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

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) over bar, is the graph in which V((G) over bar) = V(G); and for each pair of vertices u, v in (G) over bar, uv is an element of E((G) over bar) if and only if uv is not an element of E(G). G is called a self-complementary graph if G and (G) over bar are isomorphic. Let G be a self-complementary graph with the vertex set V(G)={nu 1,nu 2,..., nu(4n)}, where d(G)(nu(1))less than or equal to d(G)(nu(2)) less than or equal to... less than or equal to d(G)(nu(4n)) Let H = G[nu(1),nu(2),..., nu(2n)], H' = G[nu(2n+1), nu(2n+2),...,nu(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)). (C) 2000 Elsevier Science B.V. All rights reserved.