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arXiv · 0803.2665

Boundary chromatic polynomial

Jesper Lykke Jacobsen·Hubert Saleur
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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/n) for the usual chromatic polynomial does not extend to the case Q different from Q_s. The agreement with (scarce) existing numerical results is perfect; further numerical checks are presented here.

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Jesper Lykke Jacobsen, Hubert Saleur. 2008-03-18. Boundary chromatic polynomial. https://arxiv.org/abs/0803.2665

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