Long cycles and neighborhood union in 1-tough graphs with large degree sums

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 ${\displaystyle N{C}_{\sigma ₃-n+5}\left(G\right)}$ = ${\displaystyle max\{{\bigcup }_{i=1}^{\sigma ₃-n+5}}$ ${\displaystyle N\left(v_i\right):\{v₁,...,v_{\sigma ₃-n+5}\}}$ is an independent set of vertices}. We show that every 1-tough graph with σ₃(G) ≥ n contains a cycle of length at least ${\displaystyle min\{n,2N{C}_{\sigma ₃-n+5}\left(G\right)+2\}}$. 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].