Annulus graphs in $\mathbb R^d$

TL;DR

This paper characterizes the structure of $d$-dimensional annulus graphs in Euclidean space, showing they are uniquely determined by the ratio of their radii when this ratio is large, and analyzes their chromatic and clique number ratios.

Contribution

It provides a unique characterization of annulus graphs based on the radius ratio and bounds the chromatic-to-clique number ratio for these graphs in high dimensions.

Findings

Annulus graphs are uniquely characterized by the ratio of radii when this ratio is large.

The supremum of the chromatic number over clique number ratio grows exponentially with dimension.

The ratio of radii influences the structural properties of annulus graphs in high-dimensional spaces.

Abstract

A $d$-dimensional annulus graph with radii $R_1$ and $R_2$ (here $R_2\ge R_1\ge 0$) is a graph embeddable in $\mathbb R^d$ so that two vertices $u$ and $v$ form an edge if and only if their images in the embedding are at distance in the interval $[R_1, R_2]$. In this paper we show that the family $\mathcal A_d(R_1,R_2)$ of $d$-dimensional annulus graphs with radii $R_1$ and $R_2$ is uniquely characterised by $R_2/R_1$ when this ratio is sufficiently large. Moreover, as a step towards a better understanding of the structure of $\mathcal A_d(R_1,R_2)$, we show that $\sup_{G\in \mathcal A_d(R_1,R_2)} \chi(G)/\omega(G)$ is given by $\exp(O(d))$ for all $R_1,R_2$ satisfying $R_2\ge R_1 > 0$ and also $\exp(\Omega(d))$ if moreover $R_2/R_1\ge 1.2$.