Heavy paths and cycles in weighted graphs


Zhang, Shenggui and Li, Xueliang and Broersma, Hajo (2000) Heavy paths and cycles in weighted graphs. Discrete Mathematics, 223 (1-3). pp. 327-336. ISSN 0012-365X

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Abstract:A weighted graph is a graph in which each edge is assigned a non-negative number, called the weight. The weight of a path (cycle) is the sum of the weights of its edges. The weighted degree of a vertex is the sum of the weights of the edges incident with the vertex. A usual (unweighted) graph can be considered as a weighted graph with constant weight 1. In this paper, it is proved that for a 2-connected weighted graph, if every vertex has weighted degree at least d, then for any given vertex y, either y is contained in a cycle with weight at least 2d or every heaviest cycle is a Hamilton cycle. This result is a common generalization of Grötschel's theorem and Bondy–Fan's theorem assuring the existence of a cycle with weight at least 2d on the same condition. Also, as a tool for proving this result, we show a result concerning heavy paths joining two specific vertices and passing through one given vertex.
Item Type:Article
Copyright:© 2000 Elsevier
Electrical Engineering, Mathematics and Computer Science (EEMCS)
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Link to this item:http://purl.utwente.nl/publications/74275
Official URL:https://doi.org/10.1016/S0012-365X(99)00413-6
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