A closure concept based on neighborhood unions of independent triples

The well-known closure concept of Bondy and Chvatal is based on degree-sums of pairs of nonadjacent (independent) vertices. We show that a more general concept due to Ainouche and Christofides can be restated in terms of degree-sums of independent triples. We introduce a closure concept which is based on neighborhood unions of independent triples and which also generalizes the closure concept of Bondy and Chvatal. Let G be a 2-connected graph on n vertices and let u, o be a pair of nonadjacent vertices of G. Define &,=IN(u)nN(v)l, T,,={weV(G)-{u,v} Iu,v#N(w)} and t.,=I T,,I. We prove the following main result: If A,,>3 and ~N(u)uN(v)uN(w)~>n->.,, for at least t + 2 -i,, vertices MOE T, or if L,, < 2 and G satisfies the 1-2-3-condition (defined in Section 2) and IN(u)uN(u)uN(w)l=n-3 for all vertices wT, then G is Hamiltonian if and only if Gfuo is Hamiltonian.