Cyclically [formula omitted]-complementary uniform hypergraphs

Abstract

A cyclically t - complementary k - hypergraph is a k -uniform hypergraph with vertex set V and edge set E for which there exists a permutation θ ∈ S y m ( V ) such that the sets E , E θ , E θ 2 , … , E θ t − 1 partition the set of all k -subsets of V . Such a permutation θ is called a ( t , k ) - complementing permutation. The cyclically t -complementary k -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 p , we characterize the cycle type of the ( p r , k ) -complementing permutations θ ∈ S y m ( V ) which have order a power of p . This yields a test for determining whether a permutation in S y m ( V ) is a ( p r , k ) -complementing permutation, and an algorithm for generating all of the cyclically p r -complementing k -hypergraphs of order n , for feasible n , 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.