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

Li, MingChu and Chen, Xiaodong and Broersma, Hajo (2011) Hamiltonian connectedness in 4-connected hourglass-free claw-free graphs. Journal of Graph Theory, 68 (4). pp. 285-298. ISSN 0364-9024

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Abstract:	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].
Item Type:	Article
Copyright:	© 2011 Wiley
Faculty:	Electrical Engineering, Mathematics and Computer Science (EEMCS)
Research Group:	Discrete Mathematics and Mathematical Programming (DMMP)
Link to this item:	http://purl.utwente.nl/publications/79452
Official URL:	http://dx.doi.org/10.1002/jgt.20558
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