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| Main Author: | |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2601.07828 |
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| _version_ | 1866911369962455040 |
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| author | Ágoston, Péter |
| author_facet | Ágoston, Péter |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_07828 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the range of two-distance graphs Ágoston, Péter Combinatorics 05C10 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. |
| title | On the range of two-distance graphs |
| topic | Combinatorics 05C10 |
| url | https://arxiv.org/abs/2601.07828 |