Circumferences of 2-connected quasi-claw-free graphs

A graph G is quasi-claw-free if it satisfies the property: d(x, y) = 2 double right arrow there exists u is an element of N(x) boolean AND N(y) such that N[u] subset of N[x] boolean OR N[y]. In this paper, we prove that the circumference of a 2-connected quasi-claw-free graph G on n vertices is at least min{3 delta + 2, n} or G is an element of F, where F is a class of nonhamiltonian graphs of connectivity 2. Moreover, we prove that if n <= 4 delta, then G is hamiltonian or G is an element of F.