On cycle decompositions of complete 3-uniform hypergraphs

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 ? 2, let Zn 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 Z2m (respectively, Zm) and edge set {{2i, 2i+1, 2i+2}: i ? {0, 1, 2,..., m-1}} (respectively, {{i, i+1, i+2}: i ? Zm}). Any hypergraph isomorphic to LC(3)m (respectively, TC(3)m) is a 3-uniform loose m-cycle (respectively, 3-uniform tight m-cycle). A decomposition of K(3)n is a partition of the edge set of K(3)n. We show that there exists a decomposition of K(3)n into subhypergraphs isomorphic to LC(3)7 if and only if n ? 14 and n ? 0, 1 or 2 (mod 7). Next, we show that, for ? ? 1 and m ? {8, 16, 20, 28, 32, 40, 44}, there exists a decomposition of K(3)2?m into subhypergraphs isomorphic to TC(3)m .