  We study the zero-temperature partition function of the Potts antiferromagnet
(i.e., the chromatic polynomial) on a torus using a transfer-matrix approach.
We consider square- and triangular-lattice strips with fixed width L, arbitrary
length N, and fully periodic boundary conditions. On the mathematical side, we
obtain exact expressions for the chromatic polynomial of widths L=5,6,7 for the
square and triangular lattices. On the physical side, we obtain the exact
``phase diagrams'' for these strips of width L and infinite length, and from
these results we extract useful information about the infinite-volume phase
diagram of this model: in particular, the number and position of the different
phases.
