Loose 6-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 $LC^{(3)}_m$ denote the $3$\!-uniform hypergraph with vertex set $\mathbb{Z}_{2m}$ and edge set $\{\{2i, 2i+1, 2i+2\} :\, i\,\in\,\{0,1,2,\dots,m-1\}\}.$ Any hypergraph isomorphic to $LC^{(3)}_m$ is a {\it $3$\!-uniform loose $m$\!-cycle}. Given hypergraphs $\mathscr{K}$ and $\mathscr{H},$ a decomposition of $\mathscr{K}$ into $\mathscr{H}$ is a partition $\{\mathscr{E}_1,\mathscr{E}_2,\dots,\mathscr{E}_b\}$ of the edge set of $\mathscr{K}$ such that, for each $i\in\{1,2,\dots,b\},$ the subhypergraph induced by $\mathscr{E}_i$ is isomorphic to $\mathscr{H}.$ We show that there exists a decomposition of $K^{(3)}_n$ into $LC^{(3)}_6$ if and only if $n\,\geq\,12$ and $n\,\equiv\,0,$ $1,$ $2,$ $9,$ $10,$ $18,$ $20,$ $28$ or $29\pmod {36}.$