Reflections on graph theory

Thomassen, Carsten

doi:10.1002/jgt.3190100308

Cited by 141 publications

(57 citation statements)

References 52 publications

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“…Then G = L(S(H)) for some essentially 4-edge connected subdivision S(H) of a 4-edge connected graph H. As shown in [13], using the result of Kundu [8] that H has two edge-disjoint spanning trees, it is easy to show that H contains a spanning circuit, hence S(H) contains a dominating circuit. By a result of Harary and…”

Section: Lemma 5 Every 4-connected Inflation Is Hamiltonianmentioning

confidence: 99%

On factors of 4‐connected claw‐free graphs*

Broersma

Kriesell

Ryjáček

2001

Journal of Graph Theory

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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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Section: Lemma 5 Every 4-connected Inflation Is Hamiltonianmentioning

confidence: 99%

On factors of 4‐connected claw‐free graphs*

Broersma

Kriesell

Ryjáček

2001

Journal of Graph Theory

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show abstract

“…Two famous conjectures in the field of hamiltonicity of graphs are Thomassen's conjecture [10] that every 4-connected line graph is hamiltonian and Matthews and Sumner's conjecture [8] that every 4-connected claw-free graph is hamiltonian. Ryjáček proved these two conjectures are equivalent.…”

Section: Some Preliminariesmentioning

confidence: 99%