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arXiv · 2601.07828

On the range of two-distance graphs

Péter Ágoston
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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.

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Péter Ágoston. 2026-01-12. On the range of two-distance graphs. https://arxiv.org/abs/2601.07828

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