Abstract

Let G be a graph of order n, σ_k = min{ε_i=1^kd(ν_i): {ν₁,…, ν_k} is an independent set of vertices in G}, NC = min{|N(u) N(ν)|: uνE(G)} and NC2 = min{|N(u)N(ν)|: d(u,ν)=2}. Ore proved that G is hamiltonian if σ₂n3, while Faudree et al. proved that G is hamiltonian if G is 2-connected and . 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 on NC in the result of Faudree et al. can be lowered to , 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 σ₃n+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.

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^* This research was supported in part by the National Security Agency under grant number MDA904-89-H-2008.