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For simple graphs $X$ and $Y$ on $n$ vertices, the friends-and-strangersgraph $\mathsf{FS}(X,Y)$ is the graph whose vertex set consists of allbijections $\sigma: V(X) \to V(Y)$, 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 theconnectedness of $\mathsf{FS}(X, Y)$ when $Y$ is a complete bipartite graph. Wefurther extend this result to when $Y$ is a complete multipartite graph. Wealso determine when $\mathsf{FS}(X, Y)$ has exactly two connected componentswhere $X$ is bipartite and $Y$ is a complete bipartite graph.

Evacuating "O''- and "Y''-shaped houses on fire: the connectivity of friends-and-strangers graphs on complete multipartite graphs