On cyclic Hamiltonian decompositions of complete k-uniform hypergraphs

A decomposition $mathcal{C}={ C_1,C_2,…,C_h}$ of the complete $k$-uniform hypergraph $K^k_n$ of order $n$ is called {em cyclic Hamiltonian} if each $C_iin mathcal{C}$, $iin{1,2,ldots,h}$,  is a Hamiltonian cycle in $K^k_n$ and there exists a permutation $sigma$ of the vertex set of $K^k_n$ having exactly one cycle in its cycle decomposition such that for every cycle $C_iinmathcal{C}$ its set of edges coincides with an orbit of $langlesigmarangle$ when acting on the edge set of $K^k_n$.  In this paper it is shown that $K_n^k$ admits a cyclic Hamiltonian decomposition if and only if $n$ and $k$ are relatively prime and $lambda=min{d>1: d|n}>frac nk$.