On a conjecture of A. Ainouche and N. Christofides (Q1191686)

Let \(a\) and \(b\) be two nonadjacent vertices of a 2-connected graph \(G=(V,E)\). Let \(T=\{x\in V\mid a,b\notin N(x)\}\) and let \(\lambda^ T_ i=|\{a_ j\in T\mid N(a_ j)\cap N(a_ i)\neq\varnothing\}|\). The following conjecture of Ainouche and Christofides is proved. Theorem: If \(d(a_ i)\geq 3+\lambda^ T_ i\) for all \(a_ i\in T\), then \(G\) is Hamiltonian if and only if \(G+ab\) is Hamiltonian.