Concerning $3$-factorizations of $3K_{n,n}$

Abstract

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 $n\ge 12$. 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.