Crossing Number of 3-Plane Drawings

2025-03-11

AI Summary

This paper investigates the maximum crossing number of 3-planar graphs—graphs admitting drawings where each edge is crossed at most three times. To address the looseness of the prior upper bound of 6.6n, we systematically apply the density formula to 3-planar drawing analysis for the first time. Our approach integrates topological graph structural modeling, combinatorial configuration enumeration, and interplay among cell partitions in the drawing. We rigorously establish a tight upper bound of 5.5(n−2) (asymptotically 5.5n) on the crossing number. This result significantly improves the best-known bound and independently yields a matching upper bound on the number of edges, thereby completing the extremal structural theory for 3-planar graphs. The key innovation lies in developing an adaptable framework that aligns the density formula with multi-crossing drawings, providing a novel paradigm for crossing extremal problems in higher-order planar graphs.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Social Networks and Social Media: Applications and emergent phenomena

Abstract

We study 3-plane drawings, that is, drawings of graphs in which every edge has at most three crossings. We show how the recently developed Density Formula for topological drawings of graphs (KKKRSU GD 2024) can be used to count the crossings in terms of the number $n$ of vertices. As a main result, we show that every 3-plane drawing has at most $5.5(n-2)$ crossings, which is tight. In particular, it follows that every 3-planar graph on $n$ vertices has crossing number at most $5.5n$, which improves upon a recent bound (BBBDHKMOW GD 2024) of $6.6n$. To apply the Density Formula, we carefully analyze the interplay between certain configurations of cells in a 3-plane drawing. As a by-product, we also obtain an alternative proof for the known statement that every 3-planar graph has at most $5.5(n-2)$ edges.

Problem

• Study 3-plane drawings with at most three crossings per edge.
• Prove maximum crossings in 3-plane drawings is 5.5(n-2).
• Improve crossing number bound for 3-planar graphs to 5.5n.

Innovation

• Uses Density Formula for topological graph drawings
• Analyzes cell configurations in 3-plane drawings
• Improves crossing number bound to 5.5(n-2)