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Hamiltonian cycles in 3-connected Claw-free graphs

Guojun Li, M. Lu, Z. Liu
2002 Discrete Mathematics  
It is shown that every 3-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.  ... 
doi:10.1016/s0012-365x(01)00279-5 fatcat:talyv3a32fdk3ny5pkdfsllcda

Minimal 2-connected non-hamiltonian claw-free graphs

Jan Brousek
1998 Discrete Mathematics  
In this paper we characterize all minimal 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.  ... 
doi:10.1016/s0012-365x(98)00093-4 fatcat:wbglubng4rce5kqepbujcwjoaa

All 4-connected Line Graphs of Claw Free Graphs Are Hamiltonian Connected

Matthias Kriesell
2001 Journal of combinatorial theory. Series B (Print)  
Thomassen conjectured that every 4-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.  ... 
doi:10.1006/jctb.2001.2040 fatcat:tb27jxzje5e25gqy4zrumr3tt4

Every 3-connected claw-free Z8-free graph is Hamiltonian

Hong-Jian Lai, Liming Xiong, Huiya Yan, Jin Yan
2009 Journal of Graph Theory  
Every 3-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.  ... 
doi:10.1002/jgt.20433 fatcat:kbp76bqhdjbq3hue2irracwdoe

Hamiltonian cycles in regular 3-connected claw-free graphs

Mingchu Li
1996 Discrete Mathematics  
A known result by the author in 1991 is that every 3-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.  ... 
doi:10.1016/0012-365x(94)00348-m fatcat:wsxpv4rrxzcbdfedkrwmj5g2pq

Every 4-connected line graph of a quasi claw-free graph is hamiltonian connected

Hong-Jian Lai, Yehong Shao, Mingquan Zhan
2008 Discrete Mathematics  
In 1986, Thomassen conjectured that every 4-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).  ... 
doi:10.1016/j.disc.2007.09.045 fatcat:fpl57sb6gnamjd6yymrcsgflhy

2-connected Hamiltonian claw-free graphs involving degree sum of adjacent vertices

Tian Tian, Liming Xiong
2018 Discussiones Mathematicae Graph Theory  
We focus on the conditionσ 2 (H) ≥ 2n 5 − 1, and characterize non-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  ... 
doi:10.7151/dmgt.2125 fatcat:s3brpucb7zbhpmlsb7qv6prh6q

Supereulerian graphs with small matching number and 2-connected hamiltonian claw-free graphs

Jinquan Xu, Ping Li, Zhengke Miao, Keke Wang, Hong-Jian Lai
2014 International Journal of Computer Mathematics  
Flandrin and Li in 1989 showed that every 3-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.  ... 
doi:10.1080/00207160.2013.858808 fatcat:rm6vwchzqnblfnxic6ohyhqugy

2-connected claw-free chordal graphs are cycle extendable [article]

Deborah Arangno, David E. Brown
2016 arXiv   pre-print
We prove that any 2-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.  ... 
arXiv:1310.2901v4 fatcat:65rhyhjiafgmnpb7pozb52zkeq

Every $3$-connected, essentially $11$-connected line graph is hamiltonian

Hong-Jian Lai, Yehong Shao, Ju Zhou, Hehui Wu
2005 Discrete Mathematics & Theoretical Computer Science  
Using Ryjáček's line 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  ... 
doi:10.46298/dmtcs.3452 fatcat:am6kqfn4dvhlfgtdom2p7y6ddq

On Hamiltonicity of claw, net-free graphs [article]

Alexander Kelmans
2006 arXiv   pre-print
These results imply that a 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.  ... 
arXiv:math/0607234v1 fatcat:67nbgrb3dfccfgl7hkt655gv5a

The Ryjáček closure and a forbidden subgraph

Akira Saito, Liming Xiong
2016 Discussiones Mathematicae Graph Theory  
The Ryjáček closure is a powerful tool in the study of 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.  ... 
doi:10.7151/dmgt.1876 fatcat:x5mur74vnzc3lpkjjwvsk2rja4

Nonhamiltonian 2-connected claw-free graphs with large 4-degree sum

Wacław Frydrych
2001 Discrete Mathematics  
Let G be a 2-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.  ... 
doi:10.1016/s0012-365x(00)00436-2 fatcat:waq752u2fzdgdfx4pvwm2xehuu

Hamiltonicity of double domination critical claw-free graphs

Pawaton Kaemawichanurat
2018 Discussiones Mathematicae Graph Theory  
We show that the condition 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.  ... 
doi:10.7151/dmgt.2148 fatcat:zcqolhixk5ccdnvhhtg3zrp5dm

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 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.  ... 
doi:10.1090/amsip/039/09 fatcat:pxde4ptk7jfptarou6gs2f5iqi
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