Neighborhood unions and hamiltonicity of graphs Original Research Article

Let G be a graph of order n. In this paper, we prove that if G is a 2-connected graph of order n such that for all u, vϵV(G), dist(u,v)=2⇒| N(u)∪N(v)|⩾n+32, where dist(u, v) is the distance between u and v in G, then either G is hamiltonian, or G is a spanning subgraph of a graph in one of three families of exceptional graphs. As a corollary, we get that if G is a 3-connected graph of order n such that for all u, vϵ V(G). dist(u,v)=2⇒| N(u)∪N(v)|⩾n+32, then G is hamiltonian.