Vertex-transitive self-complementary uniform hypergraphs of prime order

Abstract

For an integer n and a prime p, let n.p/ D maxfi V pi divides ng. In this paper, we present a construction for vertex-transitive self-complementary k-uniform hypergraphs of order n for each integer n such that pn.p/ 1 .mod 2`C1/ for every prime p, where ` D max fk.2/; .kð€€€1/.2/g, and consequently we prove that the necessary conditions on the order of vertex-transitive self-complementary uniform hypergraphs of rank k D 2` or k D 2` C 1 due to Poto¬ick and ajna are sufficient. In addition, we use Burnside's characterization of transitive groups of prime degree to characterize the structure of vertex-transitive selfcomplementary k-hypergraphs which have prime order p in the case where k D 2` or k D 2` C 1 and p 1 .mod 2`C1/, and we present an algorithm to generate all of these structures. We obtain a bound on the number of distinct vertex-transitive selfcomplementary graphs of prime order p 1 .mod 4/, up to isomorphism.