Immersions of directed graphs in tournaments
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866909397772402688 |
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| author | Girão, António Hancock, Robert |
| author_facet | Girão, António Hancock, Robert |
| contents | Recently, Draganić, Munhá Correia, Sudakov and Yuster showed that every tournament on $(2+o(1))k^2$ vertices contains a $1$-subdivision of a transitive tournament on $k$ vertices, which is tight up to a constant factor. We prove a counterpart of their result for immersions. Let $f(k)$ be the smallest integer such that any tournament on at least $f(k)$ vertices must contain a $1$-immersion of a transitive tournament on $k$ vertices. We show that $f(k)=O(k)$, which is clearly tight up to a multiplicative factor. If one insists in finding an immersion of a complete directed graph on $k$ vertices then an extra condition on the tournament is necessary. Indeed, we show that every tournament with minimum out-degree at least $Ck$ must contain a $2$-immersion of a complete digraph on $k$ vertices. This is again tight up to the value of $C$ and tight on the order of the paths in the immersion. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_06204 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Immersions of directed graphs in tournaments Girão, António Hancock, Robert Combinatorics Recently, Draganić, Munhá Correia, Sudakov and Yuster showed that every tournament on $(2+o(1))k^2$ vertices contains a $1$-subdivision of a transitive tournament on $k$ vertices, which is tight up to a constant factor. We prove a counterpart of their result for immersions. Let $f(k)$ be the smallest integer such that any tournament on at least $f(k)$ vertices must contain a $1$-immersion of a transitive tournament on $k$ vertices. We show that $f(k)=O(k)$, which is clearly tight up to a multiplicative factor. If one insists in finding an immersion of a complete directed graph on $k$ vertices then an extra condition on the tournament is necessary. Indeed, we show that every tournament with minimum out-degree at least $Ck$ must contain a $2$-immersion of a complete digraph on $k$ vertices. This is again tight up to the value of $C$ and tight on the order of the paths in the immersion. |
| title | Immersions of directed graphs in tournaments |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2305.06204 |