Algorithmic exploration of the unit distance problem in the rational plane
Abstract
This paper presents reproducible experimental evidence on unit-distance graph density. Our approach is based on a novel algorithmic exploration of the rational plane for the generation of unit-distance graphs. An efficient algorithm for this utility must perform a local-breadth search on a bounded and finite set of elements and generate a graph that potentially encompasses the general properties of a unit-distance graph, not affected by restrictions on its generation. To this end, we show that our approach accomplishes this purpose by overcoming the limitations of grid-based structures, used in the literature for generating unit-distance graphs. When applied to large unit-distance graphs, our method generates large graphs with a scaling exponent of 1.17, that surpasses recent theoretical predictions on graph density growth.
Related
- Improved dimension bounds for Falconer's distance problem in the plane
- On Free and Nearly Free Conjecture of Rational Unibranched Projective Plane Curves
- Nonexistence of degree two rational multisections of conic bundles over the plane
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- A constant-factor approximation of the Gromov-Hausdorff distance in the plane
BibTeX
@misc{rodis2026algorithmic,
title = {{Algorithmic exploration of the unit distance problem in the rational plane}},
author = {Panteleimon Rodis},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.29415},
eprint = {2606.29415},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.29415},
}
RIS
TY - UNPB
TI - Algorithmic exploration of the unit distance problem in the rational plane
AU - Panteleimon Rodis
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2606.29415
UR - https://arxiv.org/abs/2606.29415
AB - This paper presents reproducible experimental evidence on unit-distance graph density. Our approach is based on a novel algorithmic exploration of the rational plane for the generation of unit-distance graphs. An efficient algorithm for this utility must perform a local-breadth search on a bounded and finite set of elements and generate a graph that potentially encompasses the general properties of a unit-distance graph, not affected by restrictions on its generation. To this end, we show that our approach accomplishes this purpose by overcoming the limitations of grid-based structures, used in the literature for generating unit-distance graphs. When applied to large unit-distance graphs, our method generates large graphs with a scaling exponent of 1.17, that surpasses recent theoretical predictions on graph density growth.
ER -