Colorings of a Hexagonal Lattice

Abstract

The number of ways WL of coloring the bonds of a hexagonal lattice of L sites (L large) with three colors so that no adjacent bonds are colored alike is calculated exactly, giving W = 1.20872 …. This is equivalent to counting the number of 4-colorings of the faces of the lattice and can also be regarded as a multiple-dimer problem. If one introduces activities corresponding to certain vertex configurations, then the system is found to have an infinite-order phase transition between two ordered states.