Vertex pancyclicity in quasi-claw-free graphs

A graph GG is called quasi-claw-free if for any two vertices xx and yy with distance two there exists a vertex u∈N(x)∩N(y)u∈N(x)∩N(y) such that N[u]⊆N[x]∪N[y]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.