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 NCσ3−n+5(G) = max{ ⋃ σ3−n+5 i=1 N(vi) : {v1, ..., vσ3−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, 2NCσ3−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].