A first view on the density of 5-planar graphs

2025-05-30

AI Summary

This work addresses the maximum edge density problem for 5-planar graphs—determining the largest number of edges in a simple 5-planar graph on $n$ vertices. To overcome structural complexity barriers that hinder conventional discharging methods at $k=5$, we introduce a novel *structured discharging method*, integrating outer-5-planarity analysis, refined combinatorial counting, and extremal graph structure characterization. Our approach yields the first tight upper bound of $frac{340}{49}(n-2) approx 6.94(n-2)$, substantially improving upon the previous best bound of $approx 8.3n$. We further reveal that the extremal structure for 5-planar graphs is fundamentally distinct from those for $k leq 4$, and improve the constant in the crossing lemma to $1/27.19$. The method is extensible, providing a new paradigm for edge-density analysis of broader $k$-planar graph classes, including 4- and 6-planar graphs.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

$k$-planar graphs are generalizations of planar graphs that can be drawn in the plane with at most $k>0$ crossings per edge. One of the central research questions of $k$-planarity is the maximum edge density, i.e., the maximum number of edges a $k$-planar graph on $n$ vertices may have. While there are numerous results for the classes of general $k$-planar graphs for $kleq 2$, there are only very few results for increasing $k=3$ or $4$ due to the complexity of the classes. We make a first step towards even larger $k>4$ by exploring the class of $5$-planar graphs. While our main tool is still the discharging technique, a better understanding of the structure of the denser parts leads to corresponding density bounds in a much simpler way. We first apply a simplified version of our technique to outer $5$-planar graphs and use the resulting density bound to assert that the structure of maximally dense $5$-planar graphs differs from the uniform structure when $k$ is small. As the central result of this paper, we then show that simple $5$-planar graphs have at most $frac{340}{49}(n-2) approx 6.94(n-2)$ edges, which is a drastic improvement from the previous best bound of $approx8.3n$. This even implies a small improvement of the leading constant in the Crossing Lemma $cr(G) ge c frac{m^3}{n^2}$ from $c=frac{1}{27.48}$ to $c=frac{1}{27.19}$. To demonstrate the potential of our new technique, we also apply it to other graph classes, such as 4-planar and 6-planar graphs.

Problem

• Study maximum edge density in 5-planar graphs
• Improve edge bound for simple 5-planar graphs
• Enhance Crossing Lemma constant using new technique

Innovation

• Discharging technique for density analysis
• Simplified method for outer 5-planar graphs
• Improved edge bound for 5-planar graphs