Decompositions and Packing of Complete 3-Uniform Hypergraph Into Hyperstar of a Certain Type

Paola Bonacini6 defined the hyperstar which is a hypergraph with \(m\) 3‐edges such that all of them have two vertices in common and is denoted by \(S^{(3)} (2,m + 2)\) . In this paper, we consider the \(S^{(3)} (2,6)\) decomposition and packing of complete 3-uniform hypergraph. Firstly, the necessary condition for the decomposition of the complete 3-uniform hypergraph into an \(S^{(3)} (2,6)\) is \(v = 0,1,2,4,6 \, (\bmod \, 8)\) , \(v \ge 6.\) Secondly, according to the recursive constructions, the required designs of small orders are found. For the hypergraphs with large orders, they can be recursively generated by some designs of small orders. Then, it is proved that the above necessary conditions are sufficient except \(v = 6\) , that is, there exists an \(S^{(3)} (2,6)\) decomposition of a complete 3-uniform hypergraph if and only if \(v = 0,1,2,4,6 \, (\bmod \, 8)\) , \(v > 6.\) Finally, we prove that there exists a maximum \(S^{(3)} (2,6){ - }\) packing, except \(v = 6,7\) . And we find the exact value of the packing number \(d(t,T,v)\) for any integer \(v \ge 6\) .