Long cycles, degree sums and neighborhood unions

Abstract

For a graph G, define the parameters α( G)=max{| S| | S is an independent set of vertices of G}, σ k ( G)=min{∑ k i=1 d( v i )|{ v 1,…, v k } is an independent set} and NC k ( G)= min{|∪ k i=1 N( v i )∥{ v 1,…, v k } is an independent set} ( k⩾2). It is shown that every 1-tough graph G of order n⩾3 with σ 3( G)⩾ n+ r⩾ n has a cycle of length at least min{ n, n+ NC r+5+∈( n+ r) ( G)- α( G)}, where ε(i)=3(⌈ 1 3 i⌉− 1 3 i) . This result extends previous results in Bauer et al. (1989/90), Faßbender (1992) and Flandrin et al. (1991). It is also shown that a 1-tough graph G of order n⩾3 with σ 3( G)⩾ n+ r⩾ n has a cycle of length at least min {n,2NC ⌊ 1 8 (n+6r+17)⌋ (G)} . Analogous results are established for 2-connected graphs.