Degrees, Neighborhood Unions and Hamiltonian Properties

Let \(G\) be a 2-connected simple graph of order \(n\) (\(\geq 3\)) with connectivity \(k\). One of our results is that if there exists an integer \(t\) such that for any distinct vertices \(u\) and \(v\), \(d(u,v) = 2\) implies \(|N(u) \bigcup N(v)| \geq n-t\), and for any independent set \(S\) of cardinality \(k+1\), \(\max\{d(u) \mid u \in S\} \geq t\), then \(G\) is hamiltonian. This unifies many known results for hamiltonian graphs. We also obtain a similar result for hamiltonian-connected graphs.