On the range of two-distance graphs

TL;DR

This paper investigates the set of possible distances in two-colored graphs mapped into the plane, proving that any finite interval of such distances can be realized by an appropriately constructed graph.

Contribution

It establishes that any finite interval of distances between 0 and infinity can be realized as the range of a two-distance graph with specific coloring.

Findings

Range of two-distance graphs consists of finitely many algebraic intervals.

Any finite positive interval can be realized as the range of a suitable graph.

The set of possible distances is flexible within finite bounds.

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.