Chords in longest cycles

Thomassen, Carsten

doi:10.1016/j.jctb.2017.09.008

Cited by 8 publications

(4 citation statements)

References 15 publications

(22 reference statements)

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“…In [13], Thomassen proved that every longest cycle in a 2-connected cubic graph has a chord. Our result shows that every longest path between two vertices in a 2-connected cubic has a chord, which generalizes Thomassen's result.…”

Section: Introductionmentioning

confidence: 99%

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Geography of Technology Transfer in China

Liu

2023

East China Normal University Scientific Reports

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Graphs can model complex relationships between objects, enabling a myriad of Web applications such as online page/article classification and social recommendation. While graph neural networks (GNNs) have emerged as a powerful tool for graph representation learning, in an end-to-end supervised setting, their performance heavily relies on a large amount of task-specific supervision. To reduce labeling requirement, the "pre-train, fine-tune" and "pre-train, prompt" paradigms have become increasingly common. In particular, prompting is a popular alternative to fine-tuning in natural language processing, which is designed to narrow the gap between pre-training and downstream objectives in a task-specific manner. However, existing study of prompting on graphs is still limited, lacking a universal treatment to appeal to different downstream tasks. In this paper, we propose GraphPrompt, a novel pre-training and prompting framework on graphs. GraphPrompt not only unifies pre-training and downstream tasks into a common task template, but also employs a learnable prompt to assist a downstream task in locating the most relevant knowledge from the pre-trained model in a task-specific manner. Finally, we conduct extensive experiments on five public datasets to evaluate and analyze GraphPrompt. CCS CONCEPTS• Computing methodologies → Learning latent representations; • Information systems → Data mining.

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Section: Introductionmentioning

confidence: 99%

“…Our result shows that every longest path between two vertices in a 2-connected cubic has a chord, which generalizes Thomassen's result. We have used some ideas of Thomassen [13] in our proofs, but we need to add new ones. Our second result in this paper is the following theorem.…”

Section: Introductionmentioning

confidence: 99%

Geography of Technology Transfer in China

Liu

2023

East China Normal University Scientific Reports

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“…Instead, we make the following conjecture about 2-connected graphs: If k = 1, this conjecture states that any cycle of maximum order in a 2-connected graph with minimum degree at least 3 will contain a chord. A recent result of Thomassen [8] extends the result of [10] and shows that every longest cycle in a 2-connected cubic graph contains a chord. This provides some evidence for the assertion that, in general, 3-connectivity is not required, and that 2-connectivity together with a minimum degree condition may be sufficient.…”

Section: Further Problemsmentioning

confidence: 59%

A Cycle of Maximum Order in a Graph of High Minimum Degree has a Chord

Harvey

2017

Electron. J. Combin.

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A well-known conjecture of Thomassen states that every cycle of maximum order in a $3$-connected graph contains a chord. While many partial results towards this conjecture have been obtained, the conjecture itself remains unsolved. In this paper, we prove a stronger result without a connectivity assumption for graphs of high minimum degree, which shows Thomassen's conjecture holds in that case. This result is within a constant factor of best possible. In the process of proving this, we prove a more general result showing that large minimum degree forces a large difference between the order of the largest cycle and the order of the largest chordless cycle.

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“…The first result in this area was Smith's Theorem (Tutte [11]) which is the instance of Thomason's Theorem when all vertices have degree 3. The second author [9] recently proved a partial generalization of Thomason's Theorem; he showed that in a graph with an odd-degree vertex and in which no two even-degree vertices are adjacent, if there is one cycle containing all the odd-degree vertices, then there is another. The first author [3] extended this to include Thomason's Theorem, proving that in a graph with an odd-degree vertex and in which no two even-degree vertices are adjacent, for any edge e, the number of cycles containing e and all the odd-degree vertices is even (also see [5]).…”

Section: Introductionmentioning

confidence: 99%