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×Ly and arbitrarily great length Lz, for sizes Lx×Ly=2×3 and 2×4 and boundary conditions (a) (FBCx,FBCy,FBCz) and (b)
(PBCx,FBCy,FBCz), where FBC (PBC) denote free (periodic) boundary
conditions. In the limit of infinite-length, Lz→∞, 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 B which
is the continuous accumulation set of zeros of the chromatic polynomial. For
the Lz→∞ limit of a given family of lattice sections, W is
analytic for real q down to a value qc. We determine the values of qc
for the lattice sections considered and address the question of the value of
qc 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 Km,m.Comment: 28 pages, latex, six postscript figures, two Mathematica file