On the range of two-distance graphs

Abstract

The topic of this paper is related to the well-known notion of unit distance graphs. Take a graph with its edges coloured red and blue such that for some $d$ it can be mapped into the plane with all vertices going to distinct points, the red edges to segments of length $1$ and the blue edges to segments of length $d$. We define the range of this graph to be the set of such numbers $d$. It is easy to show that the range of any edge-bicoloured graph consists of finitely many intervals with algebraic endpoints, and we now prove that any such set with a finite positive upper and lower bound is the range of a suitable graph.