Concentration of the largest induced tree size of $G_{n,p}$ around the standard expectation threshold
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866910045823827968 |
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| author | Hofstad, Jakob |
| author_facet | Hofstad, Jakob |
| contents | Let $T(G)$ be the size of the largest induced tree of $G$, and let $G_{n,p}$ be the binomial random graph. Kamaldinov, Skorkin, and Zhukovskii proved that $T(G_{n,p})$ equals one of two consecutive values with high probability if $p$ is constant, and more recently, Oropeza extended this result to include all vanishing $p$ such that $p > n^{-\frac{e-2}{3e-2} + ε}$, where $e$ is Euler's constant. We further extend this result to all vanishing $p$ such that $p \gg n^{-1/2} \ln^{3/2} n$, and furthermore, we show that, for $p$ such that $n^{-1} \ll p \ll n^{-1/2}, \ T(G_{n,p})$ cannot be concentrated at the standard expectation threshold. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_03076 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Concentration of the largest induced tree size of $G_{n,p}$ around the standard expectation threshold Hofstad, Jakob Combinatorics 05C80 Let $T(G)$ be the size of the largest induced tree of $G$, and let $G_{n,p}$ be the binomial random graph. Kamaldinov, Skorkin, and Zhukovskii proved that $T(G_{n,p})$ equals one of two consecutive values with high probability if $p$ is constant, and more recently, Oropeza extended this result to include all vanishing $p$ such that $p > n^{-\frac{e-2}{3e-2} + ε}$, where $e$ is Euler's constant. We further extend this result to all vanishing $p$ such that $p \gg n^{-1/2} \ln^{3/2} n$, and furthermore, we show that, for $p$ such that $n^{-1} \ll p \ll n^{-1/2}, \ T(G_{n,p})$ cannot be concentrated at the standard expectation threshold. |
| title | Concentration of the largest induced tree size of $G_{n,p}$ around the standard expectation threshold |
| topic | Combinatorics 05C80 |
| url | https://arxiv.org/abs/2603.03076 |