Sylow subgraphs of self-complementary vertex transitive graphs

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Expositiones Mathematicae


Mathematics and Computer Science


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 v there is an automorphism that carries the vertex u to v. 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 pm is the largest power of a prime p dividing the order of a self-complementary vertex-transitive graph, then pm must individually be congruent to 1 mod 4. This is accomplished by establishing the existence of a self-complementary vertex transitive subgraph of order pm, a result reminiscent of the Sylow theorems. This article is a self-contained survey, culminating with a detailed proof of Muzychuk's result.