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## Discussiones Mathematicae Graph Theory

2001 | 21 | 2 | 159-166
Tytuł artykułu

### A σ₃ type condition for heavy cycles in weighted graphs

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Treść / Zawartość
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Języki publikacji
EN
Abstrakty
EN
A weighted graph is a graph in which each edge e is assigned a non-negative number w(e), called the weight of e. The weight of a cycle is the sum of the weights of its edges. The weighted degree $d^w(v)$ of a vertex v is the sum of the weights of the edges incident with v. In this paper, we prove the following result: Suppose G is a 2-connected weighted graph which satisfies the following conditions: 1. The weighted degree sum of any three independent vertices is at least m; 2. w(xz) = w(yz) for every vertex z ∈ N(x)∩N(y) with d(x,y) = 2; 3. In every triangle T of G, either all edges of T have different weights or all edges of T have the same weight. Then G contains either a Hamilton cycle or a cycle of weight at least 2m/3. This generalizes a theorem of Fournier and Fraisse on the existence of long cycles in k-connected unweighted graphs in the case k = 2. Our proof of the above result also suggests a new proof to the theorem of Fournier and Fraisse in the case k = 2.
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EN
Kategorie tematyczne
Wydawca
Czasopismo
Rocznik
Tom
Numer
Strony
159-166
Opis fizyczny
Daty
wydano
2001
otrzymano
2000-02-07
Twórcy
autor
• Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an, Shaanxi 710072, P.R. China
autor
• Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an, Shaanxi 710072, P.R. China
autor
• Faculty of Mathematical Sciences, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands
Bibliografia
• [1] J.A. Bondy, Large cycles in graphs, Discrete Math. 1 (1971) 121-132, doi: 10.1016/0012-365X(71)90019-7.
• [2] J.A. Bondy, H.J. Broersma, J. van den Heuvel and H.J. Veldman, Heavy cycles in weighted graphs, to appear in Discuss. Math. Graph Theory, doi: 10.7151/dmgt.1154.
• [3] J.A. Bondy and G. Fan, Optimal paths and cycles in weighted graphs, Ann. Discrete Math. 41 (1989) 53-69, doi: 10.1016/S0167-5060(08)70449-7.
• [4] J.A. Bondy and U.S.R. Murty, Graph Theory with Applications (Macmillan London and Elsevier, New York, 1976).
• [5] G.A. Dirac, Some theorems on abstract graphs, Proc. London Math. Soc. 2 (3) (1952) 69-81, doi: 10.1112/plms/s3-2.1.69.
• [6] I. Fournier and P. Fraisse, On a conjecture of Bondy, J. Combin. Theory (B) 39 (1985) 17-26, doi: 10.1016/0095-8956(85)90035-8.
• [7] L. Pósa, On the circuits of finite graphs, Magyar Tud. Math. Kutató Int. Közl. 8 (1963) 355-361.
• [8] S. Zhang, X. Li and H.J. Broersma, A Fan type condition for heavy cycles in weighted graphs, to appear in Graphs and Combinatorics.
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