The Connectivity of Friends-and-Strangers Graphs on Complete Multipartite Graphs
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Author(s)
Zhu, Honglin
Date Issued
December 31, 2024
Journal
Annals of Combinatorics
Publisher
Springer International Publishing
Citation
Zhu, H. The Connectivity of Friends-and-Strangers Graphs on Complete Multipartite Graphs. Ann. Comb. 29, 691–718 (2025).
Version
Author's final manuscript
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
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DOI of Published Version
https://doi.org/10.1007/s00026-024-00740-z
