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)

Abstract

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 $n\equiv 5,8,\text{ or }11(\mathrm{mod}12)$. When $n\equiv 11(\mathrm{mod}12)$, 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 $n\equiv 5\text{ or }8(\mathrm{mod}12)$, 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.