Embedding Euclidean Distance Graphs in $\mathbb{R}^n$ and $\mathbb{Q}^n$

TL;DR

This paper investigates the embedding of Euclidean distance graphs in rational and real spaces, demonstrating the existence of certain subgraphs in rational spaces that cannot be embedded in lower-dimensional real spaces.

Contribution

It provides an affirmative answer for the existence of such subgraphs in dimensions up to five, advancing understanding of geometric graph embeddings.

Findings

Existence of subgraphs in $G(Q^n, d)$ not embeddable in $G(R^{n-1}, 1)$ for $n \,\leq\, 5$

Resolution of related questions about Euclidean distance graph embeddings

Extension of embedding results to rational and real spaces in specified dimensions

Abstract

For $S \subseteq \mathbb{R}$, positive integer $n$, and $d > 0$, let $G(S^n, d)$ be the graph whose vertex set is $S^n$ where any two vertices are adjacent if and only if they are Euclidean distance $d$ apart. The primary question we will consider in our work is as follows. Given $n$ and distance $d$ actually realized as a distance between points of the rational space $\mathbb{Q}^n$, does there exist a finite graph $G$ that appears as a subgraph of $G(\mathbb{Q}^n, d)$ but not as a subgraph of $G(\mathbb{R}^{n-1}, 1)$? We answer this question affirmatively for $n \leq 5$, and along the way, resolve a few related questions as well.