For a 1-tough graph G we define σ₃(G) = min{d(u) + d(v) + d(w):{u,v,w} is an independent set of vertices} and $NC_{σ₃-n+5}(G)$ = $max{⋃_{i = 1}^{σ₃-n+5}$ $N(v_i) : {v₁, ..., v_{σ₃-n+5}}$ is an independent set of vertices}. We show that every 1-tough graph with σ₃(G) ≥ n contains a cycle of length at least $min{n,2NC_{σ₃-n+5}(G)+2}$. 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) ≥ min{n,2NC₂(G)+2} where NC₂(G) = min{|N(u) ∪ N(v)|:d(u,v) = 2}. This strengthens a result that c(G) ≥ min{n, 2NC₂(G)} of Bauer, Fan and Veldman [3].
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