Heavy cycles in k-connected weighted graphs with large weighted degree sums

A weighted graph is one in which every edge e   is assigned a nonnegative number w(e)w(e), called the weight of e. The weight of a cycle is defined as 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 it. In this paper, we prove that: Let G be a k  -connected weighted graph with k⩾2k⩾2. Then G   contains either a Hamilton cycle or a cycle of weight at least 2m/(k+1)2m/(k+1), if G   satisfies the following conditions: (1) The weighted degree sum of any k+1k+1 pairwise nonadjacent vertices is at least m; (2) In each induced claw and each induced modified claw of G, all edges have the same weight. This generalizes an early result of Enomoto et al. on the existence of heavy cycles in k-connected weighted graphs.