  We prove that the spectral gap of the Swendsen-Wang dynamics for the
random-cluster model is larger than the spectral gap of a single-bond dynamics,
that updates only a single edge per step. For this we give a representation of
the algorithms on the joint (Potts/random-cluster) model. Furthermore we obtain
upper and lower bounds on the mixing time of the single-bond dynamics on the
discrete $d$-dimensional torus of side length $L$ at the Potts transition
temperature for $q$ large enough that are exponential in $L^{d-1}$,
complementing a result of Borgs, Chayes and Tetali.
