An efficient condition for a graph to be Hamiltonian

Abstract

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