On the range of two-distance graphs
Combinatorics
2026-01-13 v1
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 it can be mapped into the plane with all vertices going to distinct points, the red edges to segments of length and the blue edges to segments of length . We define the range of this graph to be the set of such numbers . 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.
Keywords
Cite
@article{arxiv.2601.07828,
title = {On the range of two-distance graphs},
author = {Péter Ágoston},
journal= {arXiv preprint arXiv:2601.07828},
year = {2026}
}