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doi:10.1016/0012-365X(92)00049-W 
Copyright © 1994 Published by Elsevier Science B.V. All rights reserved.
A closure concept based on neighborhood unions of independent triples
Received 18 December 1990;
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Abstract
The well-known closure concept of Bondy and Chvátal 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 Chvátal. Let G be a 2-connected graph on n vertices and let u, v be a pair of nonadjacent vertices of G. Define λuv=¦N(u)∩N(v)¦, Tuv={w
V(G)-¦u,v>{¦uv
N(w)} andtuv=¦Tuv¦. We prove the following main resul : If λuv
3 and ¦N(u)
N(v)N¦ least t + 2 - λuv vertices wT, or if λuv 
2 and G satisfies the 1-2-3-condition (defined in Section 2) and >|N(u)
N(v)N(w)λ=n - 3 for all vertices w |teT, then G is Hamiltonian if and only G + uv is Hamiltonian.
