A new Kempe invariant and the (non)-ergodicity of the Wang-Swendsen-Kotecky algorithm

TL;DR

This paper introduces a new Kempe invariant for three-colorable triangulations of surfaces and demonstrates that the Wang-Swendsen-Kotecky algorithm is non-ergodic on certain torus triangulations due to multiple Kempe classes.

Contribution

It establishes a novel Kempe invariant modulo 12 for three-colorable triangulations and shows non-ergodicity of the Wang-Swendsen-Kotecky algorithm on specific torus triangulations.

Findings

Degree of a four-coloring modulo 12 is Kempe invariant.

At least two Kempe classes exist for certain torus triangulations.

Wang-Swendsen-Kotecky algorithm is non-ergodic on these triangulations.

Abstract

We prove that for the class of three-colorable triangulations of a closed oriented surface, the degree of a four-coloring modulo 12 is an invariant under Kempe changes. We use this general result to prove that for all triangulations T(3L,3M) of the torus with 3<= L <= M, there are at least two Kempe equivalence classes. This result implies in particular that the Wang-Swendsen-Kotecky algorithm for the zero-temperature 4-state Potts antiferromagnet on these triangulations T(3L,3M) of the torus is not ergodic.