Chromatic Polynomials for Lattice Strips with Cyclic Boundary Conditions

The zero-temperature $q$-state Potts model partition function for a lattice strip of fixed width $L_y$ and arbitrary length $L_x$ has the form $P(G,q)={\sum }_{j=1}^{N_{G,\lambda }}{c}_{G,j}({\lambda }_{G,j}{)}^{L_x}$, and is equivalent to the chromatic polynomial for this graph. We present exact zero-temperature partition functions for strips of several lattices with $(FB{C}_y,PB{C}_x)$, i.e., cyclic, boundary conditions. In particular, the chromatic polynomial of a family of generalized dodecahedra graphs is calculated. The coefficient $c_{G,j}$ of degree $d$ in $q$ is $c^{(d)}=U_{2d}(\frac{\sqrt{q}}{2})$, where $U_n(x)$ is the Chebyshev polynomial of the second kind. We also present the chromatic polynomial for the strip of the square lattice with $(PB{C}_y,PB{C}_x)$, i.e., toroidal, boundary conditions and width $L_y=4$ with the property that each set of four vertical vertices forms a tetrahedron. A number of interesting and novel features of the continuous accumulation set of the chromatic zeros, $\mathcal{B}$ are found.