We can further modify each of the above crossing numbers, by applying one of the following rules:
Rule + : Consider only those drawings where two edges with a common endpoint do not cross each other.
Rule 0 : Two edges with a common endpoint are allowed to cross and their crossing counts.
Rule − : Two edges with a common endpoint are allowed to cross, but their crossing does not count.
In the previous definitions we have always used Rule 0. If we apply Rule + (Rule −) in the definition of the crossing numbers, then we indicate it by using the corresponding subscript, as shown in the table below. This gives us an array of nine different crossing numbers. It is easy to see that in a drawing of a graph, which minimizes the number of crossing points, any two edges have at most one point in common (see e.g. [RT97]). Therefore, cr+(G) =cr(G), which slightly simplifies the picture.
Rule – Rule 0 Rule +
cr(G)
cr−(G) pair-cr+(G)
pair-cr(G)
pair-cr−(G) odd-cr+(G)
odd-cr(G)
odd-cr−(G)
Figure 4.12: Modifications of the crossing number
Moving from left to right or from bottom to top in this array, the numbers do not decrease. It is not hard to generalize (1) to each of these crossing
numbers. We obtain (as in in [PT97]) that odd-cr
−(G)≥ 1 64
e3 n2,
for any graph G with n vertices and with e ≥ 4n edges. We cannot prove anything else aboutodd-cr−(G),pair-cr−(G), andcr−(G). We conjecture that these values are very close tocr(G), if not the same. That is, we believe that by letting pairs ofincidentedges cross an arbitrary number of times, we cannot effectively reduce the total number of crossings betweenindependent pairs of edges. The weakest open questions are the following.
Problem. Do there exist suitable functions f1, f2, f3 such that every graph G satisfies
(i) odd-cr(G)≤f1(odd-cr
−(G)), (ii) pair-cr(G)≤f2(pair-cr
−(G)), (iii) cr(G)≤f3(cr−(G)) ?
Remark. We can prove that the Pair Crossing Number Problem, pair-cr(G) ≤ K, is NP-hard. The proof is analogous to the proofs of the corresponding results for the crossing number (see [GJ83]) and for the odd-crossing number (see Lemma 4.6.2).
On the other hand, we could not generalize Lemma 4.6.1 forpair-cr(G).
With a completely different approach, Schaefer, Sedgwick and ˇStefankoviˇc [SSS03] managed to prove that thePair Crossing Number Problem is also in NP.
Acknowledgement. We express our gratitude to Noga Alon, Joel Spencer, and Pavel Valtr for their valuable remarks and for many interesting discus-sions on the subject.
Chapter 5
Crossing numbers of random graphs
The crossing number of G is the minimum number of crossing points in any drawing of G. We consider the following two other parameters. The rectilinear crossing number is the minimum number of crossing points in any drawing ofG, with straight line segments as edges. The pair-crossing number of G is the minimum number of pairs of crossing edges over all drawings of G. The odd-crossing number of G is the minimum number of pairs of edges that cross an odd number of times. We prove several results on the expected values of these parameters of a random graph.
5.1 Introduction
A drawingof a graphGis a mapping which assigns to each vertex a point of the plane and to each edge a simple continuous arc connecting the correspond-ing two points. We assume that in a drawcorrespond-ing no three edges (arcs) cross at the same point, and the edges do not pass through any vertex. The crossing numbercr(G) ofGis the minimum number of crossing points in any drawing of G. We consider the following two variants of the crossing number. The rectilinear crossing number lin-cr(G) is the minimum number of crossing points in any drawing of G, with straight line segments as edges. The pair-crossing number orpair-crossing number,pair-cr(G), ofG is the minimum number of crossing pairs of edges over all drawings of G. The odd-crossing numberofGis the minimum number of pairs of edges that cross an odd
num-ber of times. Clearly, odd-cr(G)≤pair-cr(G)≤cr(G)≤lin-cr(G).
The determination of the crossing numbers is extremely difficult. Even the crossing numbers of the complete graphs are not known. Let
γodd-cr = lim
These limits are known to exist [RT97] and the best known bounds are 1/30 ≤ γodd-cr ≤ 1/16, 1/30 ≤ γpair-cr ≤ 1/16, 1/20 ≤ γcr ≤ 1/16, 0.06327≤γlin-cr ≤0.0639 [AGOR06], [BDG00] (see also [G72, RT97]).
In this paper we investigate the crossing numbers ofrandom graphs. Let G = G(n, p) be a random graph with n vertices, whose edges are chosen independently with probabilityp. Let edenote the expected numberof edges of G, i.e., e = pn2. We shall always have e → ∞ (indeed, p = Ω(n−1)) so that Galmost surely has e(1 +o(1)) edges.
In [PT00a] it was shown that if e > 10n, then almost surely we have cr(G) ≥ 4000e2 . Consequently, almost surely we also have lin-cr(G) ≥ 4000e2 . As we always can draw a graph with straight lines the crossing number (in any form) is never larger than the number of pairs of edges and the expected number of pairs of edges is∼ e22 Our interest will be in those regions ofpfor which the various crossing numbers are, asymptotically, a positive proportion of the number of pairs of edges.
Let
With Theorem 5.1.1 we may express (roughly) our two central concerns.
At which p = p(n) are κlin-cr(n, p), κcr(n, p), κpair-cr(n, p) bounded away
from zero? At which p = p(n) are κlin-cr(n, p), κcr(n, p), κpair-cr(n, p), κodd-cr(n, p) close to the valuesγlin-cr, γcr, γpair-cr, γodd-cr, respectively? Our results for these three crossing numbers shall be quite different. We are uncer-tain whether or not that represents the reality of the situation. The following relatively simple result shows basically that for p= n1 all three crossing num-bers are asymptotically negligible and that for p = nc with c > 1 fixed the three crossing numbers have not reached their limiting values.
Theorem 5.1.2. 1. lim supn→∞κlin-cr(n, c/n) = 0 for c≤1 2. lim supn→∞κcr(n, c/n) = 0 for c≤1
3. lim supn→∞κpair-cr(n, c/n) = 0 for c≤1 4. lim supn→∞κodd-cr(n, c/n) = 0 for c≤1 5. limc→1lim supn→∞κlin-cr(n, c/n) = 0 6. limc→1lim supn→∞κcr(n, c/n) = 0 7. limc→1lim supn→∞κpair-cr(n, c/n) = 0 8. limc→1lim supn→∞κodd-cr(n, c/n) = 0 9. lim supn→∞κlin-cr(n, c/n)< γlin-cr for all c 10. lim supn→∞κcr(n, c/n)< γcr for all c
11. lim supn→∞κpair-cr(n, c/n)< γpair-cr for all c 12. lim supn→∞κpair-cr(n, c/n)< γodd-cr for all c
Theorem 5.1.2 gives only upper bounds for the various crossing numbers.
The main results of this paper, given in Theorems 5.1.3, 5.1.4, 5.1.5, deal with lower bounds for the three crossing numbers. Our weakest result is for the pair-crossing number and the odd-crossing number.
Theorem 5.1.3. For any ε >0, p=p(n) =nε−1, lim inf
n→∞ κpair-cr(n, p)>0, lim inf
n→∞ κodd-cr(n, p)>0.
For the crossing number we have a much stronger result.
Theorem 5.1.4. For any c >1 with p=p(n) =c/n lim inf
n→∞ κcr(n, p)>0
As lin-cr(G) ≥ cr(G) the lower bound of Theorem 5.1.4 applies also to the rectilinear crossing number. Our most surprising result is that with the rectilinear crossing number one reaches an asymptotically best limit in relatively short time.
Theorem 5.1.5. If p=p(n)≫ lnnn then
nlim→∞κlin-cr(n, p) = γlin-cr(n, p)