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 Ly and arbitrary length Lx has the form P(G,q)=∑j=1NG,λcG,j(λG,j)Lx, and is equivalent to the chromatic polynomial for this graph. We present exact zero-temperature partition functions for strips of several lattices with (FBCy,PBCx), i.e., cyclic, boundary conditions. In particular, the chromatic polynomial of a family of generalized dodecahedra graphs is calculated. The coefficient cG,j of degree d in q is , where Un(x) is the Chebyshev polynomial of the second kind. We also present the chromatic polynomial for the strip of the square lattice with (PBCy,PBCx), i.e., toroidal, boundary conditions and width Ly=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, are found.