An efficient condition for a graph to be Hamiltonian Original Research Article

Let image be a 2-connected simple graph and let image denote the distance between two vertices image in G. In this paper, it is proved: if the inequality image holds for each pair of vertices u and image with image, 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].