Forbidden subgraphs that imply hamiltonian-connectedness

It is proven that if G is a 3-connected claw-free graph which is also Z3-free (where Z3 is a triangle with a path of length 3 attached), P6-free (where P6 is a path with 6 vertices) or H1-free (where H1 consists of two disjoint triangles connected by an edge), then G is hamiltonian-connected. Also, examples will be described that determine a finite family of graphs L such that if a 3-connected graph being claw-free and L-free implies G is hamiltonian-connected, then L ∈ L.