  We apply generalisations of the Swendson-Wang and Wolff cluster algorithms,
which are based on the construction of Fortuin-Kasteleyn clusters, to the
three-dimensional $\pm 1$ random-bond Ising model. The behaviour of the model
is determined by the temperature $T$ and the concentration $p$ of negative
(anti-ferromagnetic) bonds. The ground state is ferromagnetic for $0 \le
p<p_c$, and a spin glass for $p_c < p \le 0.5$ where $p_c \simeq 0.222$. We
investigate the percolation transition of the Fortuin-Kasteleyn clusters as
function of temperature. Except for $p=0$ the Fortuin-Kasteleyn percolation
transition occurs at a higher temperature than the magnetic ordering
temperature. This was known before for $p=1/2$ but here we provide evidence for
a difference in transition temperatures even for $p$ arbitrarily small.
Furthermore, for all values of $p>0$, our data suggest that the percolation
transition is universal, irrespective of whether the ground state exhibits
ferromagnetic or spin-glass order, and is in the universality class of standard
percolation. This shows that correlations in the bond occupancy of the
Fortuin-Kasteleyn clusters are irrelevant, except for $p=0$ where the clusters
are tied to Ising correlations so the percolation transition is in the Ising
universality class.
