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

Let \( c^* = \). 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 \( (S,C^*) \) a pairing of the triples into copies of \( c^* \). If \( C \) is the collection of \( 4 \)-cycles obtained by removing the double edges from each copy of \( c^* \) 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 \( n \equiv 0,1,4\, \text{or}\, 9 \pmod{12} \geq 5 \).