A σ_k type condition for heavy cycles in weighted graphs

A weighted graph is one in which every edge e is assigned a non-negative number, called the weight, of e. For a vertex v of a weighted graph, d ^w(v) is the sum of the weights of the edges incident with v. For a subgraph H of a weighted graph G, the weight of H is the sum of the weights of the edges belonging to H. In this paper, we give a new sufficient condition for a weighted graph to have a heavy cycle. Let G be a k-connected weighted graph where 2 ≤ k. Then G contains either a Hamilton cycle or a cycle of weight at least 2m/(k+1), if G satisfies the following conditions: (1) The weighted degree sum of any k independent vertices is at least m, (2) w(xz) = w(yz) for every vertex z ε N(x) ∩ N (y) with d(x,y) = 2, arid (4) In every triangle T of G, either all edges of T have different weights or all edges of T have the same weight.