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Boundary Chromatic Polynomial

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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 Q≠Q s. The agreement with (scarce) existing numerical results is perfect; further numerical checks are presented here.

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Authors and Affiliations

  1. Laboratoire de Physique Théorique, École Normale Supérieure, 24 rue Lhomond, 75231, Paris Cedex 05, France

    Jesper Lykke Jacobsen

  2. Institut de Physique Théorique, CEA Saclay, 91191, Gif-sur-Yvette, France

    Jesper Lykke Jacobsen & Hubert Saleur

  3. Department of Physics and Astronomy, University of Southern California, Los Angeles, CA, 90089, USA

    Hubert Saleur

Authors
  1. Jesper Lykke Jacobsen
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  2. Hubert Saleur
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Correspondence to Jesper Lykke Jacobsen.

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Jacobsen, J.L., Saleur, H. Boundary Chromatic Polynomial. J Stat Phys 132, 707–719 (2008). https://doi.org/10.1007/s10955-008-9585-0

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  • DOI: https://doi.org/10.1007/s10955-008-9585-0

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