---
title: Minimum Crossing Numbers in Graph Theory
url: https://www.emergentmind.com/topics/minimum-crossing-numbers
type: topic
---

# Minimum Crossing Numbers in Graph Theory

The minimum crossing number of a graph—the least number of edge crossings in any planar drawing—is a cornerstone parameter in topological graph theory and discrete geometry, deeply connected to extremal combinatorics, algorithmic complexity, surface topology, and structural graph theory. Its study encompasses a variety of drawing models (topological, rectilinear, convex), classes of graphs (minor-closed, bounded treewidth, multipartite, crossing-critical), and complexity-theoretic as well as structural phenomena.

## 1. Foundational Definitions and Crossing Lemmas

For a finite simple graph $G$, the (standard) crossing number, denoted $\mathrm{cr}(G)$, is the minimum number of pairwise edge crossings over all drawings of $G$ in the plane such that vertices are mapped to points, edges to simple curves, and no three edges cross at a single non-endpoint. Important variants include the rectilinear crossing number $\overline{\mathrm{cr}}(G)$ (edges drawn as straight segments), convex crossing number (vertices in convex position), and minor crossing number $\mathrm{mcr}(G)$ (minimum of $\mathrm{cr}(H)$ over all $H$ such that $G$ is a minor of $H$) [1807.11617][1111.6024].

The quantitative backbone of the subject is the **Crossing Lemma**, originally due to Ajtai, Chvátal, Newborn, Szemerédi, and independently Leighton, giving for simple graphs:
\[
e \geq 4n \implies \mathrm{cr}(G) \geq \frac{1}{64} \frac{e^3}{n^2}
\]
where $n=|V(G)|$, $e=|E(G)|$ [2509.14074]. The lower bound is tight up to constant factors, as demonstrated by random geometric constructions. Numerous refinements adjust constants (down to $1/27.48$) for larger minimal degree or alternative drawing constraints.

For multigraphs, the crossing lemma generalizes to
\[
\mathrm{cr}(G) \geq \frac{1}{64} \frac{e^3}{mn^2}
\]
where each pair of vertices is joined by at most $m$ edges [2509.14074]. Several natural restrictions on the drawing (separated, locally-starlike, single-crossing) are necessary for meaningful lower bounds in the multigraph setting.

## 2. Midrange and Asymptotic Crossing Estimates

Given the intractability of exact calculation for large classes, structural bounds focus on extremal asymptotics. For large $n$ and edge numbers $n \ll e \ll n^2$, the **midrange crossing constant** $C$ is defined as
\[
C = \lim_{n \to \infty,\ n \ll e \ll n^2} \kappa(n,e) n^2 / e^3
\]
where $\kappa(n,e)$ is the minimum crossing number over all $n$-vertex graphs with at least $e$ edges [1811.08071]. Current best bounds place $C$ in $[0.034, 0.09]$. This convergence mechanism extends to broad graph classes (PST-classes), including bipartite and $K_t$-free graphs, and to rectilinear crossing numbers [1811.08071]. For bipartite graphs, Angelini et al. proved $C_{\mathrm{bip}} \geq 16/289 \approx 0.055$.

The **Pach–Spencer–Tóth theorem** resolves a general crossing lower bound for graphs with monotone forbidden substructures (e.g., $C_{2k}$-free graphs). If every subgraph $H\subseteq G$ satisfies $e(H) \leq A n(H)^{1+\alpha}$ for some $A$, then for $e \geq c n$:
\[
\mathrm{cr}(G) \geq c' \frac{e^{2+1/\alpha}}{n^{1+1/\alpha}}
\]
with effective constants $c, c'$ [2502.02301].

## 3. Structural and Algorithmic Results for Minimum Crossing Numbers

### Minor-Closed and Bounded Structure Classes

For $H$-minor-free graphs of maximum degree $\Delta$ and order $n$, Dujmović, Kawarabayashi, Mohar, and Wood establish
\[
\mathrm{cr}(G) \leq c(H) \cdot \Delta n
\]
with $c(H)$ depending on the excluded minor and arising from the Graph Minors structure theorem [1807.11617]. This is tight up to constants in both parameters. The argument exploits clique-sum decompositions, almost-embeddability, and localized routing strategies. In the rectilinear and convex settings, similar $O(\Delta n)$ bounds are achievable for broad classes—excluding single-crossing minors, or having bounded pathwidth respectively [2402.15034][1807.11617].

For rectilinear crossing number, the result generalizes and sharpens previous $O(\Delta^2 n)$ bounds to $O(\Delta n)$, including for planar-plus-apex and bounded-treewidth families, via a combination of structural theorems and explicit straight-line clique-sum constructions. Constructions based on blow-ups of $K_{3,3}$ demonstrate the asymptotic optimality [2402.15034].

### Sharp Lower Bounds in Specialized Classes

For maximal 1-plane graphs of connectivity $k\in \{3,4,5,6,7\}$ and $n$ vertices, sharp lower bounds are established:
\[
\begin{cases}
\frac{n-2}{3} &k=3\\
\frac{n-2}{2} &k=4\\
\frac{3n-6}{5} &k=5,6\\
\frac{3n}{4} &k=7
\end{cases}
\]
with infinite families achieving these for most $k$ [2504.21558]. These results reveal a hierarchy: increasing connectivity forces higher unavoidable crossing numbers, witnessed by explicit recursive or Cartesian-product-based constructions.

### Crossing-Critical Graphs

A $k$-crossing-critical graph satisfies $\mathrm{cr}(G) \geq k$ and $\mathrm{cr}(G-e) < k$ for every edge $e$. The best known general upper bound for the crossing number of $k$-critical graphs is
\[
\mathrm{cr}(G) \leq 2k + 6\sqrt{k} + 47
\]
improving over the previous $2.5k + 16$ bound of Richter–Thomassen [2003.01477]. The proof employs refined cycle-finding and partitioning procedures in the spirit of the Richter–Thomassen framework, controlled via combinatorial optimization over planarizing edge-sets and delicate redraw-and-count arguments.

The additivity property plays a central role in the construction of complex crossing-critical graphs; the minor crossing number $\mathrm{mcr}(G)$ is additive over arbitrary cuts, and the classical crossing number is additive over edge-cuts of size up to 3, facilitating construction via zip products [1111.6024].

## 4. Explicit and Extremal Constructions

Determining or bounding the minimum crossing number for specific graph families remains a focus.

- For small variants of the hypercube (crossed, locally twisted, Möbius cubes), all are planar for $n \leq 3$, while at $n=4$ the crossing numbers are tightly determined: $cr(Q_4)=cr(\mathrm{CQ}_4)=8$, $cr(\mathrm{LTQ}_4)=cr(\mathrm{MQ}_4)=10$ [1102.3483]. Methods combine region-counting, parity, and combinatorial decomposition.

- For complete multipartite graphs, Zarankiewicz's construction gives a conjectured optimal upper bound for the bipartite and tripartite cases, with tight rectilinear lower bounds achievable via flag algebra techniques [1410.0720]. For balanced $r$-partite graphs, limiting ratios between minimum and maximum crossing numbers are given by $z(r)=3(r^2-r)/8(r^2+r-3)$.

- For the complete graph $K_n$, Hill's conjecture identifies the expected value $Z(n) = \frac14 \lfloor n/2\rfloor \lfloor (n-1)/2\rfloor \lfloor (n-2)/2\rfloor \lfloor (n-3)/2\rfloor$, verified up to $n=12$ in the classical model, and for all $n$ in the $x$-monotone (semisimple and weakly semisimple) setting [1312.3679].

## 5. Crossing Numbers in Multi-Page, Monotone, and Specialized Drawing Models

- The $k$-planar crossing number $cr_k(G)$, minimizing crossings over a decomposition into $k$ edge-disjoint subgraphs, satisfies
\[
cr_k(G) \leq \left(\frac{2}{k^2} - \frac{1}{k^3}\right) cr(G)
\]
for all $k \geq 1$, with tightness up to constants [1611.05746].

- For $k$-page book drawings of $K_n$, recent results specify
\[
\nu_k(K_n) = \frac12 (n-3)(n-2k)
\]
for $2 < n/k \leq 3$, and drastically improve lower bounds for all $k \geq 14$; the gap between lower and upper asymptotic constants is reduced to approximately $0.3246$ in the ratio to $\binom{n}{4}$ [1607.00131]. The analysis leverages tight extremal functions for the maximal number of edges in convex graphs of bounded local crossing number.

- In $x$-monotone or shellable drawings of $K_n$, the precise crossing number matches the Hill bound, and a combinatorial forbidden-configuration characterization (on triples and quadruples) delineates the exact class of such drawings [1312.3679].

- For 2-page book crossing numbers, semidefinite programming techniques yield exact results for $K_n$ up to $n=18$, and establish that $\liminf_{n\to\infty} \nu_2(K_n)/Z(n) \geq 0.9253$ [1110.4824].

## 6. Surfaces, Curves, and Topological Variants

Minimum crossing numbers are also studied in the context of surfaces of higher genus, where systems of simple closed curves are considered. For genus $g=2$ and $k$ non-homotopic curves, explicit values for $\mathrm{cr}(k;g)$ up to $k=12$ are established, together with the uniqueness of minimizing systems up to homeomorphism and isotopy for $k \leq 11$ [2403.05148]. Growth is asymptotically quadratic in $k$, with uniqueness and decomposition properties depending sensitively on $k$ and $g$.

The relationship between various crossing parameters—pair-crossing, odd-crossing, minor crossing—has been systematized, with the $e^3/n^2$-bound demonstrably holding for all nine natural crossing variants for simple graphs [2509.14074].

## 7. Open Problems and Future Directions

- Resolving the precise value of the midrange crossing constant, closing the remaining constant-factor gaps.
- Determining the minimal linear coefficient for the crossing number of $k$-critical graphs—is $cr(G) \leq k + C \sqrt{k}$ universally achievable?
- Characterizing the tightest structural conditions guaranteeing $O(\Delta n)$ (or linear) bounds for rectilinear or monotone crossing numbers in various minor-closed or hereditary classes.
- Extending forbidden configuration and signature function characterizations to more general classes (pseudolinear, higher page-number, surface-embedded graphs).
- Algorithmic challenges: achieving efficient deterministic, $(\mathrm{OPT}^{O(1)})$-approximation algorithms for the minimum crossing number in general graphs; understanding the hardness landscape with respect to additive and multiplicative approximations [1012.0255].

The minimum crossing number remains a central and richly interconnected invariant, at the interface of graph theory, combinatorial geometry, and algorithms, with broad ramifications and many deep unresolved questions.

Source: https://www.emergentmind.com/topics/minimum-crossing-numbers