Annulus graphs in ${\mathbb{R}}^d$

A $d$-dimensional annulus graph with radii $R_1$ and $R_2$ (here $R_2\ge R_1\ge 0$)isagraphembeddablein$\mathbb R^d$sothattwovertices$u$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$d$-dimensionalannulusgraphswithradii$R_1$and$R_2$ is uniquely characterised by $R_2\mathrm{/}{R}_1$ 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 $\sup_{G\in \mathcal A_d(R_1,R_2)} \chi(G)/\omega(G)$isgivenby$\exp(O(d))$forall$R_1,R_2$satisfying$R_2\ge R_1 > 0$andalso$\exp(\Omega(d))$ifmoreover$R_2/R_1\ge 1.2$.