Minimum Crossing Numbers in Graph Theory

• Minimum crossing number is a graph invariant defined as the minimum number of edge intersections over all planar drawings, crucial for assessing drawing efficiency and applications in network design.
• The study employs the Crossing Lemma, midrange estimates, and various drawing models like rectilinear and convex to establish tight asymptotic bounds for different graph classes.
• Recent research leverages algorithmic decompositions and explicit constructions in specialized families, enhancing our understanding of crossing-critical graphs and computational complexity.

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 $H$) (Dujmović et al., 2018, Bokal et al., 2011).

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 $n=|V(G)|$, $e=|E(G)|$ (Toth, 17 Sep 2025). 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 (Toth, 17 Sep 2025). 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 (Czabarka et al., 2018). 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 (Czabarka et al., 2018). 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 $c, c'$ (Chen et al., 4 Feb 2025).

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 (Dujmović et al., 2018). 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 (Dujmović et al., 2024, Dujmović et al., 2018).

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 (Dujmović et al., 2024).

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 $k$ (Ouyang et al., 30 Apr 2025). 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 (Barát et al., 2020). 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 (Bokal et al., 2011).

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: $cr(Q_4)=cr(\mathrm{CQ}_4)=8$, $cr(\mathrm{LTQ}_4)=cr(\mathrm{MQ}_4)=10$ (Wang et al., 2011). 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 (Gethner et al., 2014). 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 (Balko et al., 2013).

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 (Pach et al., 2016).

• 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 $\binom{n}{4}$ (Ábrego et al., 2016). 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 (Balko et al., 2013).
• For 2-page book crossing numbers, semidefinite programming techniques yield exact results for $K_n$ up to $n=18$, and establish that $\liminf_{n\to\infty} \nu_2(K_n)/Z(n) \geq 0.9253$ (Klerk et al., 2011).

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 $k \leq 11$ (Jörg, 2024). 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 (Toth, 17 Sep 2025).

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 (Chuzhoy, 2010).

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.