More powerful closure operations on graphs

Abstract

Bondy and Chvátal have observed the following result: G=(V,E) is a simple graph of order n. If uv∉E and d(u)+d(v)⩾n, then G is Hamiltonian iff G+uv is Hamiltonian. Thus, we can obtain a graph C_n(G), named the n-closure of G, from G by successively joining pairs of non-adjacent vertices whose degree sum is at least n. Therefore, G is Hamiltonian if C_n(G) is Hamiltonian. Moreover, Bondy and Chvátal [2] generalized this idea to several properties on G. In the paper, we present some more powerful closure operations that extend the idea of Bondy and Chvátal.