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]. © 2007 Elsevier B.V. All rights reserved.
Li, S., Li, R., & Feng, J. (2007). An efficient condition for a graph to be Hamiltonian. Discrete Applied Mathematics, 155(14), 1842–1845. https://doi.org/10.1016/j.dam.2007.03.013