Unavoidable subgraphs in digraphs with large out-degrees
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arXiv
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| Auteurs principaux: | , , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866908791895752704 |
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| author | Hons, Tomáš Klimošová, Tereza Kucheriya, Gaurav Mikšaník, David Tkadlec, Josef Tyomkyn, Mykhaylo |
| author_facet | Hons, Tomáš Klimošová, Tereza Kucheriya, Gaurav Mikšaník, David Tkadlec, Josef Tyomkyn, Mykhaylo |
| contents | We ask the question, which oriented trees $T$ must be contained as subgraphs in every finite directed graph of sufficiently large minimum out-degree. We formulate the following simple condition: all vertices in $T$ of in-degree at least $2$ must be on the same 'level' in the natural height function of $T$. We prove this condition to be necessary and conjecture it to be sufficient. In support of our conjecture, we prove it for a fairly general class of trees.
An essential tool in the latter proof, and a question interesting in its own right, is finding large subdivided in-stars in a directed graph of large minimum out-degree. We conjecture that any digraph and oriented graph of minimum out-degree at least $k\ell$ and $k\ell/2$, respectively, contains the $(k-1)$-subdivision of the in-star with $\ell$ leaves as a subgraph; this would be tight and generalizes a conjecture of Thomassé. We prove this for digraphs and $k=2$ up to a factor of less than $2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_20616 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Unavoidable subgraphs in digraphs with large out-degrees Hons, Tomáš Klimošová, Tereza Kucheriya, Gaurav Mikšaník, David Tkadlec, Josef Tyomkyn, Mykhaylo Combinatorics We ask the question, which oriented trees $T$ must be contained as subgraphs in every finite directed graph of sufficiently large minimum out-degree. We formulate the following simple condition: all vertices in $T$ of in-degree at least $2$ must be on the same 'level' in the natural height function of $T$. We prove this condition to be necessary and conjecture it to be sufficient. In support of our conjecture, we prove it for a fairly general class of trees. An essential tool in the latter proof, and a question interesting in its own right, is finding large subdivided in-stars in a directed graph of large minimum out-degree. We conjecture that any digraph and oriented graph of minimum out-degree at least $k\ell$ and $k\ell/2$, respectively, contains the $(k-1)$-subdivision of the in-star with $\ell$ leaves as a subgraph; this would be tight and generalizes a conjecture of Thomassé. We prove this for digraphs and $k=2$ up to a factor of less than $2$. |
| title | Unavoidable subgraphs in digraphs with large out-degrees |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2504.20616 |