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On $k$-planar Graphs without Short Cycles

Michael A. Bekos, Prosenjit Bose, Aaron Büngener, Vida Dujmović, Michael Hoffmann, Michael Kaufmann, Pat Morin, Saeed Odak, Alexandra Weinberger

TL;DR

This work investigates the edge-density of $k$-planar graphs when short cycles are forbidden, focusing on $C_3$-free, $C_4$-free, and girth $5$ cases for $k\in\{1,2,3\}$ and extending to general $k$ via the Crossing Lemma. The authors develop and combine three core techniques—the discharging method, a non-planar density formula, and refined crossing-number bounds for $2$- and $3$-planar graphs—to derive explicit linear-in-$n$ bounds $|E|\le c\,n$ for each setting and small $k$, and to obtain $|E|\le c\sqrt{k}\,n$ for larger $k$. They provide tight or near-tight results for 1-, 2-, and 3-planar graphs under the forbidden substructures, including constructive lower bounds and detailed upper bounds for various girth constraints, along with crossings-based arguments to lift bounds to higher $k$. The findings advance the understanding of density in beyond-planar graphs and offer methods that connect edge-density, planarity relaxations, and crossing numbers, with implications for crossing lemmas, graph drawing, and related algorithmic questions. The paper also outlines open problems, such as closing gaps between bounds and exploring higher girth or different forbidden-substructure regimes.

Abstract

We study the impact of forbidding short cycles to the edge density of $k$-planar graphs; a $k$-planar graph is one that can be drawn in the plane with at most $k$ crossings per edge. Specifically, we consider three settings, according to which the forbidden substructures are $3$-cycles, $4$-cycles or both of them (i.e., girth $\ge 5$). For all three settings and all $k\in\{1,2,3\}$, we present lower and upper bounds on the maximum number of edges in any $k$-planar graph on $n$ vertices. Our bounds are of the form $c\,n$, for some explicit constant $c$ that depends on $k$ and on the setting. For general $k \geq 4$ our bounds are of the form $c\sqrt{k}n$, for some explicit constant $c$. These results are obtained by leveraging different techniques, such as the discharging method, the recently introduced density formula for non-planar graphs, and new upper bounds for the crossing number of $2$-- and $3$-planar graphs in combination with corresponding lower bounds based on the Crossing Lemma.

On $k$-planar Graphs without Short Cycles

TL;DR

This work investigates the edge-density of -planar graphs when short cycles are forbidden, focusing on -free, -free, and girth cases for and extending to general via the Crossing Lemma. The authors develop and combine three core techniques—the discharging method, a non-planar density formula, and refined crossing-number bounds for - and -planar graphs—to derive explicit linear-in- bounds for each setting and small , and to obtain for larger . They provide tight or near-tight results for 1-, 2-, and 3-planar graphs under the forbidden substructures, including constructive lower bounds and detailed upper bounds for various girth constraints, along with crossings-based arguments to lift bounds to higher . The findings advance the understanding of density in beyond-planar graphs and offer methods that connect edge-density, planarity relaxations, and crossing numbers, with implications for crossing lemmas, graph drawing, and related algorithmic questions. The paper also outlines open problems, such as closing gaps between bounds and exploring higher girth or different forbidden-substructure regimes.

Abstract

We study the impact of forbidding short cycles to the edge density of -planar graphs; a -planar graph is one that can be drawn in the plane with at most crossings per edge. Specifically, we consider three settings, according to which the forbidden substructures are -cycles, -cycles or both of them (i.e., girth ). For all three settings and all , we present lower and upper bounds on the maximum number of edges in any -planar graph on vertices. Our bounds are of the form , for some explicit constant that depends on and on the setting. For general our bounds are of the form , for some explicit constant . These results are obtained by leveraging different techniques, such as the discharging method, the recently introduced density formula for non-planar graphs, and new upper bounds for the crossing number of -- and -planar graphs in combination with corresponding lower bounds based on the Crossing Lemma.
Paper Structure (17 sections, 21 theorems, 12 equations, 2 figures, 2 tables)

This paper contains 17 sections, 21 theorems, 12 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminary Techniques and Tools
  3. The Discharging Method
  4. The Crossing Lemma
  5. Upper Bounds on the Crossing Number of 2- and 3-planar graphs
  6. 1-planar graphs
  7. C${}_\mathbf{3}$-free 1-planar graphs
  8. C${}_\mathbf{4}$-free 1-planar graphs
  9. 1-planar graphs of girth 5
  10. 2-planar graphs
  11. C${}_\mathbf{3}$-free 2-planar graphs
  12. C${}_\mathbf{4}$-free 2-planar graphs
  13. 2-planar graphs of girth 5
  14. 3-planar graphs
  15. C${}_\mathbf{3}$-free 3-planar graphs
  16. ...and 2 more sections

Key Result

Theorem 1

Let $\mathcal{X}$ be a hereditaryClosed under taking induced subgraphs. graph family and $a,b\in\mathds{R}$ such that for every $H\in\mathcal{X}$ with $\nu$ vertices and $\mu$ edges we have $\mathrm{cr}(H)\ge a\mu-b\nu$. Then for every graph $G\in\mathcal{X}$ with $n$ vertices and $m$ edges with $2a

Figures (2)

  • Figure 6: Triangles in the planarization of $1$-planar graphs of girth $5$.
  • Figure 9: A $2$-plane graph with $\approx 2.5n$ edges, shown red and blue. Gray shows the grid only.

Theorems & Definitions (22)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Theorem 9
  • Theorem 10
  • ...and 12 more