On cyclic Hamiltonian decompositions of complete -uniform hypergraphs

A decomposition C = { C 1 , C 2 , … , C h } of the complete k -uniform hypergraph K n k of order n is called cyclic Hamiltonian if each C i ∈ C , i ∈ { 1 , 2 , … , h } , is a Hamiltonian cycle in K n k and there exists a permutation σ of the vertex set of K n k having exactly one cycle in its cycle decomposition such that for every cycle C i ∈ C its set of edges coincides with an orbit of 〈 σ 〉 when acting on the edge set of K n k . 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 λ = min { d > 1 : d | n } > n k .