Uniform Geodesic Drawings of Graphs
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866916018069176320 |
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| author | Lepsveridze, Saba Solé-Pi, Oriol |
| author_facet | Lepsveridze, Saba Solé-Pi, Oriol |
| contents | We study crossing numbers of dense graph drawings whose vertices are uniformly distributed either on the unit sphere or in a compact convex planar domain. We prove a sharp inequality for weighted geodesic drawings on $\mathbb S^2$ in a continuous setting: among all measurable edge arrangements of a fixed density, the amount of crossings is minimized by connecting pairs of points within a fixed distance threshold. We also prove a planar analogue for straight-line drawings in convex planar domains. We transfer these continuous results to finite graphs using a smoothing argument. In the small density limit, we recover the conjectured midrange crossing constant lower bound of $8/(9π^2)$ for this restricted model. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_16700 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Uniform Geodesic Drawings of Graphs Lepsveridze, Saba Solé-Pi, Oriol Combinatorics 05C10 (Primary) 52C10 (Secondary) We study crossing numbers of dense graph drawings whose vertices are uniformly distributed either on the unit sphere or in a compact convex planar domain. We prove a sharp inequality for weighted geodesic drawings on $\mathbb S^2$ in a continuous setting: among all measurable edge arrangements of a fixed density, the amount of crossings is minimized by connecting pairs of points within a fixed distance threshold. We also prove a planar analogue for straight-line drawings in convex planar domains. We transfer these continuous results to finite graphs using a smoothing argument. In the small density limit, we recover the conjectured midrange crossing constant lower bound of $8/(9π^2)$ for this restricted model. |
| title | Uniform Geodesic Drawings of Graphs |
| topic | Combinatorics 05C10 (Primary) 52C10 (Secondary) |
| url | https://arxiv.org/abs/2605.16700 |