The Turán number of directed paths and oriented cycles
Abstract Brown et al. (J Combin Theory Ser B 15(1):77–93, 1973) considered Turán-type extremal problems for digraphs. However, to date there are very few results on this problem, even asymptotically. Let $$\overrightarrow{P_{2,2}}$$ be the orientation of $$C_4$$ which consists of two 2-paths with th...
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| Autor*in: |
Zhou, Wenling [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2023 |
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| Anmerkung: |
© The Author(s), under exclusive licence to Springer Nature Japan KK, part of Springer Nature 2023. 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. |
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| Übergeordnetes Werk: |
Enthalten in: Graphs and combinatorics - Springer Japan, 1985, 39(2023), 3 vom: 08. Apr. |
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| Übergeordnetes Werk: |
volume:39 ; year:2023 ; number:3 ; day:08 ; month:04 |
| Links: |
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| DOI / URN: |
10.1007/s00373-023-02647-7 |
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OLC2134363843 |
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| 520 | |a Abstract Brown et al. (J Combin Theory Ser B 15(1):77–93, 1973) considered Turán-type extremal problems for digraphs. However, to date there are very few results on this problem, even asymptotically. Let $$\overrightarrow{P_{2,2}}$$ be the orientation of $$C_4$$ which consists of two 2-paths with the same initial and terminal vertices. Huang and Lyu [Discrete Math., 343 (5) (2020)] recently determined the Turán number of $$\overrightarrow{P_{2,2}}$$, and considered it a more natural and interesting problem to determine the Turán number of directed cycles. Let $$\overrightarrow{P_k}$$ and $$\overrightarrow{C_k}$$ denote the directed path and the directed cycle of order k, respectively. In this paper we determine the maximum size of $$\overrightarrow{C_k}$$-free digraphs of order n for all $$n,k \in \mathbb {N}^*$$, as well as the extremal digraphs attaining this maximum size. Similar result is obtained for $$\overrightarrow{P_k}$$ where n is large. In addition, we generalize the result of Huang and Lyu by characterizing the extremal digraphs avoiding an arbitrary orientation of $$C_4$$ except $$\overrightarrow{P_{2,2}}$$. In particular, for oriented even cycles, we classify which oriented even cycles inherit the difficulty of their underlying graphs and which do not. | ||
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10.1007/s00373-023-02647-7 doi (DE-627)OLC2134363843 (DE-He213)s00373-023-02647-7-p DE-627 ger DE-627 rakwb eng 510 VZ 510 VZ 17,1 ssgn Zhou, Wenling verfasserin aut The Turán number of directed paths and oriented cycles 2023 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Author(s), under exclusive licence to Springer Nature Japan KK, part of Springer Nature 2023. 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Abstract Brown et al. (J Combin Theory Ser B 15(1):77–93, 1973) considered Turán-type extremal problems for digraphs. However, to date there are very few results on this problem, even asymptotically. Let $$\overrightarrow{P_{2,2}}$$ be the orientation of $$C_4$$ which consists of two 2-paths with the same initial and terminal vertices. Huang and Lyu [Discrete Math., 343 (5) (2020)] recently determined the Turán number of $$\overrightarrow{P_{2,2}}$$, and considered it a more natural and interesting problem to determine the Turán number of directed cycles. Let $$\overrightarrow{P_k}$$ and $$\overrightarrow{C_k}$$ denote the directed path and the directed cycle of order k, respectively. In this paper we determine the maximum size of $$\overrightarrow{C_k}$$-free digraphs of order n for all $$n,k \in \mathbb {N}^*$$, as well as the extremal digraphs attaining this maximum size. Similar result is obtained for $$\overrightarrow{P_k}$$ where n is large. In addition, we generalize the result of Huang and Lyu by characterizing the extremal digraphs avoiding an arbitrary orientation of $$C_4$$ except $$\overrightarrow{P_{2,2}}$$. In particular, for oriented even cycles, we classify which oriented even cycles inherit the difficulty of their underlying graphs and which do not. Turán number Directed path Directed cycle Extremal digraph Li, Binlong (orcid)0000-0002-6971-8526 aut Enthalten in Graphs and combinatorics Springer Japan, 1985 39(2023), 3 vom: 08. Apr. (DE-627)129274453 (DE-600)84314-3 (DE-576)014463903 0911-0119 nnns volume:39 year:2023 number:3 day:08 month:04 https://doi.org/10.1007/s00373-023-02647-7 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT AR 39 2023 3 08 04 |
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10.1007/s00373-023-02647-7 doi (DE-627)OLC2134363843 (DE-He213)s00373-023-02647-7-p DE-627 ger DE-627 rakwb eng 510 VZ 510 VZ 17,1 ssgn Zhou, Wenling verfasserin aut The Turán number of directed paths and oriented cycles 2023 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Author(s), under exclusive licence to Springer Nature Japan KK, part of Springer Nature 2023. 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Abstract Brown et al. (J Combin Theory Ser B 15(1):77–93, 1973) considered Turán-type extremal problems for digraphs. However, to date there are very few results on this problem, even asymptotically. Let $$\overrightarrow{P_{2,2}}$$ be the orientation of $$C_4$$ which consists of two 2-paths with the same initial and terminal vertices. Huang and Lyu [Discrete Math., 343 (5) (2020)] recently determined the Turán number of $$\overrightarrow{P_{2,2}}$$, and considered it a more natural and interesting problem to determine the Turán number of directed cycles. Let $$\overrightarrow{P_k}$$ and $$\overrightarrow{C_k}$$ denote the directed path and the directed cycle of order k, respectively. In this paper we determine the maximum size of $$\overrightarrow{C_k}$$-free digraphs of order n for all $$n,k \in \mathbb {N}^*$$, as well as the extremal digraphs attaining this maximum size. Similar result is obtained for $$\overrightarrow{P_k}$$ where n is large. In addition, we generalize the result of Huang and Lyu by characterizing the extremal digraphs avoiding an arbitrary orientation of $$C_4$$ except $$\overrightarrow{P_{2,2}}$$. In particular, for oriented even cycles, we classify which oriented even cycles inherit the difficulty of their underlying graphs and which do not. Turán number Directed path Directed cycle Extremal digraph Li, Binlong (orcid)0000-0002-6971-8526 aut Enthalten in Graphs and combinatorics Springer Japan, 1985 39(2023), 3 vom: 08. Apr. (DE-627)129274453 (DE-600)84314-3 (DE-576)014463903 0911-0119 nnns volume:39 year:2023 number:3 day:08 month:04 https://doi.org/10.1007/s00373-023-02647-7 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT AR 39 2023 3 08 04 |
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10.1007/s00373-023-02647-7 doi (DE-627)OLC2134363843 (DE-He213)s00373-023-02647-7-p DE-627 ger DE-627 rakwb eng 510 VZ 510 VZ 17,1 ssgn Zhou, Wenling verfasserin aut The Turán number of directed paths and oriented cycles 2023 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Author(s), under exclusive licence to Springer Nature Japan KK, part of Springer Nature 2023. 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Abstract Brown et al. (J Combin Theory Ser B 15(1):77–93, 1973) considered Turán-type extremal problems for digraphs. However, to date there are very few results on this problem, even asymptotically. Let $$\overrightarrow{P_{2,2}}$$ be the orientation of $$C_4$$ which consists of two 2-paths with the same initial and terminal vertices. Huang and Lyu [Discrete Math., 343 (5) (2020)] recently determined the Turán number of $$\overrightarrow{P_{2,2}}$$, and considered it a more natural and interesting problem to determine the Turán number of directed cycles. Let $$\overrightarrow{P_k}$$ and $$\overrightarrow{C_k}$$ denote the directed path and the directed cycle of order k, respectively. In this paper we determine the maximum size of $$\overrightarrow{C_k}$$-free digraphs of order n for all $$n,k \in \mathbb {N}^*$$, as well as the extremal digraphs attaining this maximum size. Similar result is obtained for $$\overrightarrow{P_k}$$ where n is large. In addition, we generalize the result of Huang and Lyu by characterizing the extremal digraphs avoiding an arbitrary orientation of $$C_4$$ except $$\overrightarrow{P_{2,2}}$$. In particular, for oriented even cycles, we classify which oriented even cycles inherit the difficulty of their underlying graphs and which do not. Turán number Directed path Directed cycle Extremal digraph Li, Binlong (orcid)0000-0002-6971-8526 aut Enthalten in Graphs and combinatorics Springer Japan, 1985 39(2023), 3 vom: 08. Apr. (DE-627)129274453 (DE-600)84314-3 (DE-576)014463903 0911-0119 nnns volume:39 year:2023 number:3 day:08 month:04 https://doi.org/10.1007/s00373-023-02647-7 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT AR 39 2023 3 08 04 |
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10.1007/s00373-023-02647-7 doi (DE-627)OLC2134363843 (DE-He213)s00373-023-02647-7-p DE-627 ger DE-627 rakwb eng 510 VZ 510 VZ 17,1 ssgn Zhou, Wenling verfasserin aut The Turán number of directed paths and oriented cycles 2023 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Author(s), under exclusive licence to Springer Nature Japan KK, part of Springer Nature 2023. 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Abstract Brown et al. (J Combin Theory Ser B 15(1):77–93, 1973) considered Turán-type extremal problems for digraphs. However, to date there are very few results on this problem, even asymptotically. Let $$\overrightarrow{P_{2,2}}$$ be the orientation of $$C_4$$ which consists of two 2-paths with the same initial and terminal vertices. Huang and Lyu [Discrete Math., 343 (5) (2020)] recently determined the Turán number of $$\overrightarrow{P_{2,2}}$$, and considered it a more natural and interesting problem to determine the Turán number of directed cycles. Let $$\overrightarrow{P_k}$$ and $$\overrightarrow{C_k}$$ denote the directed path and the directed cycle of order k, respectively. In this paper we determine the maximum size of $$\overrightarrow{C_k}$$-free digraphs of order n for all $$n,k \in \mathbb {N}^*$$, as well as the extremal digraphs attaining this maximum size. Similar result is obtained for $$\overrightarrow{P_k}$$ where n is large. In addition, we generalize the result of Huang and Lyu by characterizing the extremal digraphs avoiding an arbitrary orientation of $$C_4$$ except $$\overrightarrow{P_{2,2}}$$. In particular, for oriented even cycles, we classify which oriented even cycles inherit the difficulty of their underlying graphs and which do not. Turán number Directed path Directed cycle Extremal digraph Li, Binlong (orcid)0000-0002-6971-8526 aut Enthalten in Graphs and combinatorics Springer Japan, 1985 39(2023), 3 vom: 08. Apr. (DE-627)129274453 (DE-600)84314-3 (DE-576)014463903 0911-0119 nnns volume:39 year:2023 number:3 day:08 month:04 https://doi.org/10.1007/s00373-023-02647-7 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT AR 39 2023 3 08 04 |
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10.1007/s00373-023-02647-7 doi (DE-627)OLC2134363843 (DE-He213)s00373-023-02647-7-p DE-627 ger DE-627 rakwb eng 510 VZ 510 VZ 17,1 ssgn Zhou, Wenling verfasserin aut The Turán number of directed paths and oriented cycles 2023 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Author(s), under exclusive licence to Springer Nature Japan KK, part of Springer Nature 2023. 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Abstract Brown et al. (J Combin Theory Ser B 15(1):77–93, 1973) considered Turán-type extremal problems for digraphs. However, to date there are very few results on this problem, even asymptotically. Let $$\overrightarrow{P_{2,2}}$$ be the orientation of $$C_4$$ which consists of two 2-paths with the same initial and terminal vertices. Huang and Lyu [Discrete Math., 343 (5) (2020)] recently determined the Turán number of $$\overrightarrow{P_{2,2}}$$, and considered it a more natural and interesting problem to determine the Turán number of directed cycles. Let $$\overrightarrow{P_k}$$ and $$\overrightarrow{C_k}$$ denote the directed path and the directed cycle of order k, respectively. In this paper we determine the maximum size of $$\overrightarrow{C_k}$$-free digraphs of order n for all $$n,k \in \mathbb {N}^*$$, as well as the extremal digraphs attaining this maximum size. Similar result is obtained for $$\overrightarrow{P_k}$$ where n is large. In addition, we generalize the result of Huang and Lyu by characterizing the extremal digraphs avoiding an arbitrary orientation of $$C_4$$ except $$\overrightarrow{P_{2,2}}$$. In particular, for oriented even cycles, we classify which oriented even cycles inherit the difficulty of their underlying graphs and which do not. Turán number Directed path Directed cycle Extremal digraph Li, Binlong (orcid)0000-0002-6971-8526 aut Enthalten in Graphs and combinatorics Springer Japan, 1985 39(2023), 3 vom: 08. Apr. (DE-627)129274453 (DE-600)84314-3 (DE-576)014463903 0911-0119 nnns volume:39 year:2023 number:3 day:08 month:04 https://doi.org/10.1007/s00373-023-02647-7 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT AR 39 2023 3 08 04 |
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The Turán number of directed paths and oriented cycles |
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Zhou, Wenling |
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Graphs and combinatorics |
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Zhou, Wenling Li, Binlong |
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Zhou, Wenling |
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10.1007/s00373-023-02647-7 |
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| title_sort |
the turán number of directed paths and oriented cycles |
| title_auth |
The Turán number of directed paths and oriented cycles |
| abstract |
Abstract Brown et al. (J Combin Theory Ser B 15(1):77–93, 1973) considered Turán-type extremal problems for digraphs. However, to date there are very few results on this problem, even asymptotically. Let $$\overrightarrow{P_{2,2}}$$ be the orientation of $$C_4$$ which consists of two 2-paths with the same initial and terminal vertices. Huang and Lyu [Discrete Math., 343 (5) (2020)] recently determined the Turán number of $$\overrightarrow{P_{2,2}}$$, and considered it a more natural and interesting problem to determine the Turán number of directed cycles. Let $$\overrightarrow{P_k}$$ and $$\overrightarrow{C_k}$$ denote the directed path and the directed cycle of order k, respectively. In this paper we determine the maximum size of $$\overrightarrow{C_k}$$-free digraphs of order n for all $$n,k \in \mathbb {N}^*$$, as well as the extremal digraphs attaining this maximum size. Similar result is obtained for $$\overrightarrow{P_k}$$ where n is large. In addition, we generalize the result of Huang and Lyu by characterizing the extremal digraphs avoiding an arbitrary orientation of $$C_4$$ except $$\overrightarrow{P_{2,2}}$$. In particular, for oriented even cycles, we classify which oriented even cycles inherit the difficulty of their underlying graphs and which do not. © The Author(s), under exclusive licence to Springer Nature Japan KK, part of Springer Nature 2023. 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. |
| abstractGer |
Abstract Brown et al. (J Combin Theory Ser B 15(1):77–93, 1973) considered Turán-type extremal problems for digraphs. However, to date there are very few results on this problem, even asymptotically. Let $$\overrightarrow{P_{2,2}}$$ be the orientation of $$C_4$$ which consists of two 2-paths with the same initial and terminal vertices. Huang and Lyu [Discrete Math., 343 (5) (2020)] recently determined the Turán number of $$\overrightarrow{P_{2,2}}$$, and considered it a more natural and interesting problem to determine the Turán number of directed cycles. Let $$\overrightarrow{P_k}$$ and $$\overrightarrow{C_k}$$ denote the directed path and the directed cycle of order k, respectively. In this paper we determine the maximum size of $$\overrightarrow{C_k}$$-free digraphs of order n for all $$n,k \in \mathbb {N}^*$$, as well as the extremal digraphs attaining this maximum size. Similar result is obtained for $$\overrightarrow{P_k}$$ where n is large. In addition, we generalize the result of Huang and Lyu by characterizing the extremal digraphs avoiding an arbitrary orientation of $$C_4$$ except $$\overrightarrow{P_{2,2}}$$. In particular, for oriented even cycles, we classify which oriented even cycles inherit the difficulty of their underlying graphs and which do not. © The Author(s), under exclusive licence to Springer Nature Japan KK, part of Springer Nature 2023. 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. |
| abstract_unstemmed |
Abstract Brown et al. (J Combin Theory Ser B 15(1):77–93, 1973) considered Turán-type extremal problems for digraphs. However, to date there are very few results on this problem, even asymptotically. Let $$\overrightarrow{P_{2,2}}$$ be the orientation of $$C_4$$ which consists of two 2-paths with the same initial and terminal vertices. Huang and Lyu [Discrete Math., 343 (5) (2020)] recently determined the Turán number of $$\overrightarrow{P_{2,2}}$$, and considered it a more natural and interesting problem to determine the Turán number of directed cycles. Let $$\overrightarrow{P_k}$$ and $$\overrightarrow{C_k}$$ denote the directed path and the directed cycle of order k, respectively. In this paper we determine the maximum size of $$\overrightarrow{C_k}$$-free digraphs of order n for all $$n,k \in \mathbb {N}^*$$, as well as the extremal digraphs attaining this maximum size. Similar result is obtained for $$\overrightarrow{P_k}$$ where n is large. In addition, we generalize the result of Huang and Lyu by characterizing the extremal digraphs avoiding an arbitrary orientation of $$C_4$$ except $$\overrightarrow{P_{2,2}}$$. In particular, for oriented even cycles, we classify which oriented even cycles inherit the difficulty of their underlying graphs and which do not. © The Author(s), under exclusive licence to Springer Nature Japan KK, part of Springer Nature 2023. 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. |
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The Turán number of directed paths and oriented cycles |
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