Technology Category
Constraint Satisfaction and Optimization: Other Foundations of Constraint SatisfactionSearch and Optimization: Combinatorial OptimizationKnowledge Representation and Reasoning: Computational Complexity of Reasoning
Crossing lemmas for $k$-systems of arcs
🤖 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
Application Category
📝 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
Research questions and friction points this paper is trying to address.
Generalize crossing lemma for multi-graphs
Study non-homotopic simple arcs on surfaces
Analyze arcs crossing at most k times
Innovation
Methods, ideas, or system contributions that make the work stand out.
Generalizes crossing lemma for multi-graphs
Uses non-homotopic simple arcs
Limits pairwise crossings to k times
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