Heavy cycles in k-connected weighted graphs

A weighted graph is one in which every edge e is assigned a nonnegative number 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 where k ⩾ 2 . Then G contains either a Hamilton cycle or a cycle of weight at least 2 m / ( k + 1 ) , if G satisfies the following conditions: (1) The weighted degree sum of any k + 1 independent vertices is at least m; (2) In each induced claw, each induced modified claw and each induced P 4 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.