Hamiltonian results in K 1,3 ‐free graphs

Journal of Graph Theory

Kriesell

Ryjáček

2001

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.

“…Alternatively, as shown in [10,Lemma 2], I(G) is the line graph of the subdivision graph S(G), i.e. the graph obtained from G by subdividing each edge of G once.…”

Section: Hourglass-free Graphsmentioning

confidence: 99%

On factors of 4‐connected claw‐free graphs*

Journal of Graph Theory

Kriesell

Ryjáček

2001

“…Enomoto et al [22] proved that every 2-tough graph contains a 2-factor. Since 2k-connected claw-free graphs are k-tough by a result in [50], this implies the following.…”

Section: Related Results With a Weaker Conclusionmentioning

confidence: 75%

“…Most of the results in this survey paper are inspired by the following two conjectures that were tossed in the 1980s, and later appeared in the cited papers. The first conjecture is due to Matthews and Sumner [50].…”

Section: Introductionmentioning

confidence: 99%

How Many Conjectures Can You Stand? A Survey

Graphs and Combinatorics

Ryjáček

Vrána

2011

“…Hence the smallest number of components in a 2-factor can be seen as a measure for how close a graph is to being hamiltonian. This relates to the wellknown conjecture of Matthews and Sumner [15] stating that every 4-connected clawfree graph is hamiltonian. Little progress has been made on settling this conjecture, but it is easy to construct nonhamiltonian 3-connected claw-free graphs.…”

Section: Introductionmentioning

confidence: 62%

“…It is easy to verify that an analogous result does not hold for general graphs, not even with an arbitrarily high constant lower bound on the minimum degree or connectivity. We observe that the above theorem gives a solution to a weaker form of the conjecture of Matthews and Sumner [15] that every 4-connected claw-free graph is hamiltonian: every 4-connected claw-free graph has minimum degree at least four, and hence has a 2-factor. The connectivity condition can be relaxed, however.…”

Section: Theorem 1 ([2 4])mentioning

confidence: 84%

Sharp Upper Bounds on the Minimum Number of Components of 2-factors in Claw-free Graphs

Graphs and Combinatorics

Paulusma

Yoshimoto

2009

(2009) 'Sharp upper bounds for the minimum number of components of 2-factors in claw-free graphs. ', Graphs and combinatorics., 25 (4). pp. 427-460. Further information on publisher's website:http://dx.doi.org/10.1007/s00373-009-0855-7Publisher's copyright statement:The nal publication is available at Springer via http://dx.doi.org/10.1007/s00373-009-0855-7Additional information: Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-pro t purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. Abstract. Let G be a claw-free graph with order n and minimum degree δ. We improve results of Faudree et al. and Gould & Jacobson, and solve two open problems by proving the following two results. If δ = 4, then G has a 2-factor with at most (5n − 14)/18 components, unless G belongs to a finite class of exceptional graphs. If δ ≥ 5, then G has a 2-factor with at most (n − 3)/(δ − 1) components, unless G is a complete graph. These bounds are best possible in the sense that we cannot replace 5/18 by a smaller quotient and we cannot replace δ − 1 by δ, respectively.