The Metamorphoses of Maximum Packings of \(2K_{n}\) with Triples to Maximum Packings of \(2K_{n}\) with 4-cycles for \(n \equiv 5,8,\) and 11 (mod 12)

Gionfriddo and Lindner detailed the idea of the metamorphosis of \( 2 \)-fold triple systems with no repeated triples into \( 2 \)-fold \( 4 \)-cycle systems of all orders where each system exists in [3]. In this paper, this concept is expanded to address all orders \( n \) such that \( n \equiv 5, 8, \text{ or } 11 \pmod{12} \). When \( n \equiv 11 \pmod{12} \), a maximum packing of \( 2K_n \) with triples has a metamorphosis into a maximum packing of \( 2K_n \) with \( 4 \)-cycles, with the leave of a double edge being preserved throughout the metamorphosis. For \( n \equiv 5 \text{ or } 8 \pmod{12} \), a maximum packing of \( 2K_n \) with triples has a metamorphosis into a \( 2 \)-fold \( 4 \)-cycle system of order \( n \), except for when \( n = 5 \text{ or } 8 \), when no such metamorphosis is possible.