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)


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
Issue number4
Publication statusPublished - Dec 2011


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

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