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 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1140,
author = {Shenggui Zhang and Xueliang Li and Hajo Broersma},
title = {A s3 type condition for heavy cycles in weighted graphs},
journal = {Discussiones Mathematicae Graph Theory},
volume = {21},
year = {2001},
pages = {159-166},
zbl = {1002.05047},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1140}
}
Shenggui Zhang; Xueliang Li; Hajo Broersma. A σ₃ type condition for heavy cycles in weighted graphs. Discussiones Mathematicae Graph Theory, Tome 21 (2001) pp. 159-166. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1140/
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