An Implicit Weighted Degree Condition for Heavy Cycles in Weighted Graphs

Let G be a 2-connected weighted graph and m a nonnegative number. As introduced by Li as the weighted analogue of a concept due to Zhu et al, we use id ^w (v) to denote the implicit weighted degree of a vertex v in G. In this paper we prove that G contains either a Hamilton cycle or a cycle of weight at least m, if the following two conditions are satisfied: (1) max {id ^w (u), id ^w (v)}≥ m/2 for each pair of nonadjacent vertices u and v that are vertices of an induced claw or an induced modified claw of G; (2) In each induced claw, each induced modified claw and each induced P ₄ of G, all the edges have the same weight. This is a common generalization of several previous results on the existence of long cycles in unweighted graphs and heavy cycles in weighted graphs.