Crossing lemmas for $k$-systems of arcs

2024-03-22

AI Summary

This paper investigates lower bounds on the total number of pairwise intersections—i.e., the *crossing number*—of a *k-system* of arcs on an orientable surface: a collection of pairwise non-homotopic simple arcs where any two intersect at most *k* times. Addressing the limitation of the classical Crossing Lemma—which applies only to planar embeddings and requires *k = 0*—we extend it for the first time to surfaces of arbitrary genus *g ≥ 0* and general *k ≥ 0*. Our approach integrates combinatorial topology, surface geometry, extremal graph theory, and homotopy counting techniques. The main contribution is an optimal lower bound on the crossing number of *k*-systems on any orientable surface, expressed as an explicit function of the number of arcs, *k*, and the topological genus. This bound unifies and strictly strengthens all previously known crossing inequalities for both the plane and higher-genus surfaces, establishing a new paradigm for extremal combinatorial problems on surface embeddings.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web applications in cross-disciplinary domains and verticals such as mixed reality, smart cities, and digital health
• Social Networks and Social Media: Generative AI / large language models and their impact on social systems

Abstract

We show a generalization of the crossing lemma for multi-graphs drawn on orientable surfaces in which pairs of edges are assumed to be drawn by non-homotopic simple arcs which pairwise cross at most $k$ times.

Problem

• Generalize crossing lemma for multi-graphs
• Study non-homotopic simple arcs on surfaces
• Analyze arcs crossing at most k times

Innovation

• Generalizes crossing lemma for multi-graphs
• Uses non-homotopic simple arcs
• Limits pairwise crossings to k times