An Implicit Weighted Degree Condition For Heavy Cycles

• Autorzy: Junqing Cai , Hao Li , Wantao Ning
• Języki publikacji: EN

Abstrakty

EN

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].

Słowa kluczowe

EN

weighted graph; hamiltonian cycles; heavy cycles; implicit degree; implicit weighted degree

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 2014
• Tom: 34
• Numer: 4
• Strony: 801-810

Daty

wydano

2014-11-01

otrzymano

2011-10-31

poprawiono

2013-11-18

zaakceptowano

2013-11-18

online

2014-11-15

Twórcy

autor

Junqing Cai

• School of Management, Qufu Normal University Rizhao, 276826, China

autor

Hao Li

• Institute for Interdisciplinary Research Jianghan University, Wuhan, 430019, China LRI, UMR 8623, CNRS and Universit´e de Paris-Sud 11 F-91405 Orsay, France

autor

Wantao Ning

• Department of Mathematics and Statistic, Xidian University Xi’an, Shaanxi 710071, China

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 and U.S.R. Murty, Graph Theory with Applications (Macmillan-London, Elsevier-New York, 1976).
• [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.
• [5] H. Enomoto, J. Fujisawa and K. Ota, A σk type condition for heavy cycles in weighted graphs, Ars Combin. 76 (2005) 225-232.
• [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
• [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.
• [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
• [10] Y. Zhu, H. Li and X. Deng, Implicit-degrees and circumferences, Graphs Combin. 5 (1989) 283-290. doi:10.1007/BF01788680

Identyfikatory

DOI

10.7151/dmgt.1762