On the structure of self-complementary graphs

A graph G is self complementary if it is isomorphic to its complement G. In this paper we define bipartite self-complementary graphs, and show how they can be used to understand the structure of self-complementary graphs. For G a selfcomplementary graph of odd order, we describe a decomposition of G into edge disjoint subgraphs, one of which is a bipartite self-complementary graph of order |G|−1. A method of constructing self-complementary graphs of odd order based on this decomposition is presented. The bipartite self-complementary graphs of order up to 12 are also presented 1. SELF-COMPLEMENTARY GRAPHS All graphs in this paper are simple, finite and undirected. Let G = G(V,E) be a graph with vertex set V and edge set E. If X ⊆ V , then the induced subgraph 〈X〉 is the maximal subgraph of G with vertex set X . If X and Y are disjoint subsets of V then the bipartite induced subgraph 〈X,Y 〉, is the maximal subgraph of G with vertex set X ∪ Y where every edge joins a vertex in X to a vertex in Y . A graph G is self-complementary (s.c.) if it is isomorphic to its complement G. If G is a self-complementary graph with vertex set {1, 2, 3, · · · , n}, and φ is an isomorphism from G to G, then φ can be viewed as an element of the symmetric group Sn, and is referred to as a complementing permutation of G. The permutationsa in this paper will be expressed as the product of disjoint cycles. Fig. 1 shows a s.c. graph and its complement with associated complementing permutation.