Annulus Graphs in R^d

A d-dimensional annulus graph with radii R₁ and R₂ (here R₂≥R₁≥0) is a graph embeddable in 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 [R₁, R₂]. In this paper we show that the family A_d(R₁, R₂) of d-dimensional annulus graphs with radii R₁ and R₂ is uniquely characterised by R₂/R₁ when this ratio is sufficiently large. Moreover, as a step towards a better understanding of the structure of A_d(R1, R2), we show that supG∈A_d(R₁, R₂)χ(G)/ω(G) is given by exp(O(d)) for all R₁, R₂ satisfying R₂≥R₁>0 and also exp(Ω(d)) if moreover R₂/R₁≥1.2.