Weighted Degrees and Heavy Cycles in Weighted Graphs

A weighted graph is a graph provided with an edge-weighting function w from the edge set to nonnegative real numbers. Bondy and Fan [Annals of Discrete Math. 41 (1989), 53– 69] began the study on the existence of heavy cycles in weighted graphs. Though several results with Dirac-type degree condition can be generalized to Ore-type one in unweighted graphs, it is shown in [Bondy et al., Discuss. Math. Graph Theory 22 (2002), 7–15] that Bondy and Fan’s theorem, which uses Dirac-type condition, cannot be generalized analo-gously by using Ore-type condition. In this paper we investigate the property peculiar to weighted graphs, and prove a theorem on the existence of heavy cycles in weighted graphs under Ore-type condition, which generalizes Bondy and Fan’s theorem. Moreover, we show the existence of heavy cycles passing through some specified vertices. AMS classification. 05C45, 05C38, 05C3