Extension of several sufficient conditions for Hamiltonian graphs

Let G be a 2-connected graph of order n. Suppose that for all 3-independent sets X in G, there exists a vertex u in X such that |N(X\{u})| + d(u) ≥ n − 1. Using the concept of dual closure, we prove that 1. G is hamiltonian if and only if its 0-dual closure is either complete or the cycle C7 2. G is nonhamiltonian if and only if its 0-dual closure is either the graph (Kr ∪ Ks ∪ Kt) ∨ K2, 1 ≤ r ≤ s ≤ t or the graph ( 2 )K1 ∨Kn−1 2 . It follows that it takes a polynomial time to check the hamiltonicity or the nonhamiltonicity of a graph satisfying the above condition. From this main result we derive a large number of extensions of previous sufficient conditions for hamiltonian graphs. All these results are sharp.