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A New Kempe Invariant and the (non)-Ergodicity of the Wang-Swendsen-Kotecky Algorithm

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2009-06-05
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IOP Publishing
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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–Kotecký algorithm for the zero-temperature 4-state Potts antiferromagnet on these triangulations T(3L, 3M) of the torus is not ergodic.
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Mohar, B. & Salas, J. (2009). A new Kempe invariant and the (non)-ergodicity of the Wang–Swendsen–Kotecký algorithm. Journal of Physics A: Mathematical and Theoretical, 42(22), 225204.