Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Improving the Crossing Lemma by Characterizing Dense 2-Planar and 3-Planar Graphs

Aaron Büngener, Michael Kaufmann

Abstract

The classical Crossing Lemma by Ajtai et al.~and Leighton from 1982 gave an important lower bound of $c \frac{m^3}{n^2}$ for the number of crossings in any drawing of a given graph of $n$ vertices and $m$ edges. The original value was $c= 1/100$, which then has gradually been improved. Here, the bounds for the density of $k$-planar graphs played a central role. Our new insight is that for $k=2,3$ the $k$-planar graphs have substantially fewer edges if specific local configurations that occur in drawings of $k$-planar graphs of maximum density are forbidden. Therefore, we are able to derive better bounds for the crossing number $\text{cr}(G)$ of a given graph $G$. In particular, we achieve a bound of $\text{cr}(G) \ge \frac{37}{9}m-\frac{155}{9}(n-2)$ for the range of $5n < m \le 6n$, while our second bound $\text{cr}(G) \ge 5m - \frac{203}{9}(n-2)$ is even stronger for larger $m>6n$. For $m > 6.77n$, we finally apply the standard probabilistic proof from the BOOK and obtain an improved constant of $c>1/27.48$ in the Crossing Lemma. Note that the previous constant was $1/29$. Although this improvement is not too impressive, we consider our technique as an important new tool, which might be helpful in various other applications.

Improving the Crossing Lemma by Characterizing Dense 2-Planar and 3-Planar Graphs

Abstract

The classical Crossing Lemma by Ajtai et al.~and Leighton from 1982 gave an important lower bound of for the number of crossings in any drawing of a given graph of vertices and edges. The original value was , which then has gradually been improved. Here, the bounds for the density of -planar graphs played a central role. Our new insight is that for the -planar graphs have substantially fewer edges if specific local configurations that occur in drawings of -planar graphs of maximum density are forbidden. Therefore, we are able to derive better bounds for the crossing number of a given graph . In particular, we achieve a bound of for the range of , while our second bound is even stronger for larger . For , we finally apply the standard probabilistic proof from the BOOK and obtain an improved constant of in the Crossing Lemma. Note that the previous constant was . Although this improvement is not too impressive, we consider our technique as an important new tool, which might be helpful in various other applications.
Paper Structure (10 sections, 19 theorems, 2 equations, 2 figures)

This paper contains 10 sections, 19 theorems, 2 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Definitions and Notation
  3. Results
  4. Proof of Theorems \ref{['th:density-2-planar-forbidden-configuration']} and \ref{['th:density-3-planar-forbidden-configuration']}
  5. Proof and Discharging for Theorem \ref{['th:density-2-planar-forbidden-configuration']}
  6. Proof and Discharging for Theorem \ref{['th:density-3-planar-forbidden-configuration']}
  7. Proof of Theorem \ref{['th:linear-bounds']}
  8. Discussion
  9. Details for Section \ref{['sec:charging']}
  10. Details for Section \ref{['sec:linear-bounds']}

Key Result

Theorem 1

Any graph $G$ with $n\ge 3$ vertices that admits a 2-planar $F^2_5$-free drawing has at most $4.5(n-2)$ edges. If the drawing is also $F^2_6$-free, then $G$ has at most $\frac{13}{3}(n-2)$ edges.

Figures (2)

  • Figure 1: (a) A full 2-planar pentagon $F^2_5$, (b) a full 2-planar hexagon $F^2_6$ and (c) a full 3-planar hexagon $F_3^6$ with their boundaries (dashed).
  • Figure 2: Illustrations of the defined neighborhood-relations. (a) From top to bottom: The faces $f$ and $f'$ are 0-neighbors, 1-neighbors, 2-neighbors resp. (b) The 0-pentagon $f_2$ is the wedge-neighbor of the 1-triangle $f_0$ at its edge $e_0$. (c) The faces $f$ and $f'$ are vertex-neighbors.

Theorems & Definitions (19)

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