New sufficient condition for Hamiltonian graphs

Let G be a graph and α(G) be the independence number of G. For a vertex v ∈ V (G), d(v) and N(v) represent the degree of v and the neighborhood of v in G, respectively. In this paper, we prove that if G is a k-connected graph of order n, and if max{d(v) : v ∈ S} ≥ n/2 for every independent set S of G with |S| = k which has two distinct vertices x, y ∈ S satisfying 1 ≤ |N(x) ∩N(y)| ≤ α(G)− 1, then G is Hamiltonian.