On hamiltonicity of 2-connected claw-free graphs

A graph G has the hourglass property if every induced hourglass S (a tree with a degree sequence 22224) contains two non-adjacent vertices which have a common neighbor in G ? V (S). For an integer k ?? 4, a graph G has the single k-cycle property if every edge of G, which does not lie in a triangle, lies in a cycle C of order at most k such that C has at least $\left\lfloor {\tfrac{{\left| {V(C)} \right|}} {2}} \right\rfloor $ edges which do not lie in a triangle, and they are not adjacent. In this paper, we show that every hourglass-free claw-free graph G of ??(G) ?? 3 with the single 7-cycle property is Hamiltonian and is best possible; we also show that every claw-free graph G of ??(G) ?? 3 with the hourglass property and with single 6-cycle property is Hamiltonian.