Concerning \(3\)-factorizations of \(3K_{n,n}\)

Standard doubling and tripling constructions for block designs with block size three (triple systems) employ factorizations of complete graphs and of complete bipartite graphs. In these constructions, repeated edges in a factor lead to repeated blocks in the design. Hence the construction of triple systems with a prescribed number of repeated blocks is facilitated by determining the possible structure of repeated edges in the factors of a factorization of \(\lambda K_n\) and \(\lambda K_{n,n}\). For \(\lambda =3\), a complete determination of the possible combinations of numbers of doubly and triply repeated edges in 3-factorizations of \(\lambda K_n\) has been completed for \(n \geq12\). In this paper, we solve the analogous problem for the complete bipartite graphs in the case \(\lambda=3\). The case \(\lambda=1\) is trivial, and the case \(\lambda=2\) has been previously solved by Fu.