What question did this study set out to answer?

This research aims to generalize 1-factorisations and hyperfactorisations in the context of complete graphs.

July 4, 2026Open Access

On the association scheme of perfect matchings and their designs

Key Points

This research aims to generalize 1-factorisations and hyperfactorisations in the context of complete graphs.
Explored subsets of the association scheme from the Gelfand pair (S_2n, S_2 ≀ S_n)
Utilized group algebra ℂ[S_2n] for analysis
Applied representation theory of symmetric groups S_2n.
Unified and extended previous results by Cameron (1976)
Established new existence theorems for certain matchings
Identified non-existence conditions for specific configurations.

Abstract

We investigate generalisations of 1-factorisations and hyperfactorisations of the complete graph K 2 n . We show that they are special subsets of the association scheme obtained from the Gelfand pair ( S 2 n , S 2 ≀ S n ) . This unifies and extends results by Cameron (1976) and gives rise to new existence and non-existence results. Our methods involve working in the group algebra ℂ S 2 n and using the representation theory of S 2 n .

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