Forbidden triples for hamiltonicity

Brousek, Jan

doi:10.1016/s0012-365x(01)00326-0

Cited by 19 publications

(12 citation statements)

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“…As a related work, the relationship between forbidden subgraphs and hamiltonicity has also been studied. To the readers who are interested in this topic, we refer [1,4,5,8].…”

Section: Discussionmentioning

confidence: 99%

“…Let H = {H 1 , H 2 , H 3 }∈H 0 . We assume that H ≤ {K 1,3 }, which means that none of H 1 , H 2 , and H 3 are induced subgraphs of K 1,3 . By the definition of H 0 , there exists an integer n 0 = n 0 (H) such that every H-free connected graph G of even order with |V(G)|>n 0 has a perfect matching.…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…Two of the present authors investigated a sort of converse of the above theorem, and considered what kind of a single graph we can forbid to force the existence of a perfect matching. They proved that essentially K 1,3 is the only answer.…”

Section: Introductionmentioning

confidence: 99%

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Forbidden triples for perfect matchings

Ota¹,

Plummer²,

Saito³

2011

J. Graph Theory

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“…As a related work, the relationship between forbidden subgraphs and hamiltonicity has also been studied. To the readers who are interested in this topic, we refer [1,4,5,8].…”

Section: Discussionmentioning

confidence: 99%

Section: Proof Of Theoremmentioning

confidence: 99%

See 1 more Smart Citation

Forbidden triples for perfect matchings

Ota¹,

Plummer²,

Saito³

2011

J. Graph Theory

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“…Note that Brousek [14] gave a complete characterization of triples of connected graphs K 1,3 , X, Y such that a graph G being 2-connected and {K 1,3 , X, Y }-free implies G is hamiltonian. Clearly, if K 1,3 , S, T is a triple such that every 2-connected {K 1,3 , S, T }-heavy graph is hamiltonian, then, for some triple [14], S and T are induced subgraphs of X and Y , respectively (of course, the triples of Theorems 6.8 and 6.9 have this property).…”

Section: (Deer)mentioning

confidence: 99%

“…Clearly, if K 1,3 , S, T is a triple such that every 2-connected {K 1,3 , S, T }-heavy graph is hamiltonian, then, for some triple [14], S and T are induced subgraphs of X and Y , respectively (of course, the triples of Theorems 6.8 and 6.9 have this property). We refer the interested reader to [14] for more details.…”

Section: (Deer)mentioning

confidence: 99%

Subgraph conditions for Hamiltonian properties of graphs

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Advances on the Hamiltonian Problem - A Survey

Gould

2003

Graphs and Combinatorics

184

110

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This article is intended as a survey, updating earlier surveys in the area. For completeness of the presentation of both particular questions and the general area, it also contains material on closely related topics such as traceable, pancyclic and hamiltonian-connected graphs and digraphs. Corollary 3 [4] Let G be a graph of order n with δ(G) ≥ (2n − 1)/3 and suppose n ≥ n 1 + n 2 +. .. + n k where n i ≥ 3 for all i. Then G contains the vertex disjoint union of the cycles C n 1 ∪ C n 2 ∪. .. ∪ C n k. Clearly then, any such graph contains any 2-factor we would want and hence provides a strong analogue to Dirac's theorem. I should mention that Alon and Fischer [6] also provided a solution to the Sauer-Spencer conjecture (δ = 2n/3). Their result used work dependent on the regularity lemma and thus holds only for large graphs. Related to the last result is another old conjecture due to El-Zahar [95]. Conjecture 2 Let G be a graph of order n = n 1 + n 2 +. .. + n k with δ(G) ≥ k i=1 ⌈n i /2⌉, then G contains the 2-factor C n 1 ∪. .. ∪ C n k. Note that the graph K s−1 + K ⌈ n−s+1 2 ⌉,⌈ n−s+1 2 ⌉ has minimum degree (n + s − 1)/2 but contains no s vertex disjoint odd length cycles. Thus, the conjecture is best possible. El-Zahar [95] provided an affirmative answer to the case k = 2, while Dirac's Theorem handles k = 1. Recently, Abbasi [1] announced a solution for large n using the regularity lemma. It would still be interesting to find a solution to this beautiful conjecture for all n. It should be noted that Corrádi and Hajnal [87] provided an affirmative answer to the El-Zahar conjecture for the case that each n i = 3 An old conjecture of Erdös and Faudree [102] generalizes the Corrádi-Hajnal theorem in another direction. Conjecture 3 Let G be a graph with order n = 4k and δ(G) ≥ 2k, then G contains k vertex disjoint 4-cycles. Alon and Yuster [8] proved that for any ǫ > 0, there exists k 0 such that if G has order 4k and δ(G) ≥ (2 + ǫ)k with k ≥ k 0 , then G contains k disjoint 4-cycles. In [207], a near solution was provided by Randerath, Schiermeyer and Wang.

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Forbidden triples for hamiltonicity

Cited by 19 publications

References 2 publications

Forbidden triples for perfect matchings

Forbidden triples for perfect matchings

Subgraph conditions for Hamiltonian properties of graphs

Advances on the Hamiltonian Problem - A Survey

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