An hourglass is the only graph with degree sequence 4,2,2,2,2 (i.e. two triangles meeting in exactly one vertex). There are infinitely many claw-free graphs G such that G is not hamiltonian connected while its Ryjá? cek closure cl(G) is hamiltonian connected. This raises such a problem what conditions can guarantee that a claw-free graph G is Hamiltonian connected if and only if cl(G) is hamiltonian connected. In this paper, we will do exploration toward the direction, and show that a 3-connected {claw, (P6)2, hourglass}-free graph G with minimum degree at least 4 is Hamiltonian connected if and only if cl(G) is hamiltonian connected, where (P6)2 is the square of a path P6 on 6 vertices. Using the result, we prove that every 4-connected {claw, (P6)2,hourglass}-free graph is hamiltonian connected, hereby generalizing the result that every 4-connected hourglass-free line graph is hamiltonian connected by Kriesell [J Combinatorial Theory (B) 82 (2001), 306-315].
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