Crossing number (graph theory)

A drawing of the Heawood graph with three crossings. This is the minimum number of crossings among all drawings of this graph, so the graph has crossing number cr(G) = 3.

In graph theory, the crossing number cr(G) of a graph G is the lowest number of edge 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.^[1]

History

Zarankiewicz attempted to solve Turán's brick factory problem;^[2] his proof contained an error, but he established a valid upper bound of

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

for the crossing number of a complete bipartite graph K_m,n. The conjecture that this inequality is actually an 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.^[3] Hill and his collaborator John Ernest were two constructionist artists fascinated by mathematics who not only formulated this problem but also originated a conjectural upper bound for this crossing number, which Richard Guy published in 1960.^[3] As of March 2009, both problems remain unresolved except for a few special cases; there has been some progress on lower bounds, as reported by de Klerk et al. (2006).^[4]

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.^[5]

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.^[6]

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.^[7] In fact the problem remains NP-hard even when restricted to cubic graphs.^[8] More specifically, determining the rectilinear crossing number is complete for the existential theory of the reals.^[9]

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.^[10] It remains difficult for larger k, such as |V|/2. There are also efficient approximation algorithms for approximating cr(G) on graphs of bounded degree.^[11] 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^[12] distributed computing project.

Crossing numbers of cubic graphs

The smallest cubic graphs with crossing numbers 1–8 are known (sequence A110507 in OEIS). The smallest 1-crossing cubic 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 cubic graph is the Heawood graph, with 14 vertices. The smallest 4-crossing cubic graph is the Möbius-Kantor graph, with 16 vertices. The smallest 5-crossing cubic graph is the Pappus graph, with 18 vertices. The smallest 6-crossing cubic graph is the Desargues graph, with 20 vertices. None of the four 7-crossing cubic graphs, with 22 vertices, are well known.^[13] The smallest 8-crossing cubic 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.^[14][15]

The crossing number inequality

The very useful crossing number inequality, discovered independently by Ajtai, Chvátal, Newborn, and Szemerédi^[16] and by Leighton,^[17] asserts that if a graph G (undirected, with no loops or multiple edges) with n vertices and e edges satisfies

$e > 7.5 n,\,$

then we have

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

The constant 33.75 is the best known to date, and is due to Pach and Tóth;^[18] 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^[19] 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.^[20]

For graphs with girth larger than 2r and e ≥ 4n, Pach, Spencer and Tóth^[21] demonstrated an improvement of this inequality to

$\operatorname{cr}(G) \geq c_r\frac{e^{r+2}}{n^{r+1}}.\,$

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

${\Bbb E}(\operatorname{cr}_H) \geq {\Bbb E(e_H)} - 3 {\Bbb E}(n_H).$

Since each of the n vertices in G had a probability p of being in H, we have ${\Bbb E}(n_H) = pn$. Similarly, since each of the edges in G has a probability p² of remaining in H (since both endpoints need to stay in H), then ${\Bbb E}(e_H) = p^2 e$. Finally, every crossing in the diagram of G has a probability p⁴ of remaining in H, since every crossing involves four vertices, and so ${\Bbb E}(\operatorname{cr}_H) = p^4 \operatorname{cr}(G)$. Thus we have

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

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

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

A slight refinement of this argument allows one to replace 64 by 33.75 when e is greater than 7.5 n.^[18]

Notes

1. ^ Turán, P. (1977). "A Note of Welcome". J. Graph Theory 1: 7–9. doi:10.1002/jgt.3190010105. 
2. ^ Zarankiewicz, K. (1954). "On a Problem of P. Turán Concerning Graphs". Fund. Math. 41: 137–145. 
3. ^ ^{a b} Guy, R.K. (1960). "A combinatorial problem". Nabla (Bulletin of the Malayan Mathematical Society) 7: 68–72. 
4. ^ de Klerk, E.; Maharry, J.; Pasechnik, D. V.; Richter, B.; Salazar, G. (2006). "Improved bounds for the crossing numbers of K_m,n and K_n". SIAM Journal on Discrete Mathematics 20 (1): 189–202. doi:10.1137/S0895480104442741. http://arno.uvt.nl/show.cgi?fid=71701 .
5. ^ Barát, János; Tóth, Géza (2009). "Towards the Albertson Conjecture". arXiv:0909.0413 [math.CO]. 
6. ^ Scheinerman, Edward R.; Wilf, Herbert S. (1994). "The rectilinear crossing number of a complete graph and Sylvester's "four point problem" of geometric probability". American Mathematical Monthly 101 (10): 939–943. doi:10.2307/2975158. JSTOR 2975158. 
7. ^ Garey, M. R.; Johnson, D. S. (1983). "Crossing number is NP-complete". SIAM J. Alg. Discr. Meth. 4 (3): 312–316. doi:10.1137/0604033. MR0711340. 
8. ^ Hliněný, P. (2006). "Crossing number is hard for cubic graphs". Journal of Combinatorial Theory, Series B 96 (4): 455–471. doi:10.1016/j.jctb.2005.09.009. MR2232384. 
9. ^ Schaefer, Marcus (2010). "Complexity of Some Geometric and Topological Problems". Graph Drawing, 17th International Symposium, GS 2009, Chicago, IL, USA, September 2009, Revised Papers. Lecture Notes in Computer Science. 5849. Springer-Verlag. pp. 334–344. doi:10.1007/978-3-642-11805-0_32. ISBN 978-3-642-11804-3. http://ovid.cs.depaul.edu/documents/convex.pdf .
10. ^ Grohe, M. (2005). "Computing crossing numbers in quadratic time". J. Comput. System Sci. 68 (2): 285–302. doi:10.1016/j.jcss.2003.07.008. MR2059096 ; Kawarabayashi, Ken-ichi; Reed, Bruce (2007). "Computing crossing number in linear time". Proceedings of the 29th Annual ACM Symposium on Theory of Computing. pp. 382–390. doi:10.1145/1250790.1250848. ISBN 9781595936318. 
11. ^ Even, Guy; Guha, Sudipto; Schieber, Baruch (2003). "Improved Approximations of Crossings in Graph Drawings and VLSI Layout Areas". SIAM Journal on Computing 32 (1): 231–252. doi:10.1137/S0097539700373520. 
12. ^ Rectilinear Crossing Number on the Institute for Software Technology at Graz, University of Technology (2009).
13. ^ Weisstein, Eric W., "Graph Crossing Number" from MathWorld.
14. ^ Exoo, G.. "Rectilinear Drawings of Famous Graphs". http://isu.indstate.edu/ge/COMBIN/RECTILINEAR/. 
15. ^ Pegg, E. T.; Exoo, G. (2009). "Crossing Number Graphs". Mathematica J. 11 .
16. ^ Ajtai, M.; Chvátal, V.; Newborn, M.; Szemerédi, E. (1982). "Crossing-free subgraphs". Theory and Practice of Combinatorics. North-Holland Mathematics Studies. 60. pp. 9–12. MR0806962. 
17. ^ Leighton, T. (1983). Complexity Issues in VLSI. Foundations of Computing Series. Cambridge, MA: MIT Press. 
18. ^ ^{a b} Pach, J.; Tóth, G. (1997). "Graphs drawn with few crossings per edge". Combinatorica 17 (3): 427–439. doi:10.1007/BF01215922. MR1606052. 
19. ^ Székely, L. A. (1997). "Crossing numbers and hard Erdős problems in discrete geometry". Combinatorics, Probability and Computing 6 (3): 353–358. doi:10.1017/S0963548397002976. MR1464571. 
20. ^ Dey, T. L. (1998). "Improved bounds for planar k-sets and related problems". Discrete and Computational Geometry 19 (3): 373–382. doi:10.1007/PL00009354. MR1608878. 
21. ^ Pach, János; Spencer, Joel; Tóth, Géza (2000). "New bounds on crossing numbers". Discrete and Computational Geometry 24 (4): 623–644.