An efficient condition for a graph to be Hamiltonian

Let G=(V,E) be a 2-connected simple graph and let dG(u,v) denote the distance between two vertices u,v in G. In this paper, it is proved: if the inequality dG(u)+dG(v)≥| V(G)|-1 holds for each pair of vertices u and v with dG(u,v)=2, then G is Hamiltonian, unless G belongs to an exceptional class of graphs. The latter class is described in this paper. Our result implies the theorem of Ore [Note on Hamilton circuits, Amer. Math. Monthly 67 (1960) 55]. However, it is not included in the theorem of Fan [New sufficient conditions for cycles in graph, J. Combin. Theory Ser. B 37 (1984) 221-227].