Geodesic
Heavy cycles in weighted graphs
Discussiones Mathematicae. Graph Theory, Tome 22 (2002) no. 1, pp. 7-15.

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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.
Keywords: weighted graph, (long, optimal, Hamilton) cycle, (edge-, vertex-)weighting, weighted degree 
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Bondy, J.; Broersma, Hajo; van den Heuvel, Jan; Veldman, Henk. Heavy cycles in weighted graphs. Discussiones Mathematicae. Graph Theory, Tome 22 (2002) no. 1, pp. 7-15. http://geodesic.mathdoc.fr/item/DMGT_2002_22_1_a1/

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