The number of crossings in multigraphs with no empty lens

Let G be a multigraph with n vertices and e>4n edges, drawn in the plane such that any two parallel edges form a simple closed curve with at least one vertex in its interior and at least one vertex in its exterior. Pach and Tóth (2018) extended the Crossing Lemma of Ajtai et al. (1982) and Leighton (1983) by showing that if no two adjacent edges cross and every pair of nonadjacent edges cross at most once, then the number of edge crossings in G is at least α e^3/n^2, for a suitable constant α>0. The situation turns out to be quite different if nonparallel edges are allowed to cross any number of times. It is proved that in this case the number of crossings in G is at least α e^2.5/n^1.5. The order of magnitude of this bound cannot be improved.

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1 Introduction

In this paper, multigraphs may have parallel edges but no loops. A topological graph (or multigraph) is a graph (multigraph) drawn in the plane with the property that every vertex is represented by a point and every edge is represented by a curve (continuous arc) connecting the two points corresponding to the vertices and . We assume, for simplicity, that the points and curves are in “general position”, that is, (a) no vertex is an interior point of any edge; (b) any pair of edges intersect in at most finitely many points; (c) if two edges share an interior point, then they properly cross at this point; and (d) no 3 edges cross at the same point. Throughout this paper, every multigraph is a topological multigraph, that is, is considered with a fixed drawing that is given from the context. In notation and terminology, we then do not distinguish between the vertices (edges) and the points (curves) representing them. The number of crossing points in the considered drawing of is called its crossing number, denoted by . (I.e., is defined for topological multigraphs rather than abstract multigraphs.)

The classic “crossing lemma” of Ajtai, Chvátal, Newborn, Szemerédi [1] and Leighton [3] gives an asymptotically best-possible lower bound on the crossing number in any -vertex -edge topological graph without loops or parallel edges, provided .

Theorem A (Crossing Lemma, Ajtai et al.[1] and Leighton [3])

There is an absolute constant , such that for any -vertex -edge topological graph we have

 cr(G)≥αe3n2,provided e>4n.

In general, the Crossing Lemma does not hold for topological multigraphs with parallel edges, as for every and there are -vertex -edge topological multigraphs with . Székely proved the following variant for multigraphs by restricting the edge multiplicity, that is the maximum number of pairwise parallel edges, in to be at most .

Theorem B (Székely [6])

There is an absolute constant such that for any and any -vertex -edge multigraph with edge multiplicity at most  we have

 cr(G)≥αe3mn2,provided e>4mn.

Most recently, Pach and Tóth [5] extended the Crossing Lemma to so-called branching multigraphs. We say that a topological multigraph is

• separated if any pair of parallel edges form a simple closed curve with at least one vertex in its interior and at least one vertex in its exterior,

• single-crossing if any pair of edges cross at most once (that is, edges sharing endpoints, , may have at most points in common), and

• locally starlike if no two adjacent edges cross (that is, edges sharing endpoints, , may not cross).

A topological multigraph is branching if it is separated, single-crossing and locally starlike. Note that the edge multiplicity of a branching multigraph may be as high as .

Theorem C (Pach and Tóth [5])

There is an absolute constant such that for any -vertex -edge branching multigraph we have

 cr(G)≥αe3n2,provided e>4n.

In this paper we generalize Theorem C by showing that the Crossing Lemma holds for all topological multigraphs that are separated and locally starlike, but not necessarily single-crossing. We shall sometimes refer to the separated condition as the multigraph having “no empty lens,” where we remark that here a lens is bounded by two entire edges, rather than general edge segments as sometimes defined in the literature. We also prove a Crossing Lemma variant for separated (and not necessarily locally starlike) multigraphs, where however the term must be replaced by . Both results are best-possible up to the value of constant .

Theorem 1.1

There is an absolute constant such that for any -vertex -edge topological multigraph with we have

1. , if is separated and locally starlike.

2. , if is separated.

Moreover, both bounds are best-possible up to the constant .

We prove Theorem 1.1 in Section 3. Our arguments hold in a more general setting, which we present in Section 2. In Section 4 we use this general setting to deduce other known Crossing Lemma variants, including Theorem B. We conclude the paper with some open questions in Section 5.

2 A Generalized Crossing Lemma

In this section we consider general drawing styles and propose a generalized Crossing Lemma, which will subsume all Crossing Lemma variants mentioned here. A drawing style is a predicate over the collection of all topological drawings, i.e., for each topological drawing of a multigraph we specify whether is in drawing style or not. We say that is a multigraph in drawing style when is a topological multigraph whose drawing is in drawing style .

In order to prove our generalized Crossing Lemma, we follow the line of arguments of Pach and Tóth [5] for branching multigraphs. Their main tool is a bisection theorem for branching drawings, which easily generalizes to all separated drawings. We generalize their definition as follows.

Definition 1 (D-bisection width)

For a drawing style the -bisection width of a multigraph in drawing style is the smallest number of edges whose removal splits into two multigraphs, and , in drawing style with no edge connecting them such that .

We say that a drawing style is monotone if removing edges retains the drawing style, that is, for every multigraph in drawing style and any edge removal, the resulting multigraph with its inherited drawing from is again in drawing style . Note that we require a monotone drawing style to be retained only after removing edges, but not necessarily after removing vertices. For example, the branching drawing style is in general not maintained after removing a vertex, since a closed curve formed by a pair of parallel edges might become empty.

Given a topological multigraph , we call any operation of the following form a vertex split: (1) Replace a vertex of by two vertices and and (2) by locally modifying the edges in a small neighborhood of , connect each edge in incident to to either or in such a way that no new crossing is created. We say that a drawing style is split-compatible if performing vertex splits retains the drawing style, that is, for every multigraph in drawing style and any vertex split, the resulting multigraph with its inherited drawing from is again in drawing style .

We are now ready to state our main result.

Theorem 2.1 (Generalized Crossing Lemma)

Suppose is a monotone and split-compatible drawing style, and that there are constants and such that each of the following holds for every -vertex -edge multigraph in drawing style :

1. If , then the edge count satisfies .

2. The -bisection width satisfies .

3. The edge count satisfies .

Then there exists an absolute constant such that for any -vertex -edge multigraph in drawing style we have

 cr(G)≥αex(b)+2nx(b)+1,provided e>(k1+1)n,

where and for some constant depending only on .

Lemma 1

If there exist for arbitrarily large multigraphs in drawing style with vertices and edges such that any two edges cross at most a constant number of times, then the bound in Theorem 2.1 is asymptotically tight.

Proof

Consider such an -vertex -edge multigraph in drawing style . Clearly, there are at most crossings, while Theorem 2.1 gives with that there are at least

 Ω(ex(b)+2nx(b)+1)=Ω(ex(b)+2nb⋅x(b))=Ω(nb⋅x(b)+2bnb⋅x(b))=Ω(n2b)

crossings.

2.1 Proof of Theorem 2.1

Proof idea

Before proving Theorem 2.1, let us sketch the rough idea. Suppose, for a contradiction, that is a multigraph in drawing style with fewer than crossings, for a constant to be defined. First, we conclude from i that must have many edges. Then, by ii, the -bisection width of is small, and thus we can remove few edges from the drawing to obtain two smaller multigraphs, and , both also in drawing style , which we call parts. We then repeat splitting each large enough part into two parts each, again using ii. Note that each part has at most of the vertices of the corresponding part in the previous step. We continue until all parts are smaller than a carefully chosen threshold. As we removed relatively few edges during this decomposition algorithm, the final parts still have a lot of edges, while having few vertices each. This will contradict iii and hence complete the proof.

Now, let us start with the proof of Theorem 2.1. We define an absolute constant

 α:=122x(b)+14⋅1k22⋅1kx(b)3 (1)

Now let be a fixed multigraph in drawing style with vertices and edges. Let be an edge-maximal subgraph of on vertex set such that the inherited drawing of has no crossings. Since is monotone, is in drawing style . Hence, by i, for the number of edges in we have . Since is edge-maximal crossing-free, each edge in has at least one crossing with an edge in . Thus

 cr(~G)≥~e−e′≥~e−k1~n>~n. (2)

In case for , we get

 cr(~G)~n≥α⋅~ex(b)+2~nx(b)+1,

as desired. To prove Theorem 2.1 in the remaining case we use proof by contradiction. Therefore assume that the number of crossings in satisfies

 cr(~G)<α⋅~ex(b)+2~nx(b)+1.

Let denote the average degree of the vertices of , that is, . For every vertex whose degree, , is larger than , we perform vertex splits so as to split into vertices, each of degree at most . At the end of the procedure, we obtain a multigraph with edges, vertices, and maximum degree . Moreover, as is split-compatible, is in drawing style . For the number of crossings in , we have

 cr(G)=cr(~G)<α⋅~ex(b)+2~nx(b)+1<2x(b)+1α⋅ex(b)+2nx(b)+1. (3)

Moreover, recall that

 e>β~n>βn2for β=1α1/(x(b)+2). (4)

We break into smaller parts, according to the following procedure. At each step the parts form a partition of the entire vertex set .

Decomposition Algorithm

Step 0.

Let

Suppose that we have already executed Step , and that the resulting graph consists of parts, , each in drawing style and having at most  vertices. Assume without loss of generality that each of the first parts of has at least vertices and the remaining have fewer. Letting denote the number of vertices of the part , we have

 (4/5)i+1n(G)≤n(Gij)≤(4/5)in(G),1≤j≤mi. (5)

Hence,

 mi≤(5/4)i+1. (6)

Step .

If

 (4/5)i<1(2k3)x(b)⋅ex(b)nx(b)+1, (7)

then stop.

Else, for , delete edges from , as guaranteed by ii, such that falls into two parts, each of which is in drawing style and contains at most vertices. Let denote the resulting graph on the original set of vertices.

Clearly, each part of has at most vertices.

Suppose that the Decomposition Algorithm terminates in Step . If , then

 (4/5)k<1(2k3)x(b)⋅ex(b)nx(b)+1≤(4/5)k−1. (8)

First, we give an upper bound on the total number of edges deleted from . Using Cauchy-Schwarz inequality, we get for any nonnegative numbers ,

 m∑j=1√aj≤ ⎷mm∑j=1aj, (9)

and thus obtain that, for any ,

 mi∑j=1√cr(Gij) ⎷mimi∑j=1cr(Gij)(???)darab≤√(5/4)i+1√cr(G)√(5/4)i+1√2x(b)+1α⋅ex(b)+2nx(b)+1. (10)

Letting and denote the number of edges and maximum degree in part , respectively, we obtain similarly

 mi∑j=1√Δ(Gij)⋅e(Gij)+n(Gij) ⎷mi(mi∑j=1Δ(Gij)⋅e(Gij)+n(Gij))√(5/4)i+1√Δ(G)⋅e+n≤√(5/4)i+1√4ene+n<√(5/4)i+1√5e2n<√(5/4)i+13e√n, (11)

where we used in the last line the fact that .

Using a partial sum of a geometric series we get

 k∑i=0(√5/4)i+1=(√5/4)k+2−1√5/4−1−1<(√5/4)3√5/4−1⋅(√5/4)k−1<12⋅(√5/4)k−1 (12)

Thus, as each is in drawing style and hence ii holds for each , the total number of edges deleted during the decomposition procedure is

 k∑i=0mi∑j=1bD(Gij)≤ k2k∑i=0mi∑j=1√cr(Gij)+Δ(Gij)⋅e(Gij)+n(Gij) ≤ k2(k∑i=0mi∑j=1√cr(Gij)+k∑i=0mi∑j=1√Δ(Gij)⋅e(Gij)+n(Gij)) k2(k∑i=0√(5/4)i+1)(√2x(b)+1α⋅ex(b)+2nx(b)+1+3e√n) k2⋅12√(5/4)k−1(√2x(b)+1α⋅ex(b)+2nx(b)+1+3e√n) k2⋅12√(2k3)x(b)⋅nx(b)+1ex(b)(√2x(b)+1α⋅ex(b)+2nx(b)+1+3e√n) < k2⋅36⋅√kx(b)3⎛⎝2x(b)√αe+√2x(b)nx(b)ex(b)−2⎞⎠ k2⋅36⋅√kx(b)3⋅2x(b)⎛⎝√α+√1βx(b)⎞⎠e k2⋅36⋅√kx(b)3⋅2x(b)⎛⎜⎝√α+√αx(b)x(b)+2⎞⎟⎠e

By (13) the Decomposition Algorithm removes less than half of the edges of if . Hence, the number of edges of the graph obtained in the final step of this procedure satisfies

 e(Gk)>e2. (14)

(Note that this inequality trivially holds if the algorithm terminates in the very first step, i.e., when .)

Next we shall give an upper bound on that contradicts (14). The number of vertices of each part of satisfies

 n(Gkj)≤(4/5)kn(1(2k3)x(b)⋅ex(b)nx(b)+1)n=(e2⋅k3⋅n)x(b),1≤j≤Mk.

Hence

 n(Gkj)b−1<(e2⋅k3⋅n)x(b)(b−1)=e2⋅k3⋅n,

since and hence .

As is in drawing style iii holds for and we have

 e(Gkj)≤k3⋅n(Gkj)b

Therefore, for the total number of edges of we have

 e(Gk)=Mk∑j=1e(Gkj)

contradicting (14). This completes the proof of Theorem 2.1.∎

3 Separated Multigraphs

We derive our Crossing Lemma variants for separated multigraphs (Theorem 1.1) from the generalized Crossing Lemma (Theorem 2.1) presented in Section 2. Let us denote the separated drawing style by and the separated and locally starlike drawing style by . In order to apply Theorem 2.1, we shall find for (1) the largest number of edges in a crossing-free -vertex multigraph in drawing style , (2) an upper bound on the -bisection width of multigraphs in drawing style , and (3) an upper bound on the number of edges in any -vertex multigraph in drawing style .

As for crossing-free multigraphs and are equivalent to the branching drawing style, we can rely on the following Lemma of Pach and Tóth.

Lemma 2 (Pach and Tóth [5])

Any -vertex crossing-free branching multigraph, , has at most edges.

Corollary 1

Any -vertex crossing-free multigraph in drawing style or , , has at most edges.

Also we can derive the bounds on the -bisection width from the corresponding bound for the branching drawing style due to Pach and Tóth.

Lemma 3 (Pach and Tóth [5])

For any multigraph in the branching drawing style with vertices of degrees , and with crossings, the -bisection width of satisfies

 bD(G)≤22 ⎷cr(G)+n∑i=1d2i+n.
Lemma 4

For any multigraph in the drawing style with vertices, edges, maximum degree , and with crossings, the -bisection width of satisfies

 bD(G)≤44√cr(G)+Δ(G)⋅e+n.
Proof

Let be a multigraph in drawing style . Suppose there is a simple closed curve formed by parts of only two edges and , which does not have a vertex in its interior. This can happen between two consecutive crossings of and , or for between a common endpoint and a crossing of and . Further assume that the interior of is inclusion-minimal among all such curves, and note that this implies that an edge crosses along if and only if it crosses along . Say has at most as many crossings along as . We then reroute the part of on very closely along the part of along so as to reduce the number of crossings between and . The rerouting does not introduce new crossing pairs of edges. Hence, the resulting multigraph is again in drawing style and has at most as many crossings as . Similarly, we proceed when has no vertex in its exterior.

Thus, we can redraw to obtain a multigraph in drawing style with , such that introducing a new vertex at each crossing of creates a crossing-free multigraph that is separated, i.e., in drawing style . Now, using precisely the same proof as the proof of its special case Lemma 3 in [5], we can show that

 bD(G′)≤22 ⎷cr(G′)+n∑i=1d2i+n,

where denote the degrees of vertices in . Thus with

 n∑i=1d2i≤Δ(G)n∑i=1di≤2Δ(G)⋅e

the result follows.

Finally, let us bound the number of edges in crossing-free multigraphs. Again, we can utilize the result of Pach and Tóth for the branching drawing style.

Lemma 5 (Pach and Tóth [5])

For any -vertex -edge, , multigraph of maximum degree in the branching drawing style we have and , and both bounds are best-possible.

Lemma 6

For any -vertex -edge multigraph in drawing style of maximum degree we have

1. and if ,

2. and if if .

Moreover, each bound is best-possible.

Proof

Let be a fixed -vertex, , -edge crossing-free multigraph in drawing style .

1. Let . Clearly, every set of pairwise parallel edges contains at most edges, since every lens has to contain a vertex different from the two endpoints of these edges. This gives and . To see that these bounds are tight, consider points in the plane with no four points on a circle. Then it is easy to draw between any two points edges as circular arcs such that the resulting multigraph (which has edges) is in separating drawing style.

2. Let . Consider any fixed vertex in and remove all edges not incident to . The resulting multigraph is branching and hence by Lemma 5 has at most incident edges. Thus and . By Lemma 5, these bounds are tight, even for the more restrictive branching drawing style.

We are now ready to prove that drawing styles and fulfill the requirements of the generalized Crossing Lemma (Theorem 2.1), which lets us prove Theorem 1.1.

Proof (Proof of Theorem 1.1)

Let for a and for b. Clearly, these drawing styles are monotone, i.e., maintained when removing edges, as well as split-compatible. So it remains to determine the constants and such that i, ii, and iii hold for .

i holds with for by Corollary 1. ii holds with for by Lemma 4, which implies the same for . iii holds with and for by Lemma 6a, and with and for by Lemma 6b.

For we have . Thus Theorem 2.1 for gives an absolute constant such that for every -vertex -edge separated and locally starlike multigraph we have , provided . Moreover, by Lemma 6b there are separated multigraphs with vertices and edges, any two of which cross at most once. Hence, the term is best-possible by Lemma 1.

For we have . Thus Theorem 2.1 for gives an absolute constant such that for every -vertex -edge separated multigraph we have , provided . Moreover, by Lemma 6a there are separated multigraphs with vertices and edges, any two of which cross at most twice. Hence, the term is best-possible by Lemma 1.

4 Other Crossing Lemma Variants

We use the generalized Crossing Lemma (Theorem 2.1) to reprove existing variants of the Crossing Lemma due to Székely and Pach, Spencer, Tóth, respectively.

4.1 Low Multiplicity

Here we consider for fixed the drawing style which is characterized by the absence of pairwise parallel edges. In particular, any -vertex multigraph in drawing style has at most edges, i.e., iii holds for with and . Moreover, if is crossing-free on vertices and edges, then , i.e., i holds for with .

Finally, we claim that ii holds for with being independent of . To this end, let be any -vertex -edge multigraph in drawing style . As already noted by Székely [6], we can reroute all but one edge in each bundle in such a way that in the resulting multigraph every lens is empty, no two adjacent edges cross, and . (Simply route every edge very closely to its parallel copy with the fewest crossings.) Clearly, has drawing style .

Now, we place a new vertex in each lens of , giving a multigraph with vertices and edges, which is in the separated drawing style . By Lemma 4, there is an absolute constant such that

 bD(G′′)≤k√cr(G′′)+Δ(G′′)⋅e′′+n′′.

As , , , and we conclude that

 bDm(G)≤2k√cr(G)+Δ(G)⋅e+n.

In other words, ii holds for drawing style with an absolute constant that is independent of .

Note that for , we have . We conclude with Theorem 2.1 that there is an absolute constant such that for every and every -vertex -edge multigraph in drawing style we have

 cr(G)≥α′⋅1kx(b)3⋅ex(b)+2nx(b)+1=α′⋅e3mn2,provided e>(3m+1)n,

which is the statement of Theorem B.

4.2 High Girth

Theorem D (Pach, Spencer, Tóth [4])

For any there is an absolute constant such that for any -vertex -edge graph of girth larger than we have

 cr(G)≥αr⋅er+2nr+1,provided e>4n.

Here we consider for fixed the drawing style which is characterized by the absence of cycles of length at most . In particular, any multigraph in drawing style has neither loops nor multiple edges. Hence i holds for drawing style with . Secondly, drawing style is more restrictive than the branching drawing style and thus also ii holds for . Moreover, any -vertex graph in drawing style has edges [2], i.e., iii holds for with . Finally, is obviously a monotone and split-compatible drawing style.

Thus with , Theorem 2.1 immediately gives

 cr(G)≥αr⋅er+2nr+1,provided e>4n

for any -vertex -edge multigraph in drawing style , which is the statement of Theorem D.

5 Conclusions

Let be a topological multigraph with vertices and edges. We have shown that if is separated and locally starlike, which generalizes the result for branching multigraphs [5], which are additionally single-crossing. Moreover, if is only separated, then the lower bound drops to , which is tight up to the constant factor, too. It remains open to determine a best-possible Crossing Lemma for separated and single-crossing multigraphs. This would follow from our generalized Crossing Lemma (Theorem 2.1), where the missing ingredient is the determination of the smallest such that every separated and single-crossing multigraph on vertices has edges. It is easy to see that the maximum degree may be as high as , but we suspect that any such has edges.

Acknowledgements

This project initiated at the Dagstuhl seminar 16452 “Beyond-Planar Graphs: Algorithmics and Combinatorics,” November 2016. We would like to thank all participants, especially Stefan Felsner, Vincenzo Roselli, and Pavel Valtr, for fruitful discussions.

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