Exact Chromatic Polynomials for Toroidal Chains of Complete Graphs

TL;DR

This paper computes exact chromatic polynomials for toroidal chains of complete graphs with boundary conditions, analyzing their zero distributions and eigenvalues, and establishing bounds on crossing points.

Contribution

It provides new exact calculations of chromatic polynomials for specific graph families with periodic boundary conditions, including eigenvalue analysis and conjectures on polynomial structure.

Findings

Determined the accumulation set of chromatic zeros for large graph length.

Proved that the crossing point q_c satisfies q_c ≤ b for graphs with complete subgraphs K_b.

Conjectured the structure of the chromatic polynomial for Klein bottle boundary conditions.

Abstract

We present exact calculations of the partition function of the zero-temperature Potts antiferromagnet (equivalently, the chromatic polynomial) for graphs of arbitrarily great length composed of repeated complete subgraphs $K_b$ with $b=5,6$ which have periodic or twisted periodic boundary condition in the longitudinal direction. In the $L_x \to \infty$ limit, the continuous accumulation set of the chromatic zeros ${\cal B}$ is determined. We give some results for arbitrary $b$ including the extrema of the eigenvalues with coefficients of degree $b-1$ and the explicit forms of some classes of eigenvalues. We prove that the maximal point where ${\cal B}$ crosses the real axis, $q_c$, satisfies the inequality $q_c \le b$ for $2 \le b$, the minimum value of $q$ at which ${\cal B}$ crosses the real $q$ axis is $q=0$, and we make a conjecture concerning the structure of the chromatic polynomial for Klein bottle strips.