Tight 9-Cycle Decompositions of λ-Fold Complete 3-Uniform Hypergraphs

For $2\le t\le m$, let ${\mathbb{Z}}_m$ denote the group of integers modulo m, and let $T{C}_m^{\left(t\right)}$ denote the t-uniform hypergraph with vertex set ${\mathbb{Z}}_m$ and hyperedge set $\{\{i,i+1,i+2,\dots ,i+t-1\}:i\in {\mathbb{Z}}_m\}$. Any hypergraph isomorphic to $T{C}_m^{\left(t\right)}$ is a t-uniform tight m-cycle. In this paper, we consider the existence of tight 9-cycle decompositions of $\lambda$-fold complete 3-uniform hypergraphs. 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, we obtain the necessary and sufficient conditions for the existence of $T{C}_9^{\left(3\right)}$-decomposition of $\lambda K_n^{\left(3\right)}$. We show there exists a $T{C}_9^{\left(3\right)}$-decomposition of $\lambda K_n^{\left(3\right)}$ if and only if $\lambda n(n-1)(n-2)\equiv 0(mod54)$, $\lambda (n-1)(n-2)\equiv 0(mod6)$ and $n\ge 9$.