Hamiltonian properties of graphs with large neighborhood unions

Abstract Let G be a graph of order n , σ k = min{ ϵ i =1 k d ( ν i ): { ν 1 ,…, ν k } is an independent set of vertices in G }, NC = min{| N ( u )∪ N ( ν )|: uν ∉ E ( G )} and NC2 = min{| N ( u )∪ N ( ν )|: d ( u , ν )=2}. Ore proved that G is hamiltonian if σ 2 ⩾ n ⩾3, while Faudree et al. proved that G is hamiltonian if G is 2-connected and NC ⩾ 1 3 (2n−1) . It is shown that both results are generalized by a recent result of Bauer et al. Various other existing results in hamiltonian graph theory involving degree-sums or cardinalities of neighborhood unions are also compared in terms of generality. Furthermore, some new results are proved. In particular, it is shown that the bound 1 3 (2n−1) on NC in the result of Faudree et al. can be lowered to 1 3 (2n−1) , which is best possible. Also, G is shown to have a cycle of length at least min{ n , 2(NC2)} if G is 2-connected and σ 3 ⩾ n +2. A D λ -cycle ( D λ -path) of G is a cycle (path) C such that every component of G − V ( C ) has order smaller than λ. Sufficient conditions of Lindquester for the existence of Hamilton cycles and paths involving NC2 are extended to D λ -cycles and D λ -paths.