Geodesic
Heavy Subgraph Conditions for Longest Cycles to Be Heavy in Graphs
Discussiones Mathematicae. Graph Theory, Tome 36 (2016) no. 2, pp. 383-392.

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Let G be a graph on n vertices. A vertex of G with degree at least n/2 is called a heavy vertex, and a cycle of G which contains all the heavy vertices of G is called a heavy cycle. In this note, we characterize graphs which contain no heavy cycles. For a given graph H, we say that G is H-heavy if every induced subgraph of G isomorphic to H contains two nonadjacent vertices with degree sum at least n. We find all the connected graphs S such that a 2-connected graph G being S-heavy implies any longest cycle of G is a heavy cycle.
Keywords: heavy cycles, heavy subgraphs 
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Lia, Binlong; Zhang, Shenggui. Heavy Subgraph Conditions for Longest Cycles to Be Heavy in Graphs. Discussiones Mathematicae. Graph Theory, Tome 36 (2016) no. 2, pp. 383-392. http://geodesic.mathdoc.fr/item/DMGT_2016_36_2_a9/

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