Perfect Matching and Hamilton Tight Cycle Decomposition of Complete $n$-Balanced $r$-Partite $k$-Uniform Hypergraphs

Abstract

Let $r\ge k\ge 2$ and $K_{r,n}^{(k)}$ denote the complete $n$-balanced $r$-partite $k$-uniform hypergraph, whose vertex set consists of $r$ parts, each has $n$ vertices, and whose edge set contains all the $k$-element subsets with no two vertices from one part. A decomposition of $K_{r,n}^{(k)}$ is a partition of $E(K_{r,n}^{(k)})$. A perfect matching (resp., Hamilton tight cycle) decomposition of $K_{r,n}^{(k)}$ is a decomposition of $K_{r,n}^{(k)}$ into perfect matchings (resp., Hamilton tight cycles). In this paper, we prove that if $k\mid n$ (resp., $2\nmid k$ and $k\mid n$), then $K_{k+1,n}^{(k)}$ (resp., $K_{k+2,n}^{(k)}$) has a perfect matching decomposition. We also prove that for any integer $k\geq 2$, $K_{k+1,n}^{(k)}$ has a Hamilton tight cycle decomposition. In all cases, we use constructive methods involving number theory. In fact, we confirm two conjectures proposed by Zhang, Lu, and Liu [Appl. Math. Comput., 386 (2020), 125492].