Subdivisions in dicritical digraphs with large order or digirth
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
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2024
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| _version_ | 1866909208665915392 |
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| author | Picasarri-Arrieta, Lucas Rambaud, Clément |
| author_facet | Picasarri-Arrieta, Lucas Rambaud, Clément |
| contents | Aboulker et al. proved that a digraph with large enough dichromatic number contains any fixed digraph as a subdivision. The dichromatic number of a digraph is the smallest order of a partition of its vertex set into acyclic induced subdigraphs. A digraph is dicritical if the removal of any arc or vertex decreases its dichromatic number. In this paper we give sufficient conditions on a dicritical digraph of large order or large directed girth to contain a given digraph as a subdivision. In particular, we prove that (i) for every integers $k,\ell$, large enough dicritical digraphs with dichromatic number $k$ contain an orientation of a cycle with at least $\ell$ vertices; (ii) there are functions $f,g$ such that for every subdivision $F^*$ of a digraph $F$, digraphs with directed girth at least $f(F^*)$ and dichromatic number at least $g(F)$ contain a subdivision of $F^*$, and if $F$ is a tree, then $g(F)=|V(F)|$; (iii) there is a function $f$ such that for every subdivision $F^*$ of $TT_3$ (the transitive tournament on three vertices), digraphs with directed girth at least $f(F^*)$ and minimum out-degree at least $2$ contain $F^*$ as a subdivision. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_05938 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Subdivisions in dicritical digraphs with large order or digirth Picasarri-Arrieta, Lucas Rambaud, Clément Combinatorics Discrete Mathematics Aboulker et al. proved that a digraph with large enough dichromatic number contains any fixed digraph as a subdivision. The dichromatic number of a digraph is the smallest order of a partition of its vertex set into acyclic induced subdigraphs. A digraph is dicritical if the removal of any arc or vertex decreases its dichromatic number. In this paper we give sufficient conditions on a dicritical digraph of large order or large directed girth to contain a given digraph as a subdivision. In particular, we prove that (i) for every integers $k,\ell$, large enough dicritical digraphs with dichromatic number $k$ contain an orientation of a cycle with at least $\ell$ vertices; (ii) there are functions $f,g$ such that for every subdivision $F^*$ of a digraph $F$, digraphs with directed girth at least $f(F^*)$ and dichromatic number at least $g(F)$ contain a subdivision of $F^*$, and if $F$ is a tree, then $g(F)=|V(F)|$; (iii) there is a function $f$ such that for every subdivision $F^*$ of $TT_3$ (the transitive tournament on three vertices), digraphs with directed girth at least $f(F^*)$ and minimum out-degree at least $2$ contain $F^*$ as a subdivision. |
| title | Subdivisions in dicritical digraphs with large order or digirth |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2401.05938 |