Ore-type degree condition for heavy paths in weighted graphs

A weighted graph is one in which every edge e is assigned a nonnegative number w(e), called the weight of e. For a vertex v of a weighted graph, d(v) is the sum of the weights of the edges incident to v. And the weight of a path is the sum of the weights of the edges belonging to it. In this paper, we give a sufficient condition for a weighted graph to have a heavy path which joins two specified vertices. Let G be a 2-connected weighted graph and let x and y be distinct vertices of G. Suppose that d(u)+d(v) ≥ 2d for every pair of non-adjacent vertices u and v ∈ V (G)\{x, y}. Then x and y are joined by a path of weight at least d, or they are joined by a Hamilton path. Also, we consider the case when G has some vertices whose weighted degree are not assumed.