Repeated Edges in $3$-Factorizations

Abstract

The type of a $3$-factorization of $3K_{2n}$ is the pair $(t,s)$, where $t$ is the number of doubly repeated edges in $3$-factors, and $(\genfrac{}{}{0ex}{}{n}{2})–s$ is the number of triply repeated edges in $3$-factors. We determine the spectrum of types of $3$-factorizations of $3K_{2n}$, for all $n\ge 6$; for each $n\ge 6$, there are $43$ pairs $(t,s)$ meeting numerical conditions which are not types and all others are types. These $3$-factorizations lead to threefold triple systems of different types.