On finite graphs that are self-complementary and vertex transitive

Self-complementary graphs have been extensively investigated and effectively used in the study of Ramsey numbers. It is well-known and easily proved that there exist regular self-complementary graphs of order n if and only if n == 1 (mod 4). So it is natural to ask whether a similar result for self-complementary vertex-transitive graphs exists. More precisely, for which positive integers n do there exist self-complementary vertex-transitive graphs of order n? Recently, several long-standing open problems concerning this question have been solved, but the general question is still open. In this article we give a brief survey in this area and introduce several new problems.