Boundary chromatic polynomial

TL;DR

This paper studies proper colorings of planar graphs in an annulus with boundary conditions, relating the problem to a boundary loop model, and analyzes the phase diagram for real parameters, revealing new insights beyond classical chromatic polynomial results.

Contribution

It introduces a boundary chromatic polynomial for planar graphs with boundary conditions and analyzes its phase diagram, extending understanding beyond classical chromatic polynomial properties.

Findings

Boundary chromatic polynomial relates to boundary loop model partition function.

Beraha numbers' special role does not extend when boundary colors differ.

Numerical results agree with theoretical phase diagram predictions.

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.