Hamiltonian paths and hamiltonian connectivity in graphs

Abstract

Let G be a 2-connected graph with n vertices such that d( u)+ d( v)+ d( w)-| N( u)∩ N( v)∩ N( w)| ⩾ n + 1 holds for any triple of independent vertices u, v and w. Then for any distinct vertices u and v such that { u, v} is not a cut vertex set of G, there is a hamiltonian path between u and v. In particular, if G is 3-connected, then G is hamiltonian-connected. This is closely related to the main result in Flandrin et al. (1991) and generalizes a theorem of Ore (1963) and a theorem of Faudree et al. (1989).