Minors in small-set expanders
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909747363446784 |
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| author | Krivelevich, Michael Nenadov, Rajko |
| author_facet | Krivelevich, Michael Nenadov, Rajko |
| contents | We study large minors in small-set expanders. More precisely, we consider graphs with $n$ vertices and the property that every set of size at most $αn / t$ expands by a factor of $t$, for some (constant) $α> 0$ and large $t = t(n)$. We obtain the following:
* Improving results of Krivelevich and Sudakov, we show that a small-set expander contains a complete minor of order $\sqrt{n t / \log n}$.
* We show that a small-set expander contains every graph $H$ with $O(n \log t / \log n)$ edges and vertices as a minor. We complement this with an upper bound showing that if an $n$-vertex graph $G$ has average degree $d$, then there exists a graph with $O(n \log d / \log n)$ edges and vertices which is not a minor of $G$. This has two consequences: (i) It implies the optimality of our result in the case $t = d^c$ for some constant $c > 0$, and (ii) it shows expanders are optimal minor-universal graphs of a given average degree. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_06826 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Minors in small-set expanders Krivelevich, Michael Nenadov, Rajko Combinatorics We study large minors in small-set expanders. More precisely, we consider graphs with $n$ vertices and the property that every set of size at most $αn / t$ expands by a factor of $t$, for some (constant) $α> 0$ and large $t = t(n)$. We obtain the following: * Improving results of Krivelevich and Sudakov, we show that a small-set expander contains a complete minor of order $\sqrt{n t / \log n}$. * We show that a small-set expander contains every graph $H$ with $O(n \log t / \log n)$ edges and vertices as a minor. We complement this with an upper bound showing that if an $n$-vertex graph $G$ has average degree $d$, then there exists a graph with $O(n \log d / \log n)$ edges and vertices which is not a minor of $G$. This has two consequences: (i) It implies the optimality of our result in the case $t = d^c$ for some constant $c > 0$, and (ii) it shows expanders are optimal minor-universal graphs of a given average degree. |
| title | Minors in small-set expanders |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2503.06826 |