Local properties of graphs that induce global cycle properties

Abstract

A graph \(G\) is locally Hamiltonian if \(G[N(v)]\) is Hamiltonian for every vertex \(v\in V(G)\). In this note, we prove that every locally Hamiltonian graph with maximum degree at least \(|V(G)| - 7\) is weakly pancyclic. Moreover, we show that any connected graph \(G\) with \(\Delta(G)\leq 7\) and \(\delta(G[N(v)])\geq 3\) for every \(v\in V (G)\), is fully cycle extendable. These findings improve some known results by Tang and Vumar.

Keywords

• fully cycle extendability
• weakly pancyclicity
• locally connected