Neighborhood Union Condition with Distance for Vertex-pancyclicity

Abstract

Let \(G\) be a \(2\)-connected simple graph with order \(n\) (\(n \geq 5\)) and minimum degree \(5\). This paper proves that if for any two vertices \(u,v\) of \(G\) at distance two there holds \(|N(u) \bigcup N(v)| \geq n – \delta\), then \(G\) is vertex-pancyclic with a few exceptions.