Complete Sets of Metamorphoses: Paired Stars into 4-Cycles

Abstract

If an edge-disjoint decomposition of a complete graph of order $n$ into copies of a $3$-star (i.e., the graph $K_{1,3}$ on $4$ vertices) is taken, and if these $3$-stars can be paired up in three distinct ways to form a graph on $6$ vertices consisting of a $4$-cycle with two opposite pendant edges, such that:
(1) in each of the three pairings, there exists a metamorphosis into a $4$-cycle system; (2) taking precisely those $4$-cycles formed from the two pendant edges from each pair of $3$-stars, in each of the three metamorphoses, we again have a $4$-cycle system of the complete graph, then this is called a complete set of metamorphoses from paired $3$-stars into $4$-cycles.

We show that such a complete set of metamorphoses from paired $3$-stars into $4$-cycles exists if and only if the order of the complete graph is $1$ or $9(\mathrm{mod}24)$, and greater than $9$.