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Hamiltonian connectedness in 4-connected hourglass-free claw-free graphs. (English)
J. Graph Theory 68, No. 4, 285-298 (2011).
Summary: 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\u{c}ek 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 {\it 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 {\it 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 {\it M. Kriesell} [J. Comb. Theory, Ser. B 82, No. 2, 306‒315 (2001; Zbl 1027.05059)].