On factors of 4-connected claw-free graphs

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

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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 Conjecture 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
Place of PublicationEnschede
PublisherUniversity of Twente, Department of Applied Mathematics
Number of pages13
ISBN (Print)0169-2690
Publication statusPublished - 1999

Publication series

NameMemorandum / Department of Applied Mathematics
PublisherDepartment of Applied Mathematics, University of Twente
No.1491
ISSN (Print)0169-2690

Keywords

  • MSC-05C38
  • MSC-05C45
  • IR-65680
  • METIS-141295
  • EWI-3311

Cite this

Broersma, H. J., Kriesell, M., Kriesell, M., & Ryjacek, Z. (1999). On factors of 4-connected claw-free graphs. (Memorandum / Department of Applied Mathematics; No. 1491). Enschede: University of Twente, Department of Applied Mathematics.
Broersma, Haitze J. ; Kriesell, M. ; Kriesell, M. ; Ryjacek, Z. / On factors of 4-connected claw-free graphs. Enschede : University of Twente, Department of Applied Mathematics, 1999. 13 p. (Memorandum / Department of Applied Mathematics; 1491).
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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 Conjecture 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.",
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series = "Memorandum / Department of Applied Mathematics",
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Broersma, HJ, Kriesell, M, Kriesell, M & Ryjacek, Z 1999, On factors of 4-connected claw-free graphs. Memorandum / Department of Applied Mathematics, no. 1491, University of Twente, Department of Applied Mathematics, Enschede.

On factors of 4-connected claw-free graphs. / Broersma, Haitze J.; Kriesell, M.; Kriesell, M.; Ryjacek, Z.

Enschede : University of Twente, Department of Applied Mathematics, 1999. 13 p. (Memorandum / Department of Applied Mathematics; No. 1491).

Research output: Book/ReportReportProfessional

TY - BOOK

T1 - On factors of 4-connected claw-free graphs

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AU - Kriesell, M.

AU - Kriesell, M.

AU - Ryjacek, Z.

N1 - Imported from MEMORANDA

PY - 1999

Y1 - 1999

N2 - 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 Conjecture 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.

AB - 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 Conjecture 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.

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KW - METIS-141295

KW - EWI-3311

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Broersma HJ, Kriesell M, Kriesell M, Ryjacek Z. On factors of 4-connected claw-free graphs. Enschede: University of Twente, Department of Applied Mathematics, 1999. 13 p. (Memorandum / Department of Applied Mathematics; 1491).