🤖 AI Summary
This study addresses the problem of maximizing the density of unit distance graphs in the rational plane. To overcome the limitations of existing constructions—often constrained by conventional grid structures and unable to surpass known theoretical lower bounds—the authors propose a local breadth-first search algorithm tailored to the rational plane. This approach dispenses with fixed grid constraints and efficiently generates universal unit distance graphs within bounded finite regions. The resulting graphs exhibit significantly higher density than recent theoretical lower bounds, achieving a scaling exponent that improves upon the current best-known results. The method thus offers both a novel perspective and an effective computational tool for advancing lower-bound constructions in this domain.
📝 Abstract
This paper presents reproducible experimental evidence on unit-distance graph density that surpasses recent theoretical lower bounds. 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. Furthermore, the scaling exponent of the generated graph surpasses recent results.