  The problem of constructing an optimal co-adapted coupling for a pair of
symmetric random walks on $Z_2^d$ was considered by Connor and Jacka (2008),
and the existence of a coupling which is stochastically fastest in the class of
all such co-adapted couplings was demonstrated. In this paper we show how to
generalise this construction to an optimal co-adapted coupling for the
continuous-time symmetric random walk on $K_n^d$, where $K_n$ is the complete
graph with $n$ vertices. Moreover, we show that although this coupling is not
maximal for any $n$ (i.e. it does not achieve equality in the coupling
inequality), it does tend to a maximal coupling as $n\to\infty$.
