  We present an extensive Markov-chain Monte Carlo study of the finite-size
scaling behavior of the Fortuin-Kasteleyn Ising model on five-dimensional
hypercubic lattices with periodic boundary conditions. We observe that physical
quantities, which include the contribution of the largest cluster, exhibit
complete graph asymptotics. However, for quantities, where the contribution of
the largest cluster is removed, we observe that the scaling behavior is mainly
controlled by the Gaussian fixed point. Our results therefore suggest that
\textit{both} scaling predictions, i.e. the complete graph \textit{and} the
Gaussian fixed point asymptotics, are needed to provide a complete description
for the five-dimensional finite-size scaling behavior on the torus.
