Vertex pancyclicity in quasi-claw-free graphs

A graph G is called quasi-claw-free if for any two vertices x and y with distance two there exists a vertex u ∈ N ( x ) ∩ N ( y ) such that N [ u ] ⊆ N [ x ] ∪ N [ y ] . This concept is a natural extension of the classical claw-free graphs. In this paper, we present two sufficient conditions for vertex pancyclicity in quasi-claw-free graphs, namely, quasilocally connected and almost locally connected graphs. Our results include some well-known results on claw-free graphs as special cases. We also give an affirmative answer to a problem proposed by Ainouche.