On cycle decompositions of complete $3$\!-uniform hypergraphs

Abstract

The complete $3$\!-uniform hypergraph $K^{(3)}_n$ of order $n$ has a set $V$ of cardinality $n$ as its vertex set and the set of all $3$ element subsets of $V$ as its edge set. For $n\,\geq\,2,$ let $\mathbb{Z}_n$ denote the set of integers modulo $n.$ For $m\,>\,3,$ let $LC^{(3)}_m$ (respectively, $TC^{(3)}_m$) denote the $3$\!-uniform hypergraph with vertex set $\mathbb{Z}_{2m}$ (respectively, $\mathbb{Z}_m$) and edge set $\{\{2i, 2i+1, 2i+2\} :\, i\,\in\,\{0,1,2,\dots,m-1\}\}$ (respectively, $\{\{i, i+1, i+2\} :\, i\,\in\,\mathbb{Z}_m\}$). Any hypergraph isomorphic to $LC^{(3)}_m$ (respectively, $TC^{(3)}_m$) is a {\it $3$\!-uniform loose $m$\!-cycle} (respectively, {\it $3$\!-uniform tight $m$-cycle}). A decomposition of $K^{(3)}_n$ is a partition of the edge set of $K^{(3)}_n.$ For $m\,\in\,\{5,7\},$ we show that there exists a decomposition of $K^{(3)}_n$ into subhypergraphs isomorphic to $LC^{(3)}_m$ if and only if $n\,\geq\,2m$ and $n\,\equiv\,0,$ $1$ or $2$ (mod $m$). Next, we show that, for $\ell\,\geq\,1$ and $m\,\in\,\{8, 16, 20, 28, 32, 40,44\},$ there exists a decomposition of $K^{(3)}_{2^{\ell}m}$ into subhypergraphs isomorphic to $TC^{(3)}_m.$