Degree conditions on induced claws

Ada, Roman

doi:10.1016/j.disc.2007.10.026

Cited by 14 publications

(17 citation statements)

References 12 publications

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“…The answer is positive. Before proving this result, we first introduce the closure theory of claw-heavy graphs proposed by Čada [6].…”

Section: Remarksmentioning

confidence: 97%

Forbidden subgraphs for longest cycles to contain vertices with large degrees

Zhang

2015

Discrete Mathematics

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a b s t r a c tLet G be a graph. For a given graph H, we say that G is H-free if G contains no copies of H as an induced subgraph. Suppose that G is 2-connected, has n vertices, and α is a real number with 0 ≤ α ≤ 1. In this paper, we characterize the connected graphs R such that G being R-free implies that every longest cycle of G passes through all vertices with degree at least αn + O(1) in G.

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“…The answer is positive. Before proving this result, we first introduce the closure theory of claw-heavy graphs proposed by Čada [6].…”

Section: Remarksmentioning

confidence: 97%

Forbidden subgraphs for longest cycles to contain vertices with large degrees

Zhang

2015

Discrete Mathematics

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“…In this section, we will introduce some preliminaries for the proof of the 'if' part of Theorem 9, which are similar to the ones in [12]. We first introduceČada's closure theory of claw-oheavy graphs [8], which is an extension of the closure theory of claw-free graphs invented by Ryjáček [14].…”

Section: Preliminariesmentioning

confidence: 99%

Degree conditions restricted to induced paths for hamiltonicity of claw-heavy graphs

Ning

Zhang

2016

Acta. Math. Sin.-English Ser.

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Broersma and Veldman proved that every 2-connected claw-free and P 6 -free graph is hamiltonian. Chen et al. extended this result by proving every 2-connected clawheavy and P 6 -free graph is hamiltonian. On the other hand, Li et al. constructed a class of 2-connected graphs which are claw-heavy and P 6 -o-heavy but not hamiltonian.In this paper we further give some Ore-type degree conditions restricting to induced P 6 s of a 2-connected claw-heavy graph that can guarantee the graph to be hamiltonian. This improves some previous related results.

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“…Early examples of this method used in scientific papers can date back to 1990s [2,19,5]. In particular, Cada [10] introduced the class of o-heavy graphs by restricting Ore's condition to every induced claw of a graph. Li et al [18] extendedČada's concept of claw-o-heavy graphs to a general one.…”

Section: Introductionmentioning

confidence: 99%

Heavy subgraphs, stability and Hamiltonicity

Ning²

2017

Discuss. Math. Graph Theory

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Let G be a graph. Adopting the terminology of Broersma et al. andČada, respectively, we say that G is 2-heavy if every induced claw (K 1,3 ) of G contains two end-vertices each one has degree at least |V (G)|/2; and G is o-heavy if every induced claw of G contains two end-vertices with degree sum at least |V (G)| in G. In this paper, we introduce a new concept, and say that G is S-c-heavy if for a given graph S and every induced subgraph G ′ of G isomorphic to S and every maximal clique C of G ′ , every non-trivial component of G ′ − C contains a vertex of degree at least |V (G)|/2 in G. In terms of this concept, our original motivation that a theorem of Hu in 1999 can be stated as every 2-connected 2-heavy and N -c-heavy graph is hamiltonian, where N is the graph obtained from a triangle by adding three disjoint pendant edges. In this paper, we will characterize all connected graphs S such that every 2-connected o-heavy and S-c-heavy graph is hamiltonian. Our work results in a different proof of a stronger version of Hu's theorem. Furthermore, our main result improves or extends several previous results.

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Degree conditions on induced claws

Cited by 14 publications

References 12 publications

Forbidden subgraphs for longest cycles to contain vertices with large degrees

Forbidden subgraphs for longest cycles to contain vertices with large degrees

Degree conditions restricted to induced paths for hamiltonicity of claw-heavy graphs

Heavy subgraphs, stability and Hamiltonicity

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