The Connectivity of Friends-and-Strangers Graphs on Complete Multipartite Graphs

For simple graphs X and Y on n vertices, the friends-and-strangers graph \(\textsf{FS}(X,Y)\) is the graph whose vertex set consists of all bijections \(\sigma : V(X) \rightarrow V(Y)\), 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 \(\textsf{FS}(X, Y)\) when Y is a complete bipartite graph. We further extend this result to when Y is a complete multipartite graph. We also determine when \(\textsf{FS}(X, Y)\) has exactly two connected components where X is bipartite and Y is a complete bipartite graph.