Exact chromatic polynomials for toroidal chains of complete graphs

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 Kb with b=5,6 which have periodic or twisted periodic boundary condition in the longitudinal direction. In the Lx→∞ limit, the continuous accumulation set of the chromatic zeros 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 B crosses the real axis, qc, satisfies the inequality qc⩽b for 2⩽b, the minimum value of q at which 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.