A Note on 1 – Tough Hamiltonian Graphs

Let \(G\) be a \(1\)-tough graph of order \(n\). If \(|N(S)| \geq \frac{n + |S| – 1}{3}\)
for every non-empty subset \(S\) of the vertex set \(V(G)\) of \(G\), then \(G\) is hamiltonian.