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Heavy cycles in weighted graphs

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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.
Opis fizyczny
  • Laboratoire de Mathématiques Discrètes, Université Claude, Bernard Lyon 1, 69622 Villeurbanne Cedex, France
  • Faculty of Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands
  • Department of Mathematics and Statistics, Simon Fraser University, Burnaby, B.C., Canada V5A 1S6
  • Faculty of Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands
  • [1] B. Bollobás and A.D. Scott, A proof of a conjecture of Bondy concerning paths in weighted digraphs, J. Combin. Theory (B) 66 (1996) 283-292, doi: 10.1006/jctb.1996.0021.
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  • [3] J.A. Bondy and G. Fan, Optimal paths and cycles in weighted graphs, Annals of Discrete Math. 41 (1989) 53-69, doi: 10.1016/S0167-5060(08)70449-7.
  • [4] J.A. Bondy and G. Fan, Cycles in weighted graphs, Combinatorica 11 (1991) 191-205, doi: 10.1007/BF01205072.
  • [5] J.A. Bondy and S.C. Locke, Relative lengths of paths and cycles in 3-connected graphs, Discrete Math. 33 (1981) 111-122, doi: 10.1016/0012-365X(81)90159-X.
  • [6] J.A. Bondy and U.S.R. Murty, Graph Theory with Applications (Macmillan, London and Elsevier, New York, 1976).
  • [7] G.A. Dirac, Some theorems on abstract graphs, Proc. London Math. Soc. (3) 2 (1952) 69-81, doi: 10.1112/plms/s3-2.1.69.
  • [8] L. Pósa, On the circuits of finite graphs. Magyar Tud. Akad. Mat. Kutató Int. Közl. 8 (1963) 355-361.
  • [9] T. Spencer (Personal communication, 1992).
  • [10] Yan Lirong (Personal communication, 1990).
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