An efficient condition for a graph to be Hamiltonian

Let G=(V,E)G=(V,E) be a 2-connected simple graph and let dG(u,v)dG(u,v) denote the distance between two vertices u,vu,v in G  . In this paper, it is proved: if the inequality dG(u)+dG(v)⩾|V(G)|-1dG(u)+dG(v)⩾|V(G)|-1 holds for each pair of vertices u   and vv with dG(u,v)=2dG(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].