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Boundary Chromatic Polynomial

  • Jesper Lykke Jacobsen
  • Hubert Saleur
Article

Abstract

We consider proper colorings of planar graphs embedded in the annulus, such that vertices on one rim can take Q s colors, while all remaining vertices can take Q colors. The corresponding chromatic polynomial is related to the partition function of a boundary loop model. Using results for the latter, the phase diagram of the coloring problem (with real Q and Q s) is inferred, in the limits of two-dimensional or quasi one-dimensional infinite graphs. We find in particular that the special role played by Beraha numbers \(Q=4\cos^{2}{\pi\over n}\) for the usual chromatic polynomial does not extend to the case QQ s. The agreement with (scarce) existing numerical results is perfect; further numerical checks are presented here.

Keywords

Chromatic polynomial Boundary loop model Temperley-Lieb algebra Graph colorings 

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Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Laboratoire de Physique ThéoriqueÉcole Normale SupérieureParis Cedex 05France
  2. 2.Institut de Physique ThéoriqueCEA SaclayGif-sur-YvetteFrance
  3. 3.Department of Physics and AstronomyUniversity of Southern CaliforniaLos AngelesUSA

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