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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | https://arxiv.org/abs/2409.02812 |
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| _version_ | 1866913491541032960 |
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| author | Iršič, Vesna Portier, Julien Versteegen, Leo |
| author_facet | Iršič, Vesna Portier, Julien Versteegen, Leo |
| contents | Let $G\sim G(n,p)$ be a (hidden) Erdős-Rényi random graph with $p=(1+ \varepsilon)/n$ for some fixed constant $ \varepsilon >0$. Ferber, Krivelevich, Sudakov, and Vieira showed that to reveal a path of length $\ell=Ω\left(\frac{\log(1/ \varepsilon)}{ \varepsilon}\right)$ in $G$ with high probability, one must query the adjacency of $Ω\left(\frac{\ell}{p \varepsilon\log(1/ \varepsilon)}\right)$ pairs of vertices in $G$, where each query may depend on the outcome of all previous queries. Their result is tight up to the factor of $\log(1/ \varepsilon)$ in both $\ell$ and the number of queries, and they conjectured that this factor could be removed. We confirm their conjecture. The main ingredient in our proof is a result about path-packings in random labelled trees of independent interest. Using this, we also give a partial answer to a related question of Ferber, Krivelevich, Sudakov, and Vieira. Namely, we show that when $\ell=o\left((t/\log t)^{1/3}\right)$, the maximum number of vertices covered by edge-disjoint paths of length at least $\ell$ in a random labelled tree of size $t$ is $Θ(t/\ell)$ with high probability. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_02812 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Packing and finding paths in sparse random graphs Iršič, Vesna Portier, Julien Versteegen, Leo Combinatorics Let $G\sim G(n,p)$ be a (hidden) Erdős-Rényi random graph with $p=(1+ \varepsilon)/n$ for some fixed constant $ \varepsilon >0$. Ferber, Krivelevich, Sudakov, and Vieira showed that to reveal a path of length $\ell=Ω\left(\frac{\log(1/ \varepsilon)}{ \varepsilon}\right)$ in $G$ with high probability, one must query the adjacency of $Ω\left(\frac{\ell}{p \varepsilon\log(1/ \varepsilon)}\right)$ pairs of vertices in $G$, where each query may depend on the outcome of all previous queries. Their result is tight up to the factor of $\log(1/ \varepsilon)$ in both $\ell$ and the number of queries, and they conjectured that this factor could be removed. We confirm their conjecture. The main ingredient in our proof is a result about path-packings in random labelled trees of independent interest. Using this, we also give a partial answer to a related question of Ferber, Krivelevich, Sudakov, and Vieira. Namely, we show that when $\ell=o\left((t/\log t)^{1/3}\right)$, the maximum number of vertices covered by edge-disjoint paths of length at least $\ell$ in a random labelled tree of size $t$ is $Θ(t/\ell)$ with high probability. |
| title | Packing and finding paths in sparse random graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2409.02812 |