Claw Conditions for Heavy Cycles in Weighted Graphs

A graph is called a weighted graph when each edge e is assigned a nonnegative number w(e), 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. A 2-connected weighted graph G contains either a Hamilton cycle or a cycle of weight at least c, if G satisfies the following conditions: In every induced claw or induced modified claw F of G, (1) max{d^w(x),d^w(y)} c/2 for each non-adjacent pair of vertices x and y in F, and (2) all edges of F have the same weight.