Vertex-transitive self-complementary uniform hypergraphs of prime order

For an integer nn and a prime pp, let n(p)=max{i:pidividesn}. In this paper, we present a construction for vertex-transitive self-complementary kk-uniform hypergraphs of order nn for each integer nn such that pn(p)≡1(mod2ℓ+1) for every prime pp, where ℓ=max{k(2),(k−1)(2)}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=2ℓk=2ℓ or k=2ℓ+1k=2ℓ+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 kk-hypergraphs which have prime order pp in the case where k=2ℓk=2ℓ or k=2ℓ+1k=2ℓ+1 and p≡1(mod2ℓ+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(mod4), up to isomorphism.