Algorithmic exploration of the unit distance problem in the rational plane

Panteleimon Rodis

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arXiv · 2606.29415

Algorithmic exploration of the unit distance problem in the rational plane

Panteleimon Rodis

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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.

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Panteleimon Rodis. 2026-06-28. Algorithmic exploration of the unit distance problem in the rational plane. https://arxiv.org/abs/2606.29415

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