The existence of bimatching designs (Q2712511)

A hyperfactorization of index \(\lambda\) of the complete graph \(K_{2n}\) consists of a family of perfect matchings of \(K_{2n}\) so that every pair of independent edges of \(K_{2n}\) lies in exactly \(\lambda\) of the perfect matchings. This had been studied by Jungnickel and Vanstone. \textit{B. Alspach} and \textit{K. Heinrich} [Australas. J. Comb. 2, 39-55 (1990; Zbl 0757.05037)] have given a generalization of the notion of a hyperfactorization, called matching design. A matching design, denoted by \(\text{MATCH}(n, k, \lambda)\) is a family of \(k\)-matchings (i.e., \(k\) independent edges) of \(K_n\) so that every pair of independent edges of \(K_n\) lies in exactly \(\lambda\) members of the \(k\)-matchings. An analogous definition for the bipartite graph \(K_{n,n}\) with the property that every pair of independent edges lies in exactly \(\lambda\) of the \(k\)-matchings is called a bimatching design, denoted by \(\text{BIMATCH}(n, k, \lambda)\).NEWLINENEWLINENEWLINEIn this paper the author gives a new construction of bimatching designs and shows that the necessary conditions for the existence of a \(\text{BIMATCH}(n, k, \lambda)\) are also sufficient whenever \(k=3\) and \(4\).