Independent sets, cliques and hamiltonian graphs

Abstract

LetG be ak-connected (k ≥ 2) graph onn vertices. LetS be an independent set ofG. S is called essential if there exist two distinct vertices inS which have a common neighbor inG. LetV r, be a clique which is a complete subgraph ofG. In this paper it is proven that if every essential independent setS ofk + 1 vertices satisfiesS ∩ V r ≠ ∅, thenG is hamiltonian, orG∖{u} is hamiltonian for someu ∈ V r, orG is one of three classes of exceptional graphs. Our theorem generalizes several well-known theorems.