Vertex-transitive self-complementary uniform hypergraphs

Abstract

In this paper we examine the orders of vertex-transitive self-complementary uniform hypergraphs. In particular, we prove that if there exists a vertex-transitive self-complementary k -uniform hypergraph of order n , where k = 2 ℓ or k = 2 ℓ + 1 and n ≡ 1 ( mod 2 ℓ + 1 ) , then the highest power of any prime dividing n must be congruent to 1 modulo 2 ℓ + 1 . We show that this necessary condition is also sufficient in many cases–for example, for n a prime power, and for k = 3 and n odd–thus generalizing the result on vertex-transitive self-complementary graphs of Rao and Muzychuk. We also give sufficient conditions for the existence of vertex-transitive self-complementary uniform hypergraphs in several other cases. Since vertex-transitive self-complementary uniform hypergraphs are equivalent to a certain kind of large sets of t -designs, the results of the paper imply the corresponding results in design theory.