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arXiv · 2504.15212

A universal threshold for geometric embeddings of trees

Dylan J. Altschuler·Pandelis Dodos·Konstantin Tikhomirov·Konstantinos Tyros
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Abstract

A graph $G=(V,E)$ is geometrically embeddable into a normed space $X$ when there is a mapping $\zeta: V\to X$ such that $\|\zeta(v)-\zeta(w)\|_X\leqslant 1$ if and only if $\{v,w\}\in E$, for all distinct $v,w\in V$. Our result is the following universal threshold for the embeddability of trees. Let $\Delta \geqslant 3$, and let $N$ be sufficiently large in terms of $\Delta$. Every $N$--vertex tree of maximal degree at most $\Delta$ is embeddable into any normed space of dimension at least $64\,\frac{\log N}{\log\log N}$, and complete trees are non-embeddable into any normed space of dimension less than $\frac{1}{2}\,\frac{\log N}{\log\log N}$. In striking contrast, spectral expanders and random graphs are known to be non-embeddable in sublogarithmic dimension. Our result is based on a randomized embedding whose analysis utilizes the recent breakthroughs on Bourgain's slicing problem.

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BibTeXRIS

Dylan J. Altschuler, Pandelis Dodos, Konstantin Tikhomirov, Konstantinos Tyros. 2025-04-21. A universal threshold for geometric embeddings of trees. https://arxiv.org/abs/2504.15212

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