Geodesic
Long cycles and neighborhood union in 1-tough graphs with large degree sums
Discussiones Mathematicae. Graph Theory, Tome 18 (1998) no. 1, pp. 5-13

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For a 1-tough graph G we define σ₃(G) = mind(u) + d(v) + d(w):u,v,w is an independent set of vertices and NC_σ₃-n+5(G) = max⋃_i = 1^σ₃-n+5N(v_i) : v₁, ..., v_σ₃-n+5is an independent set of vertices. We show that every 1-tough graph with σ₃(G) ≥ n contains a cycle of length at leastminn,2NC_σ₃-n+5(G)+2. This result implies some well-known results of Faßbender [2] and of Flandrin, Jung Li [6]. The main result of this paper also implies that c(G) ≥ minn,2NC₂(G)+2 where NC₂(G) = min|N(u) ∪ N(v)|:d(u,v) = 2. This strengthens a result that c(G) ≥ minn, 2NC₂(G) of Bauer, Fan and Veldman [3].
Keywords: graphs, neighborhood, toughness, cycles 
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Hoa, Vu. Long cycles and neighborhood union in 1-tough graphs with large degree sums. Discussiones Mathematicae. Graph Theory, Tome 18 (1998) no. 1, pp. 5-13. http://geodesic.mathdoc.fr/item/DMGT_1998_18_1_a0/