Exact Chromatic Polynomials for Toroidal Chains of Complete Graphs

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.