Heavy paths and cycles in weighted graphs

A weighted graph is a graph in which each edge is assigned a non-negative number, called the weight. The weight of a path (cycle) is the sum of the weights of its edges. The weighted degree of a vertex is the sum of the weights of the edges incident with the vertex. A usual (unweighted) graph can be considered as a weighted graph with constant weight 1. In this paper, it is proved that for a 2-connected weighted graph, if every vertex has weighted degree at least d, then for any given vertex y, either y is contained in a cycle with weight at least 2d or every heaviest cycle is a Hamilton cycle. This result is a common generalization of Grötschel's theorem and Bondy-Fan's theorem assuring the existence of a cycle with weight at least 2d on the same condition. Also, as a tool for proving this result, we show a result concerning heavy paths joining two specific vertices and passing through one given vertex.