Heavy cycles passing through some specified vertices in weighted graphs

A weighted graph is one in which every edge e is assigned a nonnegative number, called the weight of e. The sum of the weights of the edges incident with a vertex v is called the weighted degree of v. The weight of a cycle is defined as the sum of the weights of its edges. In this paper, we prove that: (1) if G is a 2-connected weighted graph such that the minimum weighted degree of G is at least d, then for every given vertices x and y, either G contains a cycle of weight at least 2d passing through both of x and y or every heaviest cycle in G is a hamiltonian cycle, and (2) if G is a 2-connected weighted graph such that the weighted degree sum of every pair of nonadjacent vertices is at least s, then for every vertex y, G contains either a cycle of weight at least s passing through y or a hamiltonian cycle. AMS classification: 05C45 05C38 05C35.