Connectedness of Unit Distance Subgraphs Induced by Closed Convex Sets

TL;DR

This paper investigates the connectedness of unit distance subgraphs induced by closed convex sets in Euclidean space, revealing precise conditions for connectivity and providing diameter bounds for specific cases.

Contribution

It characterizes when such subgraphs are connected based on the radius and dimension of the convex set, a novel analysis beyond traditional chromatic number studies.

Findings

Connectedness depends on the radius and affine dimension of the convex set.

The graph is connected if radius is 0 or at least 1 with dimension ≥ 2.

Bounds for the diameter are provided for hyperrectangles with radius exactly 1.

Abstract

The unit distance graph $G_{\mathbb{R}^d}^1$ is the infinite graph whose nodes are points in $\mathbb{R}^d$, with an edge between two points if the Euclidean distance between these points is 1. The 2-dimensional version $G_{\mathbb{R}^2}^1$ of this graph is typically studied for its chromatic number, as in the Hadwiger-Nelson problem. However, other properties of unit distance graphs are rarely studied. Here, we consider the restriction of $G_{\mathbb{R}^d}^1$ to closed convex subsets $X$ of $\mathbb{R}^d$. We show that the graph $G_{\mathbb{R}^d}^1[X]$ is connected precisely when the radius of $r(X)$ of $X$ is equal to 0, or when $r(X)\geq 1$ and the affine dimension of $X$ is at least 2. For hyperrectangles, we give bounds for the graph diameter in the critical case that the radius is exactly 1.