Self-complementary graphs and Ramsey numbers Part I: the decomposition and construction of self-complementary graphs

Abstract

A new method of studying self-complementary graphs, called the decomposition method, is proposed in this paper. Let G be a simple graph. The complement of G, denoted by G ̄ , is the graph in which V( G ̄ )=V(G) ; and for each pair of vertices u,v in G ̄ , uv∈E( G ̄ ) if and only if uv∉E(G). G is called a self-complementary graph if G and G ̄ are isomorphic. Let G be a self-complementary graph with the vertex set V(G)={v 1,v 2,…,v 4n} , where d G(v 1)⩽d G(v 2)⩽⋯⩽d G(v 4n) . Let H=G[v 1,v 2,…,v 2n], H′=G[v 2n+1,v 2n+2,…,v 4n] and H ∗=G−E(H)−E(H′) . Then G=H+H′+H ∗ is called the decomposition of the self-complementary graph G. In part I of this paper, the fundamental properties of the three subgraphs H, H′ and H ∗ of the self-complementary graph G are considered in detail at first. Then the method and steps of constructing self-complementary graphs are given. In part II these results will be used to study certain Ramsey number problems (see (II)).