On self-complementary vertex-transitive graphs

In this paper we study the problem of determining positive integers n for which there exist self-complementary vertex-transitive graphs of order n. The main aim of this paper is to prove that, for distinct primes p, q, there exist self-complementary vertex-transitive graphs of order pq if and only if p, q Scz.tbnd;1 (mod 4). This provides negative answers to a long-standing open question proposed by Zelinka (1979) and a recent open question proposed by Froncek, Rosa Siran (1996).