Antidirected trees in dense digraphs
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866929545596108800 |
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| author | Stein, Maya Trujillo-Negrete, Ana |
| author_facet | Stein, Maya Trujillo-Negrete, Ana |
| contents | We show that if $D$ is an $n$-vertex digraph with more than $(k-1)n$ arcs that does not contain any of three forbidden digraphs, then $D$ contains every antidirected tree on $k$ arcs. The forbidden digraphs are those orientations of $K_{2, \lceil k/12\rceil}$ where each of the vertices in the class of size two has either out-degree $0$ or in-degree $0$. This proves a conjecture of Addario-Berry et al. for a broad class of digraphs, and generalises a result for $K_{2, \lfloor k/12\rfloor}$-free graphs by Balasubramanian and Dobson.
We also show that every digraph $D$ on $n$ vertices with more than $(k-1)n$ arcs contains every antidirected $k$-arc caterpillar, thus solving the above conjecture for caterpillars. This generalises a result of Perles. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_10750 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Antidirected trees in dense digraphs Stein, Maya Trujillo-Negrete, Ana Combinatorics 05C05, 05C20, 05C35 We show that if $D$ is an $n$-vertex digraph with more than $(k-1)n$ arcs that does not contain any of three forbidden digraphs, then $D$ contains every antidirected tree on $k$ arcs. The forbidden digraphs are those orientations of $K_{2, \lceil k/12\rceil}$ where each of the vertices in the class of size two has either out-degree $0$ or in-degree $0$. This proves a conjecture of Addario-Berry et al. for a broad class of digraphs, and generalises a result for $K_{2, \lfloor k/12\rfloor}$-free graphs by Balasubramanian and Dobson. We also show that every digraph $D$ on $n$ vertices with more than $(k-1)n$ arcs contains every antidirected $k$-arc caterpillar, thus solving the above conjecture for caterpillars. This generalises a result of Perles. |
| title | Antidirected trees in dense digraphs |
| topic | Combinatorics 05C05, 05C20, 05C35 |
| url | https://arxiv.org/abs/2404.10750 |