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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 \( 9 \pmod{24} \), and greater than \( 9 \).