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

摘要

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 ∪ F) is a 2-fold 4-cycle system. We show that the spectrum for 2-fold triple systems having a metamorphosis into a 2-fold 4-cycle system as described above is precisely the set of all n ≡ 0, 1, 4 or 9 (mod 12) ≥ 5.