The Metamorphosis of 2-Fold Triple Systems Into 2-Fold 4-Cycle Systems

Abstract

Let $c^{\ast }=$. If we remove the double edge, the result is a $4$-cycle. Let $(S,T)$ be a $2$-fold triple system without repeated triples and $(S,C^{\ast })$ a pairing of the triples into copies of $c^{\ast }$. If $C$ is the collection of $4$-cycles obtained by removing the double edges from each copy of $c^{\ast }$ and $F$ is a reassembly of these double edges into $4$-cycles, then $(S,C\cup F)$ is a $2$-fold $4$-cycle system. We show that the spectrum for $2$-fold triple systems having a \emph{metamorphosis} into a $2$-fold $4$-cycle system as described above is precisely the set of all $n\equiv 0,1,4\text{or}9(\mathrm{mod}12)\ge 5$.