A note on heavy cycles in weighted graphs

Heavy paths and cycles in weighted graphs

A weighted graph is a graph in which each edge e is assigned a non-negative number w(e), called the weight of e. In this paper, some theorems on the existence of long paths and cycles in unweighted graphs are generalized to heavy paths and cycles in weighted graphs.

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....

Heavy fans, cycles and paths in weighted graphs of large connectivity

A set of paths joining a vertex y and a vertex set L is called (y, L)-fan if any two of the paths have only y in common, and its width is the number of paths forming it. In weighted graphs, it is known that the existence of heavy fan is useful to find a heavy cycle containing some specified vertices. In this paper, we show the existence of heavy fans with large width containing some specified v...

Heavy Cycles in 2-Connected Weighted Graphs with Large Weighted Degree Sums

In this paper, we prove that a 2-connected weighted graph G contains either a Hamilton cycle or a cycle of weight at least 2m/3 if it satisfies the following conditions: (1) ∑3 i=1 d (vi) ≥ m, where v1, v2 and v3 are three pairwise nonadjacent vertices of G, and two of them are nonadjacent vertices of an induced claw or an induced modified claw; (2) In each induced claw and each induced modifie...

Heavy Paths and Cycles in Weighted Graphs and Related Topics