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Hamiltonian cycles in 3-connected Claw-free graphs

2002
*
Discrete Mathematics
*

It is shown that every 3-

doi:10.1016/s0012-365x(01)00279-5
fatcat:talyv3a32fdk3ny5pkdfsllcda
*connected**claw*-*free**graph*having at most 6 − 7 vertices is*hamiltonian*, where is the minimum degree. ... Let G be a 3-*connected**claw*-*free**graph*of order n. If n 6 3 then G is*hamiltonian**connected*. Theorem 6 (Li [4] ). Let G be a 3-*connected**claw*-*free**graph*on n vertices with n 6 5 − 5. ... Let G be a 3-*connected**claw*-*free**graph*on n vertices with n 6 6 − 7; then G is*hamiltonian*. ...##
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Minimal 2-connected non-hamiltonian claw-free graphs

1998
*
Discrete Mathematics
*

In this paper we characterize all minimal

doi:10.1016/s0012-365x(98)00093-4
fatcat:wbglubng4rce5kqepbujcwjoaa
*graphs*with respect to the property 'to be 2-*connected*, non-*hamiltonian*and*claw*-*free*'. ... We say a*graph*G is minimal with respect to a property Q if there exists no proper induced subgraph G' of G with property Q. ... Every 2-*connected*non-*hamiltonian**claw*-*free**graph*contains a*graph*from ~ as an induced subgraph. Theorem 2B. Every 2-*connected*C, ~ -*free**graph*is*hamiltonian*. Proof. ...##
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All 4-connected Line Graphs of Claw Free Graphs Are Hamiltonian Connected

2001
*
Journal of combinatorial theory. Series B (Print)
*

Thomassen conjectured that every 4-

doi:10.1006/jctb.2001.2040
fatcat:tb27jxzje5e25gqy4zrumr3tt4
*connected*line*graph*is*hamiltonian*. Here we shall see that 4-*connected*line*graphs*of*claw**free**graphs*are*hamiltonian**connected*. Academic Press ... From this it follows that all 4-*connected*line*graphs*of*claw**free**graphs*are*hamiltonian**connected*. ... Since every*claw**free**graph*satisfies the conditions of Corollary 2, we obtain the following Corollary. Corollary 2. Every 4-*connected*line*graph*of a*claw**free**graph*is*hamiltonian**connected*. ...##
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Every 3-connected claw-free Z8-free graph is Hamiltonian

2009
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Journal of Graph Theory
*

Every 3-

doi:10.1002/jgt.20433
fatcat:kbp76bqhdjbq3hue2irracwdoe
*connected*{K 1,3 , P 11 }-*free**graph*is*Hamiltonian*. A*graph*is called Eulerian if it is*connected*and every vertex has an even degree. ... In this article, we first show that every 3-edge-*connected**graph*with circumference at most 8 is supereulerian, which is then applied to show that a 3-*connected**claw*-*free**graph*without Z 8 as an induced ... Let G be a 3-*connected*simple*claw*-*free**graph*. If G is Z 8 -*free*, then G is*Hamiltonian*. At the end of Section 4, we give an example of a 3-*connected**claw*-*free*non-*Hamiltonian*Z 9 -*free**graph*. ...##
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Hamiltonian cycles in regular 3-connected claw-free graphs

1996
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Discrete Mathematics
*

A known result by the author in 1991 is that every 3-

doi:10.1016/0012-365x(94)00348-m
fatcat:wsxpv4rrxzcbdfedkrwmj5g2pq
*connected**claw*-*free**graph*on at most 66-11 vertices is*Hamiltonian*. ... In this paper it is proved that every 3-*connected*k-regular*claw*-*free**graph*on at most 7k -19 vertices is*Hamiltonian*. ... Every 3-*connected*k-regular*claw*-*free**graph*on at most 10k-1 vertices is*Hamiltonian*. ...##
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Every 4-connected line graph of a quasi claw-free graph is hamiltonian connected

2008
*
Discrete Mathematics
*

In 1986, Thomassen conjectured that every 4-

doi:10.1016/j.disc.2007.09.045
fatcat:fpl57sb6gnamjd6yymrcsgflhy
*connected*line*graph*is*hamiltonian*. In this paper we show that every 4-*connected*line*graph*of a quasi*claw*-*free**graph*is*hamiltonian**connected*. ...*graph*of a*claw*-*free**graph*is*hamiltonian**connected*[6] . ... Every 4-*connected*line*graph*of a quasi*claw*-*free**graph*is*hamiltonian**connected*. Preliminaries A subgraph H of a*graph*G is dominating if G − V (H ) is edgeless. Let v 0 , v k ∈ V (G). ...##
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2-connected Hamiltonian claw-free graphs involving degree sum of adjacent vertices

2018
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Discussiones Mathematicae Graph Theory
*

We focus on the conditionσ 2 (H) ≥ 2n 5 − 1, and characterize non-

doi:10.7151/dmgt.2125
fatcat:s3brpucb7zbhpmlsb7qv6prh6q
*Hamiltonian*2-*connected**claw*-*free**graphs*H of order n sufficiently large withσ 2 (H) ≥ 2n 5 − 1. ... For a*graph*H, defineσ 2 (H) = min{d(u) + d(v)| uv ∈ E(H)}. Let H be a 2-*connected**claw*-*free*simple*graph*of order n with δ(H) ≥ 3. In [J. ... .: 201706030019). 2-*Connected**Hamiltonian**Claw*-*Free**Graphs*Involving Degree ... 21 ...##
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Supereulerian graphs with small matching number and 2-connected hamiltonian claw-free graphs

2014
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International Journal of Computer Mathematics
*

Flandrin and Li in 1989 showed that every 3-

doi:10.1080/00207160.2013.858808
fatcat:rm6vwchzqnblfnxic6ohyhqugy
*connected**claw*-*free**graph*G with α(G) ≤ 2κ(G) is*hamiltonian*. ... Our characterization is also applied to show that every 2-*connected**claw*-*free**graph*G with α(G) ≤ 3 is*hamiltonian*, with only one well-characterized exceptional class. ... Theorem 1.3 has an application to*hamiltonian*line*graphs*and*hamiltonian**claw*-*free**graphs*. ...##
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2-connected claw-free chordal graphs are cycle extendable
[article]

2016
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arXiv
*
pre-print

We prove that any 2-

arXiv:1310.2901v4
fatcat:65rhyhjiafgmnpb7pozb52zkeq
*connected**claw*-*free*chordal*graph*is cycle extendable. ... In 1990 Hendry conjectured that any*Hamiltonian*chordal*graph*(a*Hamiltonian**graph*with no induced cycle of length greater than three) is cycle extendable, and this conjecture has been verified for*Hamiltonian*... And also a result of Oberly and Sumner [10] : Lemma 2.2 Any 2-*connected*locally*connected**graph*which is*claw*-*free*is*Hamiltonian*. ...##
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Every $3$-connected, essentially $11$-connected line graph is hamiltonian

2005
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Discrete Mathematics & Theoretical Computer Science
*

Using Ryjáček's line

doi:10.46298/dmtcs.3452
fatcat:am6kqfn4dvhlfgtdom2p7y6ddq
*graph*closure, it follows that every $3$-*connected*, essentially $11$-*connected**claw*-*free**graph*is*hamiltonian*. ... We prove that every $3$-*connected*, essentially $11$-*connected*line*graph*is*hamiltonian*. ... Ryjácek [7] introduced the line*graph*closure of a*claw*-*free**graph*and used it to show that a*claw*-*free**graph*G is*hamiltonian*if and only if it closure cl(G) is*hamiltonian*, where cl(G) is a line*graph*...##
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On Hamiltonicity of claw, net-free graphs
[article]

2006
*
arXiv
*
pre-print

These results imply that a

arXiv:math/0607234v1
fatcat:67nbgrb3dfccfgl7hkt655gv5a
*connected**claw*, net-*free**graph*has a*Hamiltonian*path and a 2-*connected**claw*, net-*free**graph*has a*Hamiltonian*cycle [D. Duffus, R.J. Gould, M.S. ... Keywords:*graph*,*claw*, net,*claw*, net-*free**graph*,*Hamiltonian*path,*Hamiltonian*cycle, polynomial-time algorithm. ... [3] (Corollary of 4.9) Every 2-*connected*{*claw*, net}-*free**graph*has a*Hamiltonian*cycle. ...##
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The Ryjáček closure and a forbidden subgraph

2016
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Discussiones Mathematicae Graph Theory
*

The Ryjáček closure is a powerful tool in the study of

doi:10.7151/dmgt.1876
fatcat:x5mur74vnzc3lpkjjwvsk2rja4
*Hamiltonian*properties of*claw*-*free**graphs*. ... Because of its usefulness, we may hope to use it in the classes of*graphs*defined by another forbidden subgraph. ... On the other hand, the assumption of local*connectedness*is affected. While not every*connected**claw*-*free**graph*is*Hamiltonian*, every*connected*clawfree*graph*G satisfies def(G) ≤ 1. ...##
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Nonhamiltonian 2-connected claw-free graphs with large 4-degree sum

2001
*
Discrete Mathematics
*

Let G be a 2-

doi:10.1016/s0012-365x(00)00436-2
fatcat:waq752u2fzdgdfx4pvwm2xehuu
*connected**claw*-*free**graph*on n vertices. Let k (G) be the minimum degree sum among k-element independent set of vertices in G. ... Moreover, it is shown that the problem*HAMILTONIAN*CYCLE restricted to*claw*-*free**graphs*G = (V; E) with 3(G)¿ 3 4 (|G| + 3) has polynomial time complexity. ... Let G be a k-*connected**claw*-*free*n-vertex*graph*. If k+1 (G)¿n − k then G is*hamiltonian*. Theorem 9 (Wu [13] and Flandrin and Li [6] ). Let G be a 3-*connected**claw*-*free*n-vertex*graph*. ...##
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Hamiltonicity of double domination critical claw-free graphs

2018
*
Discussiones Mathematicae Graph Theory
*

We show that the condition

doi:10.7151/dmgt.2148
fatcat:zcqolhixk5ccdnvhhtg3zrp5dm
*claw*-*free*when k = 4 is best possible. We further show that every 3-*connected*k-γ ×2 -critical*claw*-*free**graph*is*Hamiltonian*when 2 ≤ k ≤ 7. ... In this paper, for k ≥ 4, we provide a 2-*connected*k-γ ×2 -critical*graph*which is non-*Hamiltonian*. We prove that all 2-*connected*k-γ ×2 -critical*claw*-*free**graphs*are*Hamiltonian*when 2 ≤ k ≤ 5. ... We prove that 2-*connected*k-γ + ×2 -stable*claw*-*free**graphs*are*Hamiltonian*when 2 ≤ k ≤ 3. We also prove that 3-*connected*k-γ + ×2 -stable*claw*-*free**graphs*are*Hamiltonian*when 2 ≤ k ≤ 5. ...##
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Some problems related to hamiltonian line graphs
[chapter]

2007
*
Proceedings of the International Conference on Complex Geometry and Related Fields
*

Part of this paper summarizes some of the recent developments in the study of

doi:10.1090/amsip/039/09
fatcat:pxde4ptk7jfptarou6gs2f5iqi
*hamiltonian*line*graphs*and the related*hamiltonian**claw*-*free**graphs*. ... As in [2], κ(G), κ (G) and δ(G) represent the*connectivity*, edge-*connectivity*, and the minimum degree of a*graph*G, respectively. Definition 1.2. ... Sumner [25] ) Every*connected*, locally*connected**claw*-*free**graph*is*hamiltonian*. Definition 2.3. ...
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