Vertex-transitive self-complementary uniform hypergraphs of prime order

For an integer n and a prime p , let n ( p ) = max { i : p i divides n } . In this paper, we present a construction for vertex-transitive self-complementary k -uniform hypergraphs of order n for each integer n such that p n ( p ) ≡ 1 ( mod 2 ℓ + 1 ) for every prime p , where ℓ = max { k ( 2 ) , ( k − 1 ) ( 2 ) } , and consequently we prove that the necessary conditions on the order of vertex-transitive self-complementary uniform hypergraphs of rank k = 2 ℓ or k = 2 ℓ + 1 due to Potoňick and Šajna are sufficient. In addition, we use Burnside’s characterization of transitive groups of prime degree to characterize the structure of vertex-transitive self-complementary k -hypergraphs which have prime order p in the case where k = 2 ℓ or k = 2 ℓ + 1 and p ≡ 1 ( mod 2 ℓ + 1 ) , and we present an algorithm to generate all of these structures. We obtain a bound on the number of distinct vertex-transitive self-complementary graphs of prime order p ≡ 1 ( mod 4 ) , up to isomorphism.