Hamiltonian Graphs Involving Neighborhood Conditions

Abstract

Let \(G\) be a graph on \(n\) vertices. \(\delta\) and \(\alpha\) be the minimum degree and independence number of \(G\), respectively. We prove that if \(G\) is a \(2\)-connected graph and \(|N(x) \cup N(y)| \geq n-\delta – 1\) for each pair of nonadjacent vertices \(x,y\) with \(1 \leq |N(x) \cap N(y)| \leq \alpha – 1\), then \(G\) is hamiltonian or \(G \in \{G_1, G_2\}\) (see Figure 1.1 and Figure 1.2). As a corollary, if \(G\) is a 2-connected graph and \(|N(x) \cup N(y)| \geq n – \delta\) for each pair of nonadjacent vertices \(x,y\) with \(1 \leq |N(x) \cap N(y)| \leq \alpha – 1\), then \(G\) is hamiltonian. This result extends former results by Faudree et al. \(([5])\) and Yin \(([7])\).