Randomly perturbed digraphs also have bounded-degree spanning trees
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866917752825970688 |
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| author | Morawski, Patryk Petrova, Kalina |
| author_facet | Morawski, Patryk Petrova, Kalina |
| contents | We show that a randomly perturbed digraph, where we start with a dense digraph $D_α$ and add a small number of random edges to it, will typically contain a fixed orientation of a bounded degree spanning tree. This answers a question posed by Araujo, Balogh, Krueger, Piga and Treglown and generalizes the corresponding result for randomly perturbed graphs by Krivelevich, Kwan and Sudakov. More specifically, we prove that there exists a constant $c = c(α, Δ)$ such that if $T$ is an oriented tree with maximum degree $Δ$ and $D_α$ is an $n$-vertex digraph with minimum semidegree $αn$, then the graph obtained by adding $cn$ uniformly random edges to $D_α$ will contain $T$ with high probability. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_14648 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Randomly perturbed digraphs also have bounded-degree spanning trees Morawski, Patryk Petrova, Kalina Combinatorics 05C80, 05D40 We show that a randomly perturbed digraph, where we start with a dense digraph $D_α$ and add a small number of random edges to it, will typically contain a fixed orientation of a bounded degree spanning tree. This answers a question posed by Araujo, Balogh, Krueger, Piga and Treglown and generalizes the corresponding result for randomly perturbed graphs by Krivelevich, Kwan and Sudakov. More specifically, we prove that there exists a constant $c = c(α, Δ)$ such that if $T$ is an oriented tree with maximum degree $Δ$ and $D_α$ is an $n$-vertex digraph with minimum semidegree $αn$, then the graph obtained by adding $cn$ uniformly random edges to $D_α$ will contain $T$ with high probability. |
| title | Randomly perturbed digraphs also have bounded-degree spanning trees |
| topic | Combinatorics 05C80, 05D40 |
| url | https://arxiv.org/abs/2306.14648 |