Hamiltonian-connected self-complementary graphs

AbstractA self-complementary graph having a complementing permutation σ = [1, 2, 3, …, 4k], consisting of one cycle, and having the edges (1, 2) and (1, 3) is strongly Hamiltonian iff it has an edge between two even-labelled vertices. Some of these strongly Hamiltonian self-complementary graphs are also shown to be Hamiltonian connected

On separable self-complementary graphs

AbstractIn this paper, we describe the structure of separable self-complementary graphs

Ramsey's theorem and self-complementary graphs

AbstractIt is proved that, given any positive integer k, there exists a self-complementary graph with more than 4·214k vertices which contains no complete subgraph with k+1 vertices. An application of this result to coding theory is mentioned

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

AbstractA 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 Ḡ, is the graph in which V(Ḡ)=V(G); and for each pair of vertices u,v in Ḡ,uv∈E(Ḡ) if and only if uv∉E(G). G is called a self-complementary graph if G and Ḡ are isomorphic. Let G be a self-complementary graph with the vertex set V(G)={v1,v2,…,v4n}, where dG(v1)⩽dG(v2)⩽⋯⩽dG(v4n). Let H=G[v1,v2,…,v2n],H′=G[v2n+1,v2n+2,…,v4n] 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))