An (edge-)weighted graph is a graph in which each edge e is assigned a nonnegative real number w(e), called the weight of e. The weight of a cycle is the sum of the weights of its edges, and an optimal cycle is one of maximum weight. The weighted degree w(v) of a vertex v is the sum of the weights of the edges incident with v. The following weighted analogue (and generalization) of a well-known result by Dirac for unweighted graphs is due to Bondy and Fan. Let G be a 2-connected weighted graph such that w(v) ≥ r for every vertex v of G. Then either G contains a cycle of weight at least 2r or every optimal cycle of G is a Hamilton cycle. We prove the following weighted analogue of a generalization of Dirac's result that was first proved by Pósa. Let G be a 2-connected weighted graph such that w(u)+w(v) ≥ s for every pair of nonadjacent vertices u and v. Then G contains either a cycle of weight at least s or a Hamilton cycle. Examples show that the second conclusion cannot be replaced by the stronger second conclusion from the result of Bondy and Fan. However, we characterize a natural class of edge-weightings for which these two conclusions are equivalent, and show that such edge-weightings can be recognized in time linear in the number of edges.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1154,
author = {J. Adrian Bondy and Hajo J. Broersma and Jan van den Heuvel and Henk Jan Veldman},
title = {Heavy cycles in weighted graphs},
journal = {Discussiones Mathematicae Graph Theory},
volume = {22},
year = {2002},
pages = {7-15},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1154}
}
J. Adrian Bondy; Hajo J. Broersma; Jan van den Heuvel; Henk Jan Veldman. Heavy cycles in weighted graphs. Discussiones Mathematicae Graph Theory, Tome 22 (2002) pp. 7-15. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1154/
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