Crossing numbers of dense graphs on surfaces

2025-06-11

AI Summary

This work addresses the asymptotic characterization of the crossing number of dense graphs embedded on high-genus surfaces. We resolve a conjecture by Shahrokhi, Székely, and Vrt'o by establishing the first Ω(m²/(g log² g)) lower bound—up to poly-logarithmic factors—thereby refuting their 1996 upper-bound conjecture. Furthermore, we construct an explicit family of hyperbolic surfaces and employ probabilistic embedding techniques combined with asymptotic analysis to obtain a matching O(m²/(g log² g)) upper bound. Our approach integrates extremal graph theory, hyperbolic geometry, surface topology, and random embedding theory. Crucially, the upper and lower bounds agree up to constant factors, yielding the first exact asymptotic characterization—within constants—of the crossing number for dense graphs on surfaces of genus g as g → ∞.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Graphical Models
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Social Networks and Social Media: Applications and emergent phenomena

Abstract

In this paper, we provide upper and lower bounds on the crossing numbers of dense graphs on surfaces, which match up to constant factors. First, we prove that if $G$ is a dense enough graph with $m$ edges and $Sigma$ is a surface of genus $g$, then any drawing of $G$ on $Sigma$ incurs at least $Omega left(frac{m^2}{g} log ^2 g ight)$ crossings. The poly-logarithmic factor in this lower bound is new even in the case of complete graphs and disproves a conjecture of Shahrokhi, Sz'ekely and Vrt'o from 1996. Then we prove a geometric converse to this lower bound: we provide an explicit family of hyperbolic surfaces such that for any graph $G$, sampling the vertices uniformly at random on this surface and connecting them with shortest paths yields $Oleft(frac{m^2}{g} log ^2 g ight)$ crossings in expectation.

Problem

• Bounds on crossing numbers of dense graphs on surfaces
• Disproving a 1996 conjecture about graph crossings
• Geometric construction for expected crossing numbers

Innovation

• Proves lower bounds for dense graph crossings
• Disproves 1996 conjecture with poly-logarithmic factor
• Uses hyperbolic surfaces for geometric converse