  We present exact calculations of the partition function of the $q$-state
Potts model on (i) open, (ii) cyclic, and (iii) M\"obius strips of the
honeycomb (brick) lattice of width $L_y=2$ and arbitrarily great length. In the
infinite-length limit the thermodynamic properties are discussed. The
continuous locus of singularities of the free energy is determined in the $q$
plane for fixed temperature and in the complex temperature plane for fixed $q$
values. We also give exact calculations of the zero-temperature partition
function (chromatic polynomial) and $W(q)$, the exponent of the ground-state
entropy, for the Potts antiferromagnet for honeycomb strips of type (iv)
$L_y=3$, cyclic, (v) $L_y=3$, M\"obius, (vi) $L_y=4$, cylindrical, and (vii)
$L_y=4$, open. In the infinite-length limit we calculate $W(q)$ and determine
the continuous locus of points where it is nonanalytic. We show that our exact
calculation of the entropy for the $L_y=4$ strip with cylindrical boundary
conditions provides an extremely accurate approximation, to a few parts in
$10^5$ for moderate $q$ values, to the entropy for the full 2D honeycomb
lattice (where the latter is determined by Monte Carlo measurements since no
exact analytic form is known).
