Neighbourhood Unions and Edge-Pancyclicity

Let \(G\) be a \(2\)-connected simple graph with order \(n\) (\(n \geq 5\)) and minimum degree 6. This paper proves that if \(|N(u) \cup N(v)| \geq n – \delta + 2\) for any two nonadjacent vertices \(u, v \in V(G)\), then \(G\) is edge-pancyclic, with a few exceptions. Under the same condition, we prove that if \(u, v \in V(G)\) and \(\{u, v\}\) is not a cut set and \(N(u) \cap N(v) \neq \phi\) when \(uv \in E(G)\), then there exist \(u\)–\(v\) paths of length from \(d(u, v)\) to \(n – 1\).