Sylow subgraphs in self-complementary vertex transitive graphs

A graph is self-complementary if it is isomorphic to its complement. A graph is vertex transitive if for each choice of vertices u   and vv there is an automorphism that carries the vertex u   to vv. The number of vertices in a self-complementary vertex-transitive graph must necessarily be congruent to 1 mod 4. However, Muzychuk has shown that if pmpm is the largest power of a prime p   dividing the order of a self-complementary vertex-transitive graph, then pmpm must individually be congruent to 1 mod 4. This is accomplished by establishing the existence of a self-complementary vertex transitive subgraph of order pmpm, a result reminiscent of the Sylow theorems. This article is a self-contained survey, culminating with a detailed proof of Muzychuk's result.