Self-Complementary Hypergraphs

In this thesis, we survey the current research into self-complementary hypergraphs, and present several new results. We characterize the cycle type of the permutations on n elements with order equal to a power of 2 which are k-complementing. The k-complementing permutations map the edges of a k-uniform hypergraph to the edges of its complement. This yields a test to determine whether a finite permutation is a k-complementing permutation, and an algorithm for generating all self-complementary k-uniform hypergraphs of order n, up to isomorphism, for feasible n. We also obtain an alternative description of the known necessary and sufficient conditions on the order of a self-complementary k-uniform hypergraph in terms of the binary representation of k. We examine the orders of t-subset-regular self-complementary uniform hypergraphs. These form examples of large sets of two isomorphic t-designs. We restate the known necessary conditions on the order of these structures in terms of the binary representation of the rank k, and we construct 1-subset-regular self-complementary uniform hypergraphs to prove that these necessary conditions are sufficient for all ranks k in the case where t = 1. We construct vertex transitive self-complementary k-hypergraphs of order n for all integers n which satisfy the known necessary conditions due to Potočnik and Šajna, and consequently prove that these necessary conditions are also sufficient. We also generalize Potočnik and Šajna’s necessary conditions on the order of a vertex