  We report exact calculations of the ground state degeneracy per site
(exponent of the ground state entropy) $W(\{G\},q)$ of the $q$-state Potts
antiferromagnet on infinitely long strips with specified end graphs for free
boundary conditions in the longitudinal direction and free and periodic
boundary conditions in the transverse direction. This is equivalent to
calculating the chromatic polynomials and their asymptotic limits for these
graphs. Making the generalization from $q \in {\mathbb Z}_+$ to $q \in {\mathbb
C}$, we determine the full locus ${\cal B}$ on which $W$ is nonanalytic in the
complex $q$ plane. We report the first example for this class of strip graphs
in which ${\cal B}$ encloses regions even for planar end graphs. The bulk of
the specific strip graph that exhibits this property is a part of the $(3^3
\cdot 4^2)$ Archimedean lattice.
