A note between transitive C_4$$ -factor and oriented Ramsey number (original) (raw)

Abstract

Let \(\overrightarrow{C_{k}}\) be the transitive cycle on k vertices. Motivated by work of Balogh, Lo, and Molla [J. Combin. Theory Ser.B 124:64–87, 2017] showed that every _n_-vertex oriented graph D with \(n\in 3\mathbb {N}\) and \(\delta ^{0}(D)\ge 7n/18\) contains a \(\overrightarrow{C_{3}}\)-factor. We consider the next open subcase, and show that every _n_-vertex oriented graph D with \(n\in 4\mathbb {N}\) and \(\delta (D)\ge 3n/4\) (or \(\delta ^{0}(D)\ge 3n/8\)) contains a \(\overrightarrow{C_4}\)-factor. These two bounds are tight respectively.

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Notes

  1. Different elements in \(\mathcal {T}\) may have different orientations.

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Acknowledgements

In 1986, Thomason (1986) proved that every _n_-vertex tournament with \(n\ge 2^{128}+1\) contains every non-strongly oriented cycle of order n. Given \(\ell \ge 2^{128}+1\), let H be a non-strongly oriented cycle of order \(\ell \). It holds that every _n_-vertex oriented graph D with \(n\in \ell \mathbb {N}\) and \(\delta (D)\ge (1-\frac{1}{\ell })n\) contains an _H_-factor. The total-degree condition is tight by considering an almost balanced complete multipartite oriented graph, which is a blow-up of H. However, when \(\ell \ge 2^{128}+1\), we cannot imagine a general lower bound construction that demonstrates the semi-degree condition, which is half of the total-degree condition, being tight. Everything appears different when \(\ell \) is small, even for \(\ell =4\). In this note, Theorem 1.1 says that \(R(\overline{C_4})=8\). Hence, in the \(\overline{C_4}\)-factor case, we are not able to apply the similar arguments, and obtain a \(\overline{C_4}\)-factor. By considering an almost balanced blow-up of a cyclic triangle, we are able to build an _n_-vertex oriented graph whose minimum semi-degree is \(\lfloor \frac{n}{3}\rfloor \), and it dose not contain a \(\overline{C_4}\)-factor.

Funding

Funded by Basic Research Program of Jiangsu (No.BK20251044), National Natural Science Foundation of China (No.12501483), National Key Research and Development Program of China (No.2024YFA1013900), and Natural Science Foundation of the Jiangsu Higher Education Institutions of China (No.25KJB110003). Funded by National Natural Science Foundation of China (No.12431013).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, China
    Ming Chen, Zhengke Miao & Shan Zhou

Authors

  1. Ming Chen
  2. Zhengke Miao
  3. Shan Zhou

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Correspondence toMing Chen.

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Chen, M., Miao, Z. & Zhou, S. A note between transitive \(C_4\)-factor and oriented Ramsey number.J Comb Optim 51, 15 (2026). https://doi.org/10.1007/s10878-026-01393-9

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