Hamiltonian properties of graphs with large neighborhood unions

Bauer, D., G. Fan and H.J. Veldman, Hamiltonian properties of graphs with large neighborhood unions, Discrete Mathematics 96 (1991) 33-49. Let G be a graph of order n, a k =min{~ki=ld(vi): {V 1 . . . . . Vn} is an independent set of vertices in G}, NC=min{IN(u) 13N(v) l :uv~E(G)} and NC2=min{IN(u) t3 N(v)l: d(u, v)=2}. O.~ proved that G is hamiltonian if o2~>n ~>3, while Faudree et al. proved that G is hamiltonian if G is 2-connected and NC ~> -~(2n 1). It is shown that both results are generalized by a recent result ef Bauer et al. Various other existing results in hamiltonian graph theory involving degree-sums or cardinalities of neighborhood unions are also compdred in terms of generality. Furthermore, some new results are proved. In particular, it is shown that the bound ~(2n 1) on NC in the result of Faudree et al. can be lowered to ~ ( 2 n 3), 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 03 ~> n + 2. A Dx-cycle (Dx-path) of G is a cycle (path) C such that every component of G V(C) has order smaller than 2. Sufficient conditions of Lindquester for the existence of Hamilton cycles and paths involving NC2 are extended to Dx-cycles and D~-paths.