  We prove comparison results for the Swendsen-Wang (SW) dynamics, the
heat-bath (HB) dynamics for the Potts model and the single-bond (SB) dynamics
for the random-cluster model on arbitrary graphs. In particular, we prove that
rapid mixing of HB implies rapid mixing of SW on graphs with bounded maximum
degree and that rapid mixing of SW and rapid mixing of SB are equivalent.
Additionally, the spectral gap of SW and SB on planar graphs is bounded from
above and from below by the spectral gap of these dynamics on the corresponding
dual graph with suitably changed temperature.
  As a consequence we obtain rapid mixing of the Swendsen-Wang dynamics for the
Potts model on the two-dimensional square lattice at all non-critical
temperatures as well as rapid mixing for the two-dimensional Ising model at all
temperatures. Furthermore, we obtain new results for general graphs at high or
low enough temperatures.
