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

MingChu Li, Xiaodong Chen, Haitze J. Broersma

Research output: Contribution to journalArticleAcademicpeer-review

4 Citations (Scopus)

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].
Original languageUndefined
Pages (from-to)285-298
Number of pages14
JournalJournal of graph theory
Volume68
Issue number4
DOIs
Publication statusPublished - Dec 2011

Keywords

  • EWI-21342
  • MSC-05C
  • Claw-free graph
  • hourglass-free graph
  • IR-79452
  • Hamiltonian connectedness

Cite this

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title = "Hamiltonian connectedness in 4-connected hourglass-free claw-free graphs",
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 Ryj{\'a}c̆ek 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].",
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author = "MingChu Li and Xiaodong Chen and Broersma, {Haitze J.}",
year = "2011",
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volume = "68",
pages = "285--298",
journal = "Journal of graph theory",
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Hamiltonian connectedness in 4-connected hourglass-free claw-free graphs. / Li, MingChu; Chen, Xiaodong; Broersma, Haitze J.

In: Journal of graph theory, Vol. 68, No. 4, 12.2011, p. 285-298.

Research output: Contribution to journalArticleAcademicpeer-review

TY - JOUR

T1 - Hamiltonian connectedness in 4-connected hourglass-free claw-free graphs

AU - Li, MingChu

AU - Chen, Xiaodong

AU - Broersma, Haitze J.

PY - 2011/12

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N2 - 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].

AB - 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].

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KW - hourglass-free graph

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KW - Hamiltonian connectedness

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