Loose 6-cycle decompositions of complete 3-uniform hypergraphs

The complete 3-uniform hypergraph K_n⁽³⁾ 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 ≥ 2 , let Z_n denote the set of integers modulo n . For m > 3 , let LC_m⁽³⁾ denote the 3-uniform hypergraph with vertex set Z₂ₘ and edge set \{\{2i, 2i+1, 2i+2\} : i ∈ \{0, 1, 2, …, m-1\}\} . Any hypergraph isomorphic to LC_m⁽³⁾ is a 3-uniform loose m -cycle. Given hypergraphs K and H , a decomposition of K into H is a partition \{E_1, E_2, …, E_b\} of the edge set of K such that, for each i ∈ \{1, 2, …, b\} , the subhypergraph induced by E_i is isomorphic to H . We show that there exists a decomposition of K_n⁽³⁾ into LC_6⁽³⁾ if and only if n ≥ 12 and n ≡ 0, 1, 2, 9, 10, 18, 20, 28 or 29 {36} .