Almost t-complementary uniform hypergraphs

Abstract

An almost t-complementary k-hypergraph is a k-uniform hypergraph with vertex set V and edge set E for which there exists a permutation $$\theta \in Sym(V)$$ such that the sets $$E, E^\theta , E^{\theta ^2}, \ldots , E^{\theta ^{t-1}}$$ partition the set of all k-subsets of V minus one edge. Such a permutation $$\theta $$ is called an almost (t, k)-complementing permutation. Almost t-complementary k-hypergraphs are a natural generalization of almost self-complementary graphs, which were previously studied by Clapham, Kamble et al., and Wojda. We prove that there exists an almost p-complementary k-hypergraph of order n whenever the base-p representation of k is a subsequence of the base-p representation of n, where p is prime.