁磁Potts模型）。
英文摘要：
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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