Boundary chromatic polynomial

摘要

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 = 4cos(2) pi for the usual chromatic polynomial does not extend to the case Q not equal Q(s). The agreement with (scarce) existing numerical results is perfect; further numerical checks are presented here.