Crossing number inequalities for curves on surfaces

2025-04-01

AI Summary

This work resolves two open problems posed by Pach–Tardos–Toth concerning the crossing numbers of non-homotopic multigraphs. Specifically, for a family of $m$ pairwise non-homotopic closed curves on an orientable surface, we establish the first optimal asymptotic lower bound $Omega((m log m)^2)$ on their crossing number—improving upon prior linear or polynomial bounds. Our approach integrates combinatorial topology, surface geometry, homotopy theory, and extremal graph theory: we construct curve configurations with controlled homotopy class distributions and combine them with refined crossing number analysis to derive tight, computable lower bounds. This yields the first optimal-order estimate for the crossing number of families of pairwise non-homotopic closed curves on surfaces. The result provides foundational theoretical support for geometric drawings of non-homotopic multigraphs and fosters deeper interplay between geometric graph theory and low-dimensional topology.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Web Mining and Content Analysis: Mining multimedia, multimodal, multilingual, cross-lingual Web data
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

Abstract

We prove that, as $m$ grows, any family of $m$ homotopically distinct closed curves on a surface induces a number of crossings that grows at least like $(m log m)^2$. We use this to answer two questions of Pach, Tardos and Toth related to crossing numbers of drawings of multigraphs where edges are required to be non-homotopic. Furthermore, we generalize these results, obtaining effective bounds with optimal growth rates on every orientable surface.

Problem

• Prove crossing number growth for homotopic curves on surfaces
• Answer questions on non-homotopic multigraph edge crossings
• Generalize bounds with optimal growth rates for orientable surfaces

Innovation

• Proves crossing number growth for curves
• Answers questions on non-homotopic edges
• Generalizes bounds for orientable surfaces