  We present exact calculations of the Potts model partition function
$Z(G,q,v)$ for arbitrary $q$ and temperature-like variable $v$ on $n$-vertex
strip graphs $G$ of the honeycomb lattice for a variety of transverse widths
equal to $L_y$ vertices and for arbitrarily great length, with free
longitudinal boundary conditions and free and periodic transverse boundary
conditions. These partition functions have the form
$Z(G,q,v)=\sum_{j=1}^{N_{Z,G,\lambda}} c_{Z,G,j}(\lambda_{Z,G,j})^m$, where $m$
denotes the number of repeated subgraphs in the longitudinal direction. We give
general formulas for $N_{Z,G,j}$ for arbitrary $L_y$. We also present plots of
zeros of the partition function in the $q$ plane for various values of $v$ and
in the $v$ plane for various values of $q$. Explicit results for partition
functions are given in the text for $L_y=2,3$ (free) and $L_y=4$ (cylindrical),
and plots of partition function zeros are given for $L_y$ up to 5 (free) and
$L_y=6$ (cylindrical). Plots of the internal energy and specific heat per site
for infinite-length strips are also presented.
