On factors of 4-connected claw-free graphs

Haitze J. Broersma, M. Kriesell, M. Kriesell, Z. Ryjacek

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31 Citations (Scopus)

Abstract

We consider the existence of several different kinds of factors in 4-connected claw-free graphs. This is motivated by the following two conjectures which are in fact equivalent by a recent result of the third author. Conjecture 1 (Thomassen): Every 4-connected line graph is hamiltonian, i.e., has a connected 2-factor. Conjecture 2 (Matthews and Sumner): Every 4-connected claw-free graph is hamiltonian. We first show that Conjecture 2 is true within the class of hourglass-free graphs, i.e., graphs that do not contain an induced subgraph isomorphic to two triangles meeting in exactly one vertex. Next we show that a weaker form of Conjecture 2 is true, in which the conclusion is replaced by the conclusion that there exists a connected spanning subgraph in which each vertex has degree two or four. Finally we show that Conjectures 1 and 2 are equivalent to seemingly weaker conjectures in which the conclusion is replaced by the conclusion that there exists a spanning subgraph consisting of a bounded number of paths.
Original languageUndefined
Pages (from-to)125-136
Number of pages12
JournalJournal of graph theory
Volume37
Issue number2
DOIs
Publication statusPublished - 2001

Keywords

  • METIS-201534
  • Line graph
  • Factor
  • (Hamilton) cycle
  • Claw-free graph
  • IR-71757
  • Hamilton path

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