Cyclically tt-complementary uniform hypergraphs

A cyclically  tt-complementary   kk-hypergraph   is a kk-uniform hypergraph with vertex set VV and edge set EE for which there exists a permutation θ∈Sym(V)θ∈Sym(V) such that the sets E,Eθ,Eθ2,…,Eθt−1E,Eθ,Eθ2,…,Eθt−1 partition the set of all kk-subsets of VV. Such a permutation θθ is called a (t,k)(t,k)-complementing permutation  . The cyclically tt-complementary kk-hypergraphs are a natural and useful generalization of the self-complementary graphs, which have been studied extensively in the past due to their important connection to the graph isomorphism problem.For a prime pp, we characterize the cycle type of the (pr,k)(pr,k)-complementing permutations θ∈Sym(V)θ∈Sym(V) which have order a power of pp. This yields a test for determining whether a permutation in Sym(V)Sym(V) is a (pr,k)(pr,k)-complementing permutation, and an algorithm for generating all of the cyclically prpr-complementing kk-hypergraphs of order nn, for feasible nn, up to isomorphism. We also obtain some necessary and sufficient conditions on the order of these structures. This generalizes previous results due to Ringel, Sachs, Adamus, Orchel, Szymański, Wojda, Zwonek, and Bernaldez.