Maximum Packing of λ -Fold Complete 3-Uniform Hypergraph with a Special Tetrahedron

Let K v ( 3 ) = ( V , E ) be the complete 3-uniform hypergraph, where the vertex set is V = { x 1 , x 2 , ⋯ , x v } , in which the edge set E is of all triples. Let S T denote the special tetrahedron with four edges, where each edge contains three vertices of degree 2. In this paper, we consider the decomposition and packing of a complete 3-uniform hypergraph of an λ - fold special tetrahedron. Firstly, the necessary conditions for the existence of the λ - fold S T - decomposition are discussed in four distinct cases. Secondly, according to the recursive constructions, the required designs of small orders are found. For hypergraphs with large orders, they can be recursively generated using some designs of small orders. Then, it is proven that the above necessary conditions are sufficient. Finally, we prove that a maximum S T - packing of a complete 3-uniform hypergraph K v ( 3 ) exists for all v ≥ 6 and λ ≥ 1 .