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9-Connected Claw-Free Graphs Are Hamilton-Connected

1999
*
Journal of combinatorial theory. Series B (Print)
*

*claw*-

*free*

*graphs*

*are*

*Hamilton*-

*connected*. ... The main result of this note verifies the conjecture for k=

*9*. Theorem 3. Every

*9*-

*connected*

*claw*-

*free*

*graph*is

*Hamilton*-

*connected*. ...

##
###
Hamiltonicity in claw-free graphs

1991
*
Journal of combinatorial theory. Series B (Print)
*

We strengthen this result for CN-

doi:10.1016/0095-8956(91)90074-t
fatcat:t6g6t3xbpnfgvhnxmbwxq6zcca
*free**graphs*of higher*connectivity*. For example, we show 3-*connected*CN-*free**graphs**are*both*Hamilton**connected*and pancyclic. ... A*graph*is*claw*-*free*if it contains no induced subgraph isomorphic to a K"3. This paper studies*hamiltonicity*in two subclasses of*claw*-*free**graphs*. ... We show that these*graphs**are**Hamilton**connected*. ...##
###
How Many Conjectures Can You Stand? A Survey

2011
*
Graphs and Combinatorics
*

We survey results and open problems in hamiltonian

doi:10.1007/s00373-011-1090-6
fatcat:jj7oxvyx5vhrzdjtv2qyj7nysy
*graph*theory centered around two conjectures of the 1980s that*are*still open: every 4-*connected**claw*-*free**graph*(line*graph*) is hamiltonian. ... Conjecture 23 Every 4-*connected**claw*-*free**graph*is*Hamilton*-*connected*. ... Conjecture 25 Every 4-*connected**claw*-*free**graph*is 1-*Hamilton*-*connected*. ...##
###
Closure Concepts: A Survey

2000
*
Graphs and Combinatorics
*

As a consequence, for any

doi:10.1007/s003730050002
fatcat:6qncwca2bvhtvh7oqfvoskidhq
*graph*G with cl n G K n (and n 3), a*Hamilton*cycle can be found in polynomial time, whereas this problem is NP-hard for general*graphs*. ... The ®rst and now well-known result of this type was established by Bondy and Chva Âtal in a paper published in 1976: If u and v*are*two nonadjacent vertices with degree sum n in a*graph*G on n vertices ... Every*9*-*connected**claw*-*free**graph*is*Hamilton*-*connected*. This result motivates the following question. ...##
###
Forbidden triples for hamiltonicity

2002
*
Discrete Mathematics
*

In this paper we characterize all triples of

doi:10.1016/s0012-365x(01)00326-0
fatcat:siajq7tfwvenpgdbsqgeergnai
*connected**graphs*C; X; Y (where C is a*claw*) and such that every 2-*connected*CXY -*free**graph*G is hamiltonian.This result together with a previous result by ... Faudree, Gould, Jacobson, and Lesniak give a full characterization of triples of forbidden subgraphs implying*hamiltonicity*of 2-*connected**graphs*. ... The*graphs*P 3; 3; 3 and P T; T; T*are*2-*connected*, non-hamiltonian and*claw*-*free*and thus the set of CXY -*free**graphs*contains neither P 3; 3; 3 nor P T; T; T . ...##
###
Extremal problems on the Hamiltonicity of claw-free graphs

2018
*
Discrete Mathematics
*

Similar results for the traceability of

doi:10.1016/j.disc.2018.06.023
fatcat:ywmjhwndlzacnopugbh2gk64qq
*connected**claw*-*free**graphs**are*also obtained. ... Our tools include Ryjáček's*claw*-*free*closure theory and Brousek's characterization of minimal 2-*connected**claw*-*free*non-Hamiltonian*graphs*. ... We prove the following lemma on*Hamiltonicity*of closed*claw*-*free**graphs*. Lemma 1. Let G be a 2-*connected*closed*claw*-*free**graph*on n vertices. ...##
###
On some intriguing problems in hamiltonian graph theory—a survey

2002
*
Discrete Mathematics
*

We survey results and open problems in hamiltonian

doi:10.1016/s0012-365x(01)00325-9
fatcat:wcggfjuntbgrjmegqldzdz55hy
*graph*theory centered around three themes: regular*graphs*, t-tough*graphs*, and*claw*-*free**graphs*. ... Then; the following statements*are*equivalent: (1) Every k-*connected**claw*-*free**graph*is hamiltonian. (2) Every k-*connected**claw*-*free**graph*has an f(n)-path-factor. (3) Every k-*connected**claw*-*free**graph*... Presently, the best su cient minimum degree condition for*hamiltonicity*of 3-*connected**claw*-*free**graphs*we*are*aware of is due to Favaron and Fraisse [31] ; using the*claw*-*free*closure and a relationship ...##
###
Forbidden subgraphs, stability and hamiltonicity

1999
*
Discrete Mathematics
*

We characterize all

doi:10.1016/s0012-365x(99)90053-5
fatcat:s7toibwmkbf7to5t6g3ivcmo6e
*connected**graph*.,; A such that the class of all CA-*free**graphs*(where C denotes the*claw*) is stable. ... Using this result, we prove that every 2-*connected*and CHPs-*free*, CHZs-*free*or CHNt.l.4-*free**graph*is either hamiltonian or belongs to some classes of exceptional*graphs*(all of them having"*connectivity*... Ever), non-*Hamilton*/an 2-*connected**claw*-*free**graph*contains an induced subyraph F E ,~. ...##
###
Forbidden subgraphs, stability and hamiltonicity

1999
*
Discrete Mathematics
*

We characterize all

doi:10.1016/s0012-365x(98)00229-5
fatcat:etu2wmh74randnlzfpveua3frq
*connected**graph*.,; A such that the class of all CA-*free**graphs*(where C denotes the*claw*) is stable. ... Using this result, we prove that every 2-*connected*and CHPs-*free*, CHZs-*free*or CHNt.l.4-*free**graph*is either hamiltonian or belongs to some classes of exceptional*graphs*(all of them having"*connectivity*... Ever), non-*Hamilton*/an 2-*connected**claw*-*free**graph*contains an induced subyraph F E ,~. ...##
###
Spectral radius and traceability of connected claw-free graphs

2016
*
Filomat
*

Let G be a

doi:10.2298/fil1609445n
fatcat:fhyupk5dxneydivvtepv7dyqa4
*connected**claw*-*free**graph*on n vertices and G be its complement*graph*. Let μ(G) be the spectral radius of G. ... Our works*are*counterparts on*claw*-*free**graphs*of previous theorems due to Lu et al., and Fiedler and Nikiforov, respectively. ... There*are*examples of 3-*connected*non-Hamiltonian*claw*-*free*(even line)*graphs*, but it is a long-standing conjecture that all 4-*connected**claw*-*free**graphs**are*Hamiltonian (and then, traceable). ...##
###
Line graphs of multigraphs and Hamilton-connectedness of claw-free graphs

2010
*
Journal of Graph Theory
*

) is equivalent with the statement that every 4-

doi:10.1002/jgt.20498
fatcat:wehpsrgwebaalezmeqvmuqplpq
*connected**claw*-*free**graph*is*Hamilton*-*connected*. ... As an application, we show that every 7-*connected**claw*-*free**graph*is*Hamilton*-*connected*, and we show that the wellknown conjecture by Matthews and Sumner (every 4-*connected*clawfree*graph*is hamiltonian ...*claw*-*free**graphs*(there The existence of a*connectivity*bound for*Hamilton*-*connectedness*in*claw*-*free**graphs*was established by Brandt [5] who proved that every 9connected*claw*-*free**graph*is*Hamilton*-*connected*...##
###
On implicit heavy subgraphs and Hamiltonicity of 2-connected graphs

2018
*
Discussiones Mathematicae Graph Theory
*

A

doi:10.7151/dmgt.2170
fatcat:pcpeongzezc7lg37tyazctgdam
*graph*G of order n is implicit*claw*-heavy if in every induced copy of K 1,3 in G there*are*two non-adjacent vertices with sum of their implicit degrees at least n. ... Math. 219 (2017) 126-131] and complete the characterizations of pairs of o-heavy and f-heavy subgraphs for*hamiltonicity*of 2-*connected**graphs*. ... If G is {P 7 , D}-*free*or {P 7 , H}-*free*, then G is hamiltonian. Corollary 8 (Ning [29] ). Let G be a 2-*connected*,*claw*-f-heavy*graph*. ...##
###
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

doi:10.1090/amsip/039/09
fatcat:pxde4ptk7jfptarou6gs2f5iqi
*graphs*and the related hamiltonian*claw*-*free**graphs*. ... Definitions and Terminology*Graphs*considered here*are*finite and loopless. Unless otherwise noted, we follow [2] for notations and terms. ... On the other hand, we shall check that each G(k) is*claw*-*free*3-*connected*(hence 3-edge-*connected*) and N 2 -locally*connected*. Since line*graphs**are**claw*-*free*, G(k) must also be*claw*-*free*. ...##
###
Hamilton Circuits and Essential Girth of Claw Free Graphs

2015
*
Graphs and Combinatorics
*

In this paper, we prove that every 2-

doi:10.1007/s00373-015-1559-9
fatcat:ariofyhwzve3jpvg6pkmuligxu
*connected*{K 1,3 , N 1,1,g e (G)−4 }-*free*(and {K 1,3 , N 1,0,g e (G)−3 }-*free*)*graph*G contains a*Hamilton*circuit. ... Abstract Let G be a K 1,3 -*free**graph*. A circuit of G is essential if it contains a non-locally*connected*vertex v and passes through both components of N (v). ... Here is the main theorem of this paper. (2) If a 2-*connected**graph*G is*claw*-*free*and N 1,0,g e (G)−3 -*free*, then G contains a*Hamilton*circuit (see Fig. 2 ). ...##
###
Hamiltonicity of 3-connected quasi-claw-free graphs

2003
*
Discrete Mathematics
*

A

doi:10.1016/s0012-365x(02)00882-8
fatcat:kkleoznz5vgzze7fyqcrmp5kr4
*graph*G is called quasi-*claw*-*free*if it satisÿes the property: . Let G be a 3-*connected*quasi-*claw*-*free**graph*of order n. If (G) ¿ (n + 5)=5, then G is hamiltonian. ... If H is not*hamilton*-*connected*, then |V (C)| ¿ 4 (G) − 5. Proof of Theorem 2 Proof of Theorem 2. Let G be a 3-*connected*quasi-*claw*-*free*nonhamiltonian*graph*satisfying the conditions in Theorem 2. ... Ainouche [1] extended a variety of results, in particular, hamiltonian results on*claw*-*free**graphs*to quasi-*claw*-*free**graphs*. ...
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