Heavy cycles in weighted graphs

• Autorzy: J. Adrian Bondy , Hajo J. Broersma , Jan van den Heuvel , Henk Jan Veldman
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

weighted graph; (long, optimal, Hamilton) cycle; (edge-, vertex-)weighting; weighted degree

Kategorie tematyczne

• 05C35: Extremal problems
• 05C45: Eulerian and Hamiltonian graphs
• 05C38: Paths and cycles

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 2002
• Tom: 22
• Numer: 1
• Strony: 7-15

Daty

wydano

2002

otrzymano

2000-06-27

poprawiono

2001-05-22

Twórcy

autor

J. Adrian Bondy

• Laboratoire de Mathématiques Discrètes, Université Claude, Bernard Lyon 1, 69622 Villeurbanne Cedex, France

autor

Hajo J. Broersma

• Faculty of Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands

autor

Jan van den Heuvel

• Department of Mathematics and Statistics, Simon Fraser University, Burnaby, B.C., Canada V5A 1S6

autor

Henk Jan Veldman

• Faculty of Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands

Bibliografia

• [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.
• [2] J.A. Bondy, Basic graph theory: paths and circuits, in: R.L. Graham, M. Grötschel and L. Lovász, eds., Handbook of Combinatorics (North-Holland, Amsterdam, 1995) 3-110.
• [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).

Identyfikatory

DOI

10.7151/dmgt.1154