INDEPENDENT SETS, CLIQUES AND HAMILTONIAN GRAPHS

Let G be a k-connected (k greater than or equal to 2) graph on n vertices. Let S be an independent set of G. S is called essential if there exist two distinct vertices in S which have a common neighbor in G. Let V-r be a clique which is a complete subgraph of G. In this paper it is proven that if every essential independent set S of k + 1 vertices satisfies S boolean AND V-r not equal empty set, then G is hamiltonian, or G\(u) is hamiltonian for some u is an element of V-r or G is one of three classes of exceptional graphs. Our theorem generalizes several well-known theorems.