Chvátal-Erdős conditions for paths and cycles in graphs and digraphs. A survey

Jackson, Bill; Ordaz, Óscar

doi:10.1016/0012-365x(90)90130-a

Cited by 38 publications

(16 citation statements)

References 17 publications

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“…The proof involves a 'reduction' of the problem to the undirected case. As observed by Thomassen and Chakroun (see [36] again), there are non-Hamiltonian digraphs with κ(G) = α 2 (G) = 2 and κ(G) = α 2 (G) = 3. But it could well be that every digraph satisfying κ(G) ≥ α 2 (G) ≥ 4 has a Hamilton cycle.…”

Section: 4supporting

confidence: 54%

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A survey on Hamilton cycles in directed graphs

Kühn

Osthus

2012

European Journal of Combinatorics

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We survey some recent results on long-standing conjectures regarding Hamilton cycles in directed graphs, oriented graphs and tournaments. We also combine some of these to prove the following approximate result towards Kelly's conjecture on Hamilton decompositions of regular tournaments: the edges of every regular tournament can be covered by a set of Hamilton cycles which are 'almost' edge-disjoint. We also highlight the role that the notion of 'robust expansion' plays in several of the proofs. New and old open problems are discussed.

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Section: 4supporting

confidence: 54%

“…However, even the following basic question (already discussed e.g. in [36]) is still open. Question 9.…”

Section: 4mentioning

confidence: 99%

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A survey on Hamilton cycles in directed graphs

Kühn

Osthus

2012

European Journal of Combinatorics

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“…Furthermore, Lou showed the following fact which was conjectured in the survey [9] by Jackson and Ordaz. Theorem 1.3 (Lou [11]).…”

Section: Introduction a The Chvá Tal-erdö S Condition And Cyclesmentioning

confidence: 76%

A 2‐factor with two components of a graph satisfying the Chvátal‐Erdös condition

Kaneko

Yoshimoto

2003

Journal of Graph Theory

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“…A survey was presented by B. Jackson and O. Ordaz [5]. For the length of a longest cycle, J. Fouquet and J. Jolivet made the following conjecture which is still open.…”

Section: Theorem 1 [2]mentioning

confidence: 95%

On the Thomassen's conjecture*

2001

Journal of Graph Theory

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C. Thomassen proposed a conjecture: Let G be a k-connected graph with the stability number ! k, then G has a cycle C containing k independent vertices and all their neighbors. In this paper, we will obtain the following result: Let G be a k-connected graph with stability number ¼ k þ3 and C any longest cycle of G, then C contains k independent vertices and all their neighbors. This solves Thomassen's conjecture for the case ¼ k þ3.

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Chvátal-Erdős conditions for paths and cycles in graphs and digraphs. A survey

Cited by 38 publications

References 17 publications

A survey on Hamilton cycles in directed graphs

A survey on Hamilton cycles in directed graphs

A 2‐factor with two components of a graph satisfying the Chvátal‐Erdös condition

On the Thomassen's conjecture*

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