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

publication date

  • June 2009

start page

  • 225204

end page

  • 225232

issue

  • 22

volume

  • 42

International Standard Serial Number (ISSN)

  • 1751-8113

Electronic International Standard Serial Number (EISSN)

  • 1751-8121

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-Kotecký algorithm for the zero-temperature 4-state Potts antiferromagnet on these triangulations T(3L, 3M) of the torus is not ergodic.