  For simple graphs $X$ and $Y$ on $n$ vertices, the friends-and-strangers
graph $\mathsf{FS}(X,Y)$ is the graph whose vertex set consists of all
bijections $\sigma: V(X) \to V(Y)$, where two bijections $\sigma$ and $\sigma'$
are adjacent if and only if they agree on all but two adjacent vertices $a, b
\in V(X)$ such that $\sigma(a), \sigma(b) \in V(Y)$ are adjacent in $Y$.
Resolving a conjecture of Wang, Lu, and Chen, we completely characterize the
connectedness of $\mathsf{FS}(X, Y)$ when $Y$ is a complete bipartite graph. We
further extend this result to when $Y$ is a complete multipartite graph. We
also determine when $\mathsf{FS}(X, Y)$ has exactly two connected components
where $X$ is bipartite and $Y$ is a complete bipartite graph.
