The existence of non-trivial hyperfactorizations of \(K_{2n}\) (Q1180405)

The paper concerns nontrivial hyperfactorizations of the complete graph \(K_{2n}\). A \(\lambda\)-hyperfactorization of \(K_{2n}\) is a collection of linear factors of \(K_{2n}\) such that each pair of non-adjacent edges appears exactly in \(\lambda\) factors. In general, some factors may appear more than once. If each linear factor of \(K_{2n}\) appears at most once in a \(\lambda\)-hyperfactorization, then this \(\lambda\)-hyperfactorization is called simple. A \(\lambda\)-hyperfactorization is called trivial, if each of its factors appears in it with the same multiplicity. The main theorem states that there exists a nontrivial simple hyperfactorization of \(K_{2n}\) for all \(n\geq 5\). Then a construction is described which allows (under certain conditions) to produce a simple \((\lambda+\mu)\)- hyperfactorization of \(K_{2n}\) from simple \(\lambda\)- hyperfactorizations and \(\mu\)-hyperfactorizations.