Crossing number (graph theory)

In graph theory, the crossing number cr(G) of a graph G is the lowest number of crossings of a planar drawing of the graph G. For instance, a graph is planar if and only if its crossing number is zero. The concept originated in Turán's brick factory problem, in which Pál Turán asked to determine the crossing number of the complete bipartite graph K_m,n.

History

The first approach to understanding the crossing number was by Zarankiewicz, who attempted to solve Turán's brick factory problem. His proof contained an error, but he established a valid upper bound of

$\scriptstyle cr(K_{m,n})\leq \lfloor n/2\rfloor \lfloor (n-1)/2\rfloor \lfloor m/2\rfloor \lfloor (m-1)/2\rfloor ,$

and the conjectured equality is now known as Zarankiewicz' Crossing Number Conjecture.

The problem of determining the crossing number of the complete graph was first posed by Anthony Hill, and appears in print in 1960. Interestingly, Anthony Hill and John Ernest were two collaborating constructivist artists fascinated by mathematics who not only formulated this problem but also originated a conjectural upper bound for this crossing number co-published with Richard Guy in 1960. As of March 2009, both problems remain unresolved except for a few special cases; at least there has been some progress on lower bounds, as reported by Klerk et al. in 2006 .

The Albertson conjecture, formulated by Michael O. Albertson in 2007, states that, among all graphs with chromatic number n, the complete graph K_n has the minimum number of crossings. That is, if Guy's conjecture on the crossing number of the complete graph is valid, every n-chromatic graph has crossing number at least equal to the formula in the conjecture. It is now known to hold for n ≤ 16.

Variations

Without further qualification, the crossing number allows drawings in which the edges may be represented by arbitrary curves; the rectilinear crossing number requires all edges to be straight line segments, and may differ from the crossing number. In particular, the rectilinear crossing number of a complete graph is essentially the same as the minimum number of convex quadrilaterals determined by a set of n points in general position, closely related to the Happy Ending problem.

Complexity

In general, determining the crossing number of a graph is hard; Garey and Johnson showed in 1983 that it is an NP-hard problem. In fact the problem remains NP-hard even when restricted to cubic graphs.

On the positive side, there are efficient algorithms for determining if the crossing number is less than a fixed constant k — in other words, the problem is fixed-parameter tractable. It remains difficult for larger k, such as |V|/2. There are also efficient approximation algorithms for approximating cr(G)+n on graphs of bounded degree. In practice heuristic algorithms are used, such as the simple algorithm which starts with no edges and continually adds each new edge in a way that produces the fewest additional crossings possible. These algorithms are used in the Rectilinear Crossing Number distributed computing project.

Crossing numbers of cubic graphs

The smallest cubic graphs with crossing numbers 1–8 are known (sequence A110507 in the OEIS). The smallest 1-crossing graph is the complete bipartite graph K_3,3, with 6 vertices. The smallest 2-crossing cubic graph is the Petersen graph, with 10 vertices. The smallest 3-crossing graph is the Heawood graph, with 14 vertices. The smallest 4-crossing graph is the Möbius-Kantor graph, with 16 vertices. The smallest 5-crossing graph is the Pappus graph, with 18 vertices. The smallest 6-crossing graph is the Desargues graph, with 20 vertices. None of the four 7-crossing graphs, with 22 vertices, are well known. The smallest 8-crossing graph is the McGee graph or (3,7)-cage graph, with 24 vertices.

In 2009, Exoo conjectured that the smallest cubic graph with crossing number 11 is the Coxeter graph, the smallest cubic graph with crossing number 13 is the Tutte–Coxeter graph and the smallest cubic graph with crossing number 170 is the Tutte 12-cage.

The crossing number inequality

The very useful crossing number inequality, discovered independently by Ajtai, Chvátal, Newborn, and Szemerédi and by Leighton, asserts that if a graph G (undirected, with no loops or multiple edges) with n vertices and e edges satisfies

$e>7.5n,\,$

then we have

$\operatorname {cr} (G)\geq {\frac {e^{3}}{33.75n^{2}}}.\,$

The constant 33.75 is the best known to date, and is due to Pach and Tóth; the constant 7.5 can be lowered to 4, but at the expense of replacing 33.75 with the worse constant of 64.

The motivation of Leighton in studying crossing numbers was for applications to VLSI design in theoretical computer science. Later, Székely also realized that this inequality yielded very simple proofs of some important theorems in incidence geometry, such as Beck's theorem and the Szemerédi-Trotter theorem, and Tamal Dey used it to prove upper bounds on geometric k-sets.

Proof of crossing number inequality

We first give a preliminary estimate: for any graph G with n vertices and e edges, we have

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

To prove this, consider a diagram of G which has exactly cr(G) crossings. Each of these crossings can be removed by removing an edge from G. Thus we can find a graph with at least $e-\operatorname {cr} (G)$ edges and n vertices with no crossings, and is thus a planar graph. But from Euler's formula we must then have $e-\operatorname {cr} (G)\leq 3n$, and the claim follows. (In fact we have $e-\operatorname {cr} (G)\leq 3n-6$ for n ≥ 3).

To obtain the actual crossing number inequality, we now use a probabilistic argument. We let 0 < p < 1 be a probability parameter to be chosen later, and construct a random subgraph H of G by allowing each vertex of G to lie in H independently with probability p, and allowing an edge of G to lie in H if and only if its two vertices were chosen to lie in H. Let $e_{H}$ denote the number of edges of H, and let $n_{H}$ denote the number of vertices.

Now consider a diagram of G with cr(G) crossings. We may assume that any two edges in this diagram with a common vertex are disjoint, otherwise we could interchange the intersecting parts of the two edges and reduce the crossing number by one. Thus every crossing in this diagram involves four distinct vertices of G.

Since H is a subgraph of G, this diagram contains a diagram of H; let $\operatorname {cr} _{H}$ denote the number of crossings of this random graph. By the preliminary crossing number inequality, we have

$\operatorname {cr} _{H}\geq e_{H}-3n_{H}.$

Taking expectations we obtain

${\mathbb {E}}(\operatorname {cr} _{H})\geq {{\mathbb {E}}(e_{H})}-3{\mathbb {E}}(n_{H}).$

Since each of the n vertices in G had a probability p of being in H, we have ${\mathbb {E}}(n_{H})=pn$. Similarly, since each of the edges in G has a probability $p^{2}$ of remaining in H (since both endpoints need to stay in H), then ${\mathbb {E}}(e_{H})=p^{2}e$. Finally, every crossing in the diagram of G has a probability $p^{4}$ of remaining in H, since every crossing involves four vertices, and so ${\mathbb {E}}(\operatorname {cr} _{H})=p^{4}\operatorname {cr} (G)$. Thus we have

$p^{4}\operatorname {cr} (G)\geq p^{2}e-3pn.\,$

If we now set p to equal 4n/e (which is less than one, since we assume that e is greater than 4n), we obtain after some algebra

$\operatorname {cr} (G)\geq e^{3}/64n^{2}.\,$

A slight refinement of this argument allows one to replace 64 by 33.75 when e is greater than 7.5 n.