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

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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].
Keywords: weighted graph, hamiltonian cycles, heavy cycles, implicit degree, implicit weighted degree 
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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/