Complete sets of metamorphoses: Twofold 4-cycle systems into twofold 6-cycle systems

Let (X,C) denote a twofold k-cycle system with an even number of cycles. If these k-cycles can be paired together so that: (i) each pair contains a common edge; (ii) removal of the repeated common edge from each pair leaves a (2k−2)-cycle; (iii) all the repeated edges, once removed, can be rearranged exactly into a collection of further (2k−2)-cycles; then this is a metamorphosis of a twofold k-cycle system into a twofold (2k−2)-cycle system. The existence of such metamorphoses has been dealt with for the case of 3-cycles (Gionfriddo and Lindner, 2003) [3] and 4-cycles (Yazıcı, 2005) [7].If a twofold k-cycle system (X,C) of order n exists, which has not just one but has k different metamorphoses, from k different pairings of its cycles, into twofold (2k−2)-cycle systems, such that the collection of all removed double edges from all k metamorphoses precisely covers 2Kn, we call this a complete set of twofold paired k-cycle metamorphoses into twofold (2k−2)-cycle systems.In this paper, we show that there exists a twofold 4-cycle system (X,C) of order n with a complete set of metamorphoses into twofold 6-cycle systems if and only if n≡0,1,9,16 (mod 24), n≠9.