- Crossing number (graph theory)
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.
for the crossing number of a complete bipartite graph Km,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. 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. 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).
The Albertson conjecture, formulated by Michael O. Albertson in 2007, states that, among all graphs with chromatic number n, the complete graph Kn 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.
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.
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. More specifically, determining the rectilinear crossing number is complete for the existential theory of the reals.
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) 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 OEIS). The smallest 1-crossing cubic graph is the complete bipartite graph K3,3, 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. 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.
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
then we have
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
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 edges and n vertices with no crossings, and is thus a planar graph. But from Euler's formula we must then have , and the claim follows. (In fact we have 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 eH denote the number of edges of H, and let nH 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 denote the number of crossings of this random graph. By the preliminary crossing number inequality, we have
Taking expectations we obtain
Since each of the n vertices in G had a probability p of being in H, we have . 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 . Finally, every crossing in the diagram of G has a probability p4 of remaining in H, since every crossing involves four vertices, and so . Thus we have
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
A slight refinement of this argument allows one to replace 64 by 33.75 when e is greater than 7.5 n.
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