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 ( 2 k − 2 ) -cycle; (iii) all the repeated edges, once removed, can be rearranged exactly into a collection of further ( 2 k − 2 ) -cycles; then this is a metamorphosis of a twofold k -cycle system into a twofold ( 2 k − 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]. wofold 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 ( 2 k − 2 ) -cycle systems, such that the collection of all removed double edges from all k metamorphoses precisely covers 2 K n , we call this a complete set of twofold paired k -cycle metamorphoses into twofold ( 2 k − 2 ) -cycle systems. s 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 .