Hamiltonian connectedness in 4‐connected hourglass‐free claw‐free graphs

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 Ryjác̆ek 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, ( P 6 ) 2 , hourglass}‐free graph G with minimum degree at least 4 is hamiltonian connected if and only if cl ( G ) is hamiltonian connected, where ( P 6 ) 2 is the square of a path P 6 on 6 vertices. Using the result, we prove that every 4‐connected { claw , ( P 6 ) 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]. © 2010 Wiley Periodicals, Inc. J Graph Theory 68:285‐298, 2011