Exact T=0 partition functions for Potts antiferromagnets on sections of the simple cubic lattice

We present exact solutions for the zero-temperature partition function of the q-state Potts antiferromagnet (equivalently, the chromatic polynomial P) on tube sections of the simple cubic lattice of fixed transverse size Lx x L(y) and arbitrarily great length L(z), for sizes Lx x L(y)=2 x 3 and 2 x 4 and boundary conditions (a) (FBC(x),FBC(y),FBC(z)) and (b) (PBC(x),FBC(y),FBC(z)), where FBC (PBC) denote free (periodic) boundary conditions. In the limit of infinite length, L(z)-->infinity, we calculate the resultant ground-state degeneracy per site W (=exponent of the ground-state entropy). Generalizing q from Z+ to C, we determine the analytic structure of W and the related singular locus Beta which is the continuous accumulation set of zeros of the chromatic polynomial. For the L(z)-->infinity limit of a given family of lattice sections, W is analytic for real q down to a value q(c). We determine the values of q(c) for the lattice sections considered and address the question of the value of q(c) for a d-dimensional Cartesian lattice. Analogous results are presented for a tube of arbitrarily great length whose transverse cross section is formed from the complete bipartite graph K(m,m).