Geodesic
An implicit weighted degree condition for heavy cycles
Discussiones Mathematicae. Graph Theory, Tome 34 (2014) no. 4, pp. 801-810.

Voir la notice de l'article provenant de la source Library of Science

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].
Keywords: weighted graph, hamiltonian cycles, heavy cycles, implicit degree, implicit weighted degree 
@article{DMGT_2014_34_4_a9,
     author = {Cai, Junqing and Li, Hao and Ning, Wantao},
     title = {An implicit weighted degree condition for heavy cycles},
     journal = {Discussiones Mathematicae. Graph Theory},
     pages = {801--810},
     publisher = {mathdoc},
     volume = {34},
     number = {4},
     year = {2014},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DMGT_2014_34_4_a9/}
}
TY  - JOUR
AU  - Cai, Junqing
AU  - Li, Hao
AU  - Ning, Wantao
TI  - An implicit weighted degree condition for heavy cycles
JO  - Discussiones Mathematicae. Graph Theory
PY  - 2014
SP  - 801
EP  - 810
VL  - 34
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DMGT_2014_34_4_a9/
LA  - en
ID  - DMGT_2014_34_4_a9
ER  - 
%0 Journal Article
%A Cai, Junqing
%A Li, Hao
%A Ning, Wantao
%T An implicit weighted degree condition for heavy cycles
%J Discussiones Mathematicae. Graph Theory
%D 2014
%P 801-810
%V 34
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DMGT_2014_34_4_a9/
%G en
%F DMGT_2014_34_4_a9
Cai, Junqing; Li, Hao; Ning, Wantao. An implicit weighted degree condition for heavy cycles. Discussiones Mathematicae. Graph Theory, Tome 34 (2014) no. 4, pp. 801-810. http://geodesic.mathdoc.fr/item/DMGT_2014_34_4_a9/

[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 $σ_k$ 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 $σ_3$ 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