The Cycle Decomposition of Multiple Complete 3-Uniform Hypergraphs

This paper investigates the decomposition of the λ-fold complete 3-uniform hypergraph λKν(3) into 4-cycles, denoted as Sλ(3,Γ5,1,v). Using the Γ5,1-structure as a model, we develop recursive construction techniques that exploit symmetric properties and provide explicit designs for small orders. These recursive frameworks enable the systematic generation of large-order hypergraph designs from smaller building blocks, illustrating the symmetric inheritance of structural properties. We establish that the necessary conditions for such a decomposition are also sufficient: an Sλ(3,Γ5,1,v) exists if and only if 24∣λv(v−1)(v−2),2∣λ(v−1)(v−2),andv≥5. This result highlights the deep interplay between combinatorial design theory and symmetry in hypergraph decompositions.