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

Abstract

For simple graphs X and Y on n vertices, the friends-and-strangers graph FS ( X , Y ) is the graph whose vertex set consists of all bijections σ : V ( X ) → V ( Y ) , where two bijections σ and σ ′ are adjacent if and only if they agree on all but two adjacent vertices a , b ∈ V ( X ) such that σ ( a ) , σ ( b ) ∈ V ( Y ) are adjacent in Y. Resolving a conjecture of Wang, Lu, and Chen, we completely characterize the connectedness of 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 FS ( X , Y ) has exactly two connected components where X is bipartite and Y is a complete bipartite graph.