Cyclic partitions of complete nonuniform hypergraphs and complete multipartite hypergraphs

A cyclic q-partition of a hypergraph (V,E) is a partition of the edge set E of the form {F, F , F θ 2 , . . . , F θ q−1 } for some permutation θ of the vertex set V . Let Vn = {1, 2, . . . , n}. For a positive integer k, ( Vn k ) denotes the set of all k-subsets of Vn. For a nonempty subset K of Vn−1, we let K n denote the hypergraph ( Vn, ⋃ k∈K ( Vn k )) . In this paper, we find a necessary and sufficient condition on n, q and k for the existence of a cyclic q-partition of Kk n . In particular, we prove that if p is prime then there is a cyclic p-partition of Kk n if and only if p divides n, where β = blogp kc. As an application of this result, we obtain two sufficient conditions on n1, n2, . . . , nt, k, α and a prime p for the existence of a cyclic p-partition of the complete t-partite k-uniform hypergraph K n1,n2,...,nt .