  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
$(FBC_y,PBC_x)$, 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 $(PBC_y,PBC_x)$, 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, ${\cal B}$ are found.
