# A σ₃ type condition for heavy cycles in weighted graphs

• Volume: 21, Issue: 2, page 159-166
• ISSN: 2083-5892

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## Abstract

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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 ${d}^{w}\left(v\right)$ 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.

## How to cite

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Shenggui Zhang, Xueliang Li, and Hajo Broersma. "A σ₃ type condition for heavy cycles in weighted graphs." Discussiones Mathematicae Graph Theory 21.2 (2001): 159-166. <http://eudml.org/doc/270379>.

@article{ShengguiZhang2001,
abstract = {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.},
author = {Shenggui Zhang, Xueliang Li, Hajo Broersma},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {weighted graph; (long, heavy, Hamilton) cycle; weighted degree; (weighted) degree sum; degree sum; Hamilton cycle},
language = {eng},
number = {2},
pages = {159-166},
title = {A σ₃ type condition for heavy cycles in weighted graphs},
url = {http://eudml.org/doc/270379},
volume = {21},
year = {2001},
}

TY - JOUR
AU - Shenggui Zhang
AU - Xueliang Li
AU - Hajo Broersma
TI - A σ₃ type condition for heavy cycles in weighted graphs
JO - Discussiones Mathematicae Graph Theory
PY - 2001
VL - 21
IS - 2
SP - 159
EP - 166
AB - 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.
LA - eng
KW - weighted graph; (long, heavy, Hamilton) cycle; weighted degree; (weighted) degree sum; degree sum; Hamilton cycle
UR - http://eudml.org/doc/270379
ER -

## References

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1. [1] J.A. Bondy, Large cycles in graphs, Discrete Math. 1 (1971) 121-132, doi: 10.1016/0012-365X(71)90019-7. Zbl0224.05120
2. [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. Zbl1012.05104
3. [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. Zbl0673.05056
4. [4] J.A. Bondy and U.S.R. Murty, Graph Theory with Applications (Macmillan London and Elsevier, New York, 1976).
5. [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. Zbl0047.17001
6. [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. Zbl0576.05035
7. [7] L. Pósa, On the circuits of finite graphs, Magyar Tud. Math. Kutató Int. Közl. 8 (1963) 355-361. Zbl0133.16702
8. [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. Zbl0994.05090

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