# An Implicit Weighted Degree Condition For Heavy Cycles

Junqing Cai; Hao Li; Wantao Ning

Discussiones Mathematicae Graph Theory (2014)

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

## Access Full Article

top## Abstract

top## How to cite

topJunqing 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

top- [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 and U.S.R. Murty, Graph Theory with Applications (Macmillan-London, Elsevier-New York, 1976). Zbl1226.05083
- [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
- [4] G.A. Dirac, Some theorems on abstract graphs, Proc. Lond. Math. Soc. 2 (1952) 69-81. Zbl0047.17001
- [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] 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] P. Li, Implicit weighted degree condition for heavy paths in weighted graphs, J. Shandong Univ. (Nat. Sci.) 18 (2003) 11-13.
- [8] L. Pósa, On the circuits of finite graphs, Magyar Tud. Akad. Mat. Kutató Int. Közl 8 (1963) 355-361. Zbl0133.16702
- [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] Y. Zhu, H. Li and X. Deng, Implicit-degrees and circumferences, Graphs Combin. 5 (1989) 283-290. doi:10.1007/BF01788680

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.