Every 4-connected line graph of a quasi claw-free graph is hamiltonian connected

Let G   be a graph. For u,v∈V(G)u,v∈V(G) with distG(u,v)=2distG(u,v)=2, denote JG(u,v)={w∈NG(u)∩NG(v)|NG(w)⊆NG(u)∪NG(v)∪{u,v}}JG(u,v)={w∈NG(u)∩NG(v)|NG(w)⊆NG(u)∪NG(v)∪{u,v}}. A graph G   is called quasi claw-free if JG(u,v)≠∅JG(u,v)≠∅ for any u,v∈V(G)u,v∈V(G) with distG(u,v)=2distG(u,v)=2. In 1986, Thomassen conjectured that every 4-connected line graph is hamiltonian. In this paper we show that every 4-connected line graph of a quasi claw-free graph is hamiltonian connected.