# An Implicit Weighted Degree Condition For Heavy Cycles

• Volume: 34, Issue: 4, page 801-810
• ISSN: 2083-5892

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## Abstract

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For a vertex v in a weighted graph G, idw(v) denotes the implicit weighted degree of v. In this paper, we obtain the following result: Let G be a 2-connected weighted graph which satisfies the following conditions: (a) The implicit weighted degree sum of any three independent vertices is at least t; (b) w(xz) = w(yz) for every vertex z ∈ N(x) ∩ N(y) with xy /∈ E(G); (c) 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 hamiltonian cycle or a cycle of weight at least 2t/3. This generalizes the result of Zhang et al. [9].

## How to cite

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Junqing Cai, Hao Li, and Wantao Ning. "An Implicit Weighted Degree Condition For Heavy Cycles." Discussiones Mathematicae Graph Theory 34.4 (2014): 801-810. <http://eudml.org/doc/269817>.

@article{JunqingCai2014,
abstract = {For a vertex v in a weighted graph G, idw(v) denotes the implicit weighted degree of v. In this paper, we obtain the following result: Let G be a 2-connected weighted graph which satisfies the following conditions: (a) The implicit weighted degree sum of any three independent vertices is at least t; (b) w(xz) = w(yz) for every vertex z ∈ N(x) ∩ N(y) with xy /∈ E(G); (c) 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 hamiltonian cycle or a cycle of weight at least 2t/3. This generalizes the result of Zhang et al. [9].},
author = {Junqing Cai, Hao Li, Wantao Ning},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {weighted graph; hamiltonian cycles; heavy cycles; implicit degree; implicit weighted degree; Hamiltonian cycles},
language = {eng},
number = {4},
pages = {801-810},
title = {An Implicit Weighted Degree Condition For Heavy Cycles},
url = {http://eudml.org/doc/269817},
volume = {34},
year = {2014},
}

TY - JOUR
AU - Junqing Cai
AU - Hao Li
AU - Wantao Ning
TI - An Implicit Weighted Degree Condition For Heavy Cycles
JO - Discussiones Mathematicae Graph Theory
PY - 2014
VL - 34
IS - 4
SP - 801
EP - 810
AB - For a vertex v in a weighted graph G, idw(v) denotes the implicit weighted degree of v. In this paper, we obtain the following result: Let G be a 2-connected weighted graph which satisfies the following conditions: (a) The implicit weighted degree sum of any three independent vertices is at least t; (b) w(xz) = w(yz) for every vertex z ∈ N(x) ∩ N(y) with xy /∈ E(G); (c) 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 hamiltonian cycle or a cycle of weight at least 2t/3. This generalizes the result of Zhang et al. [9].
LA - eng
KW - weighted graph; hamiltonian cycles; heavy cycles; implicit degree; implicit weighted degree; Hamiltonian cycles
UR - http://eudml.org/doc/269817
ER -

## References

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3. [3] V. Chvátal and P. Erdős, A note on hamiltonian circuits, Discrete Math. 2 (1972) 111-113. doi:10.1016/0012-365X(72)90079-9
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5. [5] H. Enomoto, J. Fujisawa and K. Ota, A σk type condition for heavy cycles in weighted graphs, Ars Combin. 76 (2005) 225-232. Zbl1164.05389
6. [6] I. Fournier and P. Fraisse, On a conjecture of Bondy, J. Combin. Theory (B) 39 (1985) 17-26. doi:10.16/0095-8956(85)90035-8 Zbl0576.05035
7. [7] P. Li, Implicit weighted degree condition for heavy paths in weighted graphs, J. Shandong Univ. (Nat. Sci.) 18 (2003) 11-13.
8. [8] L. Pósa, On the circuits of finite graphs, Magyar Tud. Akad. Mat. Kutató Int. Közl 8 (1963) 355-361. Zbl0133.16702
9. [9] S. Zhang, X. Li and H. Broersma, A σ3 type condition for heavy cycles in weighted graphs, Discuss. Math. Graph Theory 21 (2001) 159-166. doi:10.7151/dmgt.1140 Zbl1002.05047
10. [10] Y. Zhu, H. Li and X. Deng, Implicit-degrees and circumferences, Graphs Combin. 5 (1989) 283-290. doi:10.1007/BF01788680

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