Crossing Lemma: Lower Bounds in Graph Drawing

• Crossing Lemma is a foundational inequality in graph theory that establishes a cubic lower bound on edge crossings in dense graphs using probabilistic methods.
• It plays a crucial role in applications such as VLSI design, incidence geometry, and algorithmic graph drawing by setting complexity thresholds in circuit layouts and geometric configurations.
• Recent refinements extend the lemma to multigraphs, weighted graphs, and surfaces beyond the plane, improving constants and broadening its theoretical and practical reach.

The Crossing Lemma is a foundational inequality in extremal graph theory and discrete geometry, providing a lower bound on the number of edge crossings in any drawing of a sufficiently dense graph in the plane. Its classical form, refinements, and generalizations exert wide-ranging influence on incidence geometry, algorithmic graph drawing, VLSI layout, and the broader theory of geometric and topological graphs.

1. Classical Crossing Lemma: Statement and “Book Proof”

The original Crossing Lemma was proved independently by Ajtai–Chvátal–Newborn–Szemerédi and by Leighton in the early 1980s. It asserts that for any simple graph $G$ with $n$ vertices and $e \geq 4n$ edges, the (minimum) number of edge crossings $\operatorname{cr}(G)$ in any drawing of $G$ in the plane satisfies

$$\operatorname{cr}(G) \geq \frac{1}{64} \cdot \frac{e^3}{n^2}.$$

A canonical proof, frequently referred to as “from The Book” (Chazelle–Sharir–Welzl), proceeds in two steps:

1. An elementary bound:

$\operatorname{cr}(G) \geq e - 3n.$

1. A probabilistic argument:
   • Randomly select each vertex independently with probability $p$.
   • The expected sizes: vertices $pn$, edges $p^2e$, crossings $p^4\operatorname{cr}(G)$.
   • Choosing $p = \frac{4n}{e}$ balances the terms, yielding the cubic lower bound.

These arguments not only establish the base result but also set the template for almost all subsequent improvements and generalizations.

2. Application Domains

The Crossing Lemma’s broad applicability includes:

• VLSI and Network Design: Imposing lower bounds on the wiring or edge-complexity in circuit layout and network drawings.
• Incidence Geometry: Providing the key technical step in Szekély’s proof of the Szemerédi–Trotter theorem, establishing sharp upper bounds on incidences between points and lines.
• Distance Graphs: Underpinning bounds on unit distances (Elekes, Solymosi) and distinct distances in the plane.
• Forbidden Subgraph Problems: Providing thresholds for monotone properties and applications via bisection width and related decompositions.
• Algorithmic Graph Drawing and Approximation: Certifying the density threshold past which efficient crossing minimization and approximation are impossible for general graphs.

The lemma and its variants are therefore frequently invoked in “divide-and-conquer” or probabilistic settings in combinatorial geometry and computational topology.

3. Improved Crossing Lemma Constants and Midrange Crossing Constant

Refining the constants in the lemma is an active area:

• For $e \geq 6.95n$,

$$\operatorname{cr}(G) \geq \frac{1}{29} \frac{e^3}{n^2}.$$

• For $e \geq 17.16n$,

$$\operatorname{cr}(G) \geq \frac{1}{31.1}\frac{e^3}{n^2}.$$

• The latest improvement (Büngener–Kaufmann) yields for $e \geq 6.77n$,

$$\operatorname{cr}(G) \geq \frac{1}{27.48}\frac{e^3}{n^2}.$$

The midrange crossing constant

$$\lim_{n \to \infty,\, n \ll e \ll n^2} \kappa(n,e) \frac{n^2}{e^3} = c,$$

has been estimated as $0.036 < c < 0.09$. The value of the best possible constant is conjectured to be close to these bounds and is crucial for sharp quantitative incidence results (Toth, 17 Sep 2025).

4. Crossing Number Variants

Multiple variants of crossing number exist and satisfy their own “crossing lemma” formulations:

• Pair Crossing Number: Counts pairs of edges that cross, not individual crossing points. In good drawings, these coincide, but care must be taken regarding higher-multiplicity crossings.
• Odd Crossing Number: Counts only pairs of edges that cross an odd number of times. By the Hanani–Tutte theorem, if every pair of independent edges crosses evenly, the graph is planar.
• Adjacency Rules (+, $0$, $-$): Modulate how adjacent crossings are counted, as captured in various tables in the literature (Toth, 17 Sep 2025).

The lemma holds in the basic form for all variants, but improved constants (via probabilistic sampling arguments) only apply for those variants where adjacent-edge crossings are suitably controlled. For example, the $pcr_+$ and odd crossing number with the extra plus rule permit improved bounds, while others do not.

5. Crossing Lemma for Multigraphs

The extension to multigraphs without additional constraints fails: parallel edges can evade any crossing. Several remedies restore cubic lower bounds:

• Bounded Multiplicity: If no pair of vertices is connected by more than $m$ edges,

$$\operatorname{cr}(G) \geq \frac{1}{64}\frac{e^3}{m n^2}.$$

• “Branching” or Separated Drawings: If each pair of parallel edges forms a lens that contains at least one vertex in its interior, edges are non-homotopic, and every pair of independent edges crosses at most once, the bound reverts to

$\operatorname{cr}(G) \geq c'\frac{e^3}{n^2},$

for some $c' > 0$ independent of $m$ (Pach et al., 2018, Kaufmann et al., 2018).

• Non-homotopic Drawings: If no two parallel edges are homotopic (i.e., cannot be deformed into each other in the punctured plane), then for $m > 5n$,

$$\operatorname{cr}(G) \geq \frac{1}{40}\frac{m^2}{n}.$$

These structural conditions are particularly relevant in VLSI and physical modeling, where “empty lenses” or redundant bundles are physically or combinatorially undesirable.

6. Generalizations: Surfaces, Weighted and Random Settings

Recent works have pushed the lemma beyond planar graphs:

• Drawings on Surfaces: For multi-graphs drawn on orientable surfaces, with edges as non-homotopic simple arcs that pairwise cross at most $k$ times ($k$-systems), the lower bound is

$$\operatorname{Cr}(A) \geq c_k \frac{m^{2+1/k}}{n^{1+1/k}},$$

with genus and puncture-dependent refinements (Hubard et al., 22 Mar 2024).

• Random Graphs and Weighted Crossings: For edge-weighted graphs with independent random weights (e.g., uniformly in [0,1]), the expected weighted crossing number satisfies

$$\mathbb{E}[\operatorname{cr}(G, w)] \geq \frac{32}{31827} \frac{m^3}{n^2}.$$

This robustifies the classical bound under random perturbations of edge importance (Mohar et al., 2010).

• Odd Crossing Numbers and Even Adjacencies: When adjacent edges are permitted to cross an even number of times, the crossing lemma constant may be improved, e.g.,

${_{*}}(G) \geq \frac{1}{54} \frac{m^3}{n^2},$

for $m \geq 6n$ (Karl et al., 2022).

These extensions tie the combinatorics of dense (multi-)graphs to geometric and topological parameters of their drawings and have applications in high-genus embeddability, probabilistic combinatorics, and algorithmic graph theory.

7. Incidence Geometry, Structure Theorems, and Further Directions

Through Szekély’s paradigm, the crossing lemma serves as the key tool for converting geometric configurations into combinatorial incidence bounds, such as the Szemerédi–Trotter theorem (bounding point-line incidences). Further refinements via monotone property versions, bisection width arguments, and structure theorems for dense $k$-planar graphs (e.g., improved constants when local forbidden configurations are excluded (Büngener et al., 3 Sep 2024)) drive advances in extremal and algorithmic geometry.

While steady progress in constants and generalizations continues, several key directions remain:

• Precise constants for the midrange regime, especially for sparse but not too sparse graphs.
• Extensions to higher-dimensional crossing phenomena (e.g., space crossing numbers (Bukh et al., 2011)) and embeddings in surfaces of higher genus.
• Richer structure theory for drawings with density and local configuration constraints in beyond-planar and multi-layer models (Angelini et al., 2020).
• Advances in understanding rectilinear crossing numbers in forbidden minor and bounded treewidth graph families, where tight linear bounds have only recently been achieved (Dujmović et al., 23 Feb 2024).

Key Formulas and Variants

Setting	Crossing Lemma Lower Bound	Condition(s)
Simple Planar Graph	$$\operatorname{cr}(G)\geq\frac{1}{64}\frac{e^3}{n^2}$$	$e \geq 4n$
Improved Constant	$$\operatorname{cr}(G)\geq\frac{1}{27.48}\frac{e^3}{n^2}$$	$e \geq 6.77n$
Multigraphs (mult. $\leq m$)	$$\operatorname{cr}(G)\geq\frac{1}{64}\frac{e^3}{m n^2}$$	$e \geq 4mn$
Separated Multigraphs	$\operatorname{cr}(G)\geq c' \frac{e^3}{n^2}$	“no empty lens”, single-crossing
Weighted (uniform [0,1])	$$\mathbb{E}[\operatorname{cr}(G,w)] \geq \frac{32}{31827}\frac{m^3}{n^2}$$	independent uniform edge weights
$k$-system, Surface Genus 0	$$\operatorname{Cr}(A) \geq c_k \frac{m^{2 + 1/k}}{n^{1 + 1/k}}$$	non-homotopic arcs, $\leq k$ crossings

Conclusion

The Crossing Lemma, in its original form and via numerous extensions, is the central asymptotic tool for certifying the complexity of dense graphs, the limitations of certain drawing styles, and the transition between planar and nonplanar geometry. Its influence pervades extremal combinatorics, incidence geometry, and computational applications in graph drawing, with generalizations continuing to emerge as the structure of geometric, topological, and weighted graphs is further explored (Toth, 17 Sep 2025).