Vertex-disjoint cycles of different lengths in tournaments
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866916148841283584 |
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| author | Bai, Yandong Jia, Wenpei |
| author_facet | Bai, Yandong Jia, Wenpei |
| contents | Bermond and Thomassen conjectured in 1981 that every digraph with minimum outdegree at least $2k-1$ contains $k$ vertex-disjoint cycles,here $k$ is a positive integer. Lichiardopol conjectured in 2014 that for every positive integer $k$ there exists an integer $g(k)$ such that every digraph with minimum outdegree at least $g(k)$ contains $k$ vertex-disjoint cycles of different lengths. Recently, Chen and Chang proved in [J. Graph Theory 105 (2) (2024) 297-314] that for $k\geqslant 3$ every tournament with minimum outdegree at least $2k-1$ contains $k$ vertex-disjoint cycles in which two of them have different lengths. Motivated by the above two conjectures and related results, we investigate vertex-disjoint cycles of different lengths in tournaments, and show that when $k\geqslant 5$ every tournament with minimum outdegree at least $2k-1$ contains $k$ vertex-disjoint cycles in which three of them have different lengths. In addition, we show that every tournament with minimum outdegree at least $6$ contains three vertex-disjoint cycles of different lengths and the minimum outdegree condition is sharp. This answers a question proposed by Chen and Chang. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_03692 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Vertex-disjoint cycles of different lengths in tournaments Bai, Yandong Jia, Wenpei Combinatorics Bermond and Thomassen conjectured in 1981 that every digraph with minimum outdegree at least $2k-1$ contains $k$ vertex-disjoint cycles,here $k$ is a positive integer. Lichiardopol conjectured in 2014 that for every positive integer $k$ there exists an integer $g(k)$ such that every digraph with minimum outdegree at least $g(k)$ contains $k$ vertex-disjoint cycles of different lengths. Recently, Chen and Chang proved in [J. Graph Theory 105 (2) (2024) 297-314] that for $k\geqslant 3$ every tournament with minimum outdegree at least $2k-1$ contains $k$ vertex-disjoint cycles in which two of them have different lengths. Motivated by the above two conjectures and related results, we investigate vertex-disjoint cycles of different lengths in tournaments, and show that when $k\geqslant 5$ every tournament with minimum outdegree at least $2k-1$ contains $k$ vertex-disjoint cycles in which three of them have different lengths. In addition, we show that every tournament with minimum outdegree at least $6$ contains three vertex-disjoint cycles of different lengths and the minimum outdegree condition is sharp. This answers a question proposed by Chen and Chang. |
| title | Vertex-disjoint cycles of different lengths in tournaments |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2403.03692 |