Fully cycle extendibility of Quasi-Claw free graphs

Abstract

A graph G is Quasi-claw free if for any two vertices x and y with distance 2, there exists a vertex u ∈ N(x) ∩ N(y) such that N(u) ⊆ N[x] ∪ N[y]. In [1], Ainouche conjectured that every connected, locally connected Quasi-Claw free graph is vertex pancyclic. In this paper, we prove that if G is connected, locally connected Quasi-Claw free then G is Fully cycle extendible. Several results exist to respond to Ainouche’s conjecture [1], but our result enhances them presenting a significant improvement. Moreover, it is a pioneer result in fully cycle extendibility dealing with this class of graphs.