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 $C_i\in \mathcal{C}$, $i\in\{1,2,\ldots,h\}$,  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 $C_i\in\mathcal{C}$ its set of edges coincides with an orbit of $\langle\sigma\rangle$ 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 $\lambda=\min\{d>1:\ d|n\}>\frac nk$.