  We present exact calculations of the zero-temperature partition function for
the q-state Potts antiferromagnet (equivalently, the chromatic polynomial) for
two families of arbitrarily long strip graphs of the square lattice with
periodic boundary conditions in the transverse direction and (i) periodic (ii)
twisted periodic boundary conditions in the longitudinal direction, so that the
strip graphs are embedded on a (i) torus (ii) Klein bottle. In the limit of
infinite length, we calculate the exponent of the entropy, W(q), show it to be
the same for (i) and (ii), and determine its analytic structure.
