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9-Connected Claw-Free Graphs Are Hamilton-Connected
1999
Journal of combinatorial theory. Series B (Print)
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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. ...
doi:10.1006/jctb.1998.1871
fatcat:shaho73nlnha3ghkes3in4ribe
Hamiltonicity in claw-free graphs
1991
Journal of combinatorial theory. Series B (Print)
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We strengthen this result for CN-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. ...
doi:10.1016/0095-8956(91)90074-t
fatcat:t6g6t3xbpnfgvhnxmbwxq6zcca
How Many Conjectures Can You Stand? A Survey
2011
Graphs and Combinatorics
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We survey results and open problems in hamiltonian 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. ...
doi:10.1007/s00373-011-1090-6
fatcat:jj7oxvyx5vhrzdjtv2qyj7nysy
Closure Concepts: A Survey
2000
Graphs and Combinatorics
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As a consequence, for any 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. ...
doi:10.1007/s003730050002
fatcat:6qncwca2bvhtvh7oqfvoskidhq
Forbidden triples for hamiltonicity
2002
Discrete Mathematics
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In this paper we characterize all triples of 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 . ...
doi:10.1016/s0012-365x(01)00326-0
fatcat:siajq7tfwvenpgdbsqgeergnai
Extremal problems on the Hamiltonicity of claw-free graphs
2018
Discrete Mathematics
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2016 pre-print arXiv:1504.04195v2 fatcat:prjfwsecnjflla7agonfxyxb7eOther Versions
Similar results for the traceability of 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. ...
doi:10.1016/j.disc.2018.06.023
fatcat:ywmjhwndlzacnopugbh2gk64qq
On some intriguing problems in hamiltonian graph theory—a survey
2002
Discrete Mathematics
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We survey results and open problems in hamiltonian 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 ...
doi:10.1016/s0012-365x(01)00325-9
fatcat:wcggfjuntbgrjmegqldzdz55hy
Forbidden subgraphs, stability and hamiltonicity
1999
Discrete Mathematics
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We characterize all 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 ,~. ...
doi:10.1016/s0012-365x(99)90053-5
fatcat:s7toibwmkbf7to5t6g3ivcmo6e
Forbidden subgraphs, stability and hamiltonicity
1999
Discrete Mathematics
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We characterize all 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 ,~. ...
doi:10.1016/s0012-365x(98)00229-5
fatcat:etu2wmh74randnlzfpveua3frq
Spectral radius and traceability of connected claw-free graphs
2016
Filomat
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Let G be a 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). ...
doi:10.2298/fil1609445n
fatcat:fhyupk5dxneydivvtepv7dyqa4
Line graphs of multigraphs and Hamilton-connectedness of claw-free graphs
2010
Journal of Graph Theory
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) is equivalent with the statement that every 4-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 ...
doi:10.1002/jgt.20498
fatcat:wehpsrgwebaalezmeqvmuqplpq
On implicit heavy subgraphs and Hamiltonicity of 2-connected graphs
2018
Discussiones Mathematicae Graph Theory
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A 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. ...
doi:10.7151/dmgt.2170
fatcat:pcpeongzezc7lg37tyazctgdam
Some problems related to hamiltonian line graphs
[chapter]
2007
Proceedings of the International Conference on Complex Geometry and Related Fields
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Part of this paper summarizes some of the recent developments in the study of hamiltonian line 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. ...
doi:10.1090/amsip/039/09
fatcat:pxde4ptk7jfptarou6gs2f5iqi
Hamilton Circuits and Essential Girth of Claw Free Graphs
2015
Graphs and Combinatorics
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In this paper, we prove that every 2-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 ). ...
doi:10.1007/s00373-015-1559-9
fatcat:ariofyhwzve3jpvg6pkmuligxu
Hamiltonicity of 3-connected quasi-claw-free graphs
2003
Discrete Mathematics
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A 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. ...
doi:10.1016/s0012-365x(02)00882-8
fatcat:kkleoznz5vgzze7fyqcrmp5kr4
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