Neighborhood Unions and Highly Hamiltonian Graphs

In this paper we examine bounds on \(|N(x) \cup N(y)|\) (for nonadjacent pairs \(x,y \in V(G)\)) that imply certain strong Hamiltonian properties in graphs. In particular, we show that if \(G\) is a 2-connected graph of order \(n\) and if for all pairs of distinct nonadjacent vertices \(x, y \in V(G)\),

1. \(|N(z) \cup N(y)| \geq \frac{2n+5}{3}\), then \(G\) is pancyclic.
2. \(|N(z) \cup N(y)| \geq n-t\) and \(\delta(G) \geq t\), then \(G\) is Hamiltonian.
3. \(|N(z) \cup N(y)| \geq n-2\), then \(G\) is vertex pancyclic.