Crossing Numbers and Hard Erd Os Problems in Discrete Geometry

Home , Discrete geometry

Combinatorics, Probability and Computing 1993 11, 1{10

Crossing Numb ers and Hard Erd}os

Problems in Discrete Geometry

L ASZLO SZEKELY

Department of Computer Science, Eotvos University,

Budap est, Muzeum krt 6{8, 1088 Hungary

Received

We show that an old but not well-known lower b ound for the crossing numberofa

graph yields short pro ofs for a numb er of b ounds in discrete plane geometry, which

were considered hard b efore: the numb er of incidences among p oints and lines, the

maximum numb er of unit distances among n p oints, the minimum numb er of distinct

distances among n p oints.

\A statement ab out curves is not interesting unless it is already interesting in the case

of a circle." H. Steinhaus

The main aim of this pap er is to derive short pro ofs for a number of theorems in

discrete geometry from Theorem 1.

Theorem 1. Leighton [16], Ajtai, Chv atal, Newb orn, and Szemer edi [1] For

any simple graph G with n vertices and e 4n edges, the crossing number of G on the

3 2

plane is at least e =100n .

For many graphs, Theorem 1 is tight within a constant multiplicative factor. This

result was conjectured by Erd}os and Guy [10, 12], and rst proved by Leighton [16],

who was unaware of the conjecture, and indep endently by Ajtai, Chv atal, Newb orn, and

Szemer edi [1]. The theorem is still hardly known, some distinguished mathematicians

even recently thought of the Erd}os{Guy conjecture as an op en problem. Shahrokhi,

S ykora, Sz ekely, and Vrto generalized Theorem 1 for compact 2-dimensional manifolds

with a transparent pro of Theorem 5, [21,22].

The original pro ofs of the applications of Theorem 1 shown here Theorems 2, 3, 4, 6

used sophisticated to ols like the covering lemma [25]. Although some of those theorems

were given simpler pro ofs and generalizations which used metho ds from the theory of VC

dimension and extremal graph theory, see Clarkson, Edelsbrunner, Guibas, Sharir and

Welzl[6], Furedi and Pach [11], Pach and Agarwal [18], Pach and Sharir [19], the simpler

pro ofs still lacked the simplicity and generality shown here. I b elieve that the notion of

crossing number is a central one for discrete geometry, and that the right branch of graph theory to b e applied to discrete geometry is \extremal top ological graph theory"

2 L aszl oSz ekely

rather than classical extremal graph theory. Other lower b ounds for the crossing number

see [22] mayhave other striking applications in discrete geometry.

A drawing of a graph over a surface represents edges by curves such that edges do not

pass through vertices and no three edges meet in a common internal p oint. The crossing

number of a graph over a surface is de ned as the minimum numb er of crossings of edges

over all drawings of the graph. It is not dicult to see that if we allow the intersection

of many edges in an internal p oint but count crossings of pairs of edges, then wedonot

change the de nition of the crossing numb er. We denote by cr G the planar crossing

number of a graph G.We shall always use i to denote the numb er of incidences b etween

the given sets of p oints and lines or curves in a particular situation under discussion.

We shall write c for a constant; as usual, di erent o ccurences of c may denote di erent

constants.

Theorem 2. Szemer edi and Trotter [24] For n points and l lines in the Euclidean

2=3

plane, the number of incidences among the points and lines is at most c[nl + n + l ].

Pro of. Wemay assume without loss of generality that all lines are incident to at least

one p oint. De ne a graph G drawn in the plane such that the vertex set of G is the set of

our n given p oints, and join two p oints with an edge drawn as a straight line segment, if

the p oints are consecutive p oints on one of the lines. This drawing shows that cr G l .

The numberofpoints on any of the lines is one greater than the numb er of edges drawn

along that line. Therefore the number of incidences among the p oints and the lines is

at most ` greater than the numb er of edges in G. Theorem 1 nishes the pro of: either

3 2

4n i l or cr G ci l =n .

For n = l the statement of Theorem 2 was conjectured by Erd}os [9]. Although Theo-

rem 3 b elow is known to b e a simple corollary of Theorem 2, we prove it, since we use it

in the pro of of Theorem 4.

Theorem 3. Szemer edi and Trotter [24] Let 2 k n. For n points in the

2 3

Euclidean plane, the number l of lines containing at least k of them is at most cn =k .

Pro of. Initiate the construction of the graph G from the pro of of Theorem 1, taking

only lines passing through at least k p oints. Note that G has at least l k 1 edges.

2 3 2 3 2

Hence, wehave either l >ce =n >c[lk1] =n or l k 1 < 4n. In the rst case we

2 3

are at home, and so are we in the second, as l<4n=k 1

Theorem 3, whichwas conjectured by Erd}os and Purdy [9], is known to b e tight for

p p p

n for the p oints of the n n grid [2]. In 1946 Erd}os [8] conjectured 2 k

1+o1

that the number of unit distances among n p oints is at most n and proved that

3=2

this numb er is at most cn . In 1973 J ozsa and Szemer edi [13] improved this b ound to

3=2 1:44:::

on . In 1984 Beck and Sp encer [4] further improved the b ound to n . Finally,

4=3

Sp encer, Szemer edi, and Trotter [23]achieved the b est known b ound, cn .

Theorem 4. Sp encer, Szemer edi, and Trotter [23] The number of unit distances

4=3

among n points in the plane is at most cn .

Crossing Numbers and HardErd}os Problems in Discrete Geometry 3

Pro of. Draw a graph G in the plane in the following way. The vertex set is the set of

given p oints. Draw a unit circle around each p oint, in this way consecutive p oints on the

unit circles are connected by circular arcs: these are the edges of a multigraph drawn in

the plane. Discard the circles that contain at most two p oints, and for anytwo p oints

joined by some circular arcs, keep precisely one of these arcs. Let G be the resulting

graph. The numb er of edges of this graph is at most O n less than the numb er of unit

distances. The number of crossings of G in this drawing is at most 2n , since anytwo

circles intersect in at most 2 p oints. An application of Theorem 1 nishes the pro of.

Based on earlier work of Kainen [14] and Kainen and White [15], Shahrokhi, Syk ora,

Sz ekely, and Vrto [21,22] found the following generalization of Theorem 1.

Theorem 5. Shahrokhi, S ykora, Sz ekely, and Vrto [21, 22] Let G b a simple

graph G drawn on an orientable or non-orientable compact 2-manifold of genus g . Assume

that G has n vertices, e edges, and e 8n. Then the number of crossings in the drawing

3 2 2 2 2

of G is at least ce =n ,if n =e g , and is at least ce =g +1,if n =e g e=64.

Theorem 5 is known to b e tight within a factor of O log g + 2 for some graphs [21].

We obtain the following generalization of Theorem 2 by combining Theorems 5 and 1.

Theorem 6. Suppose that we are given l simple curves and p points on an orientable

or non-orientable compact 2-manifold of genus g . Assume that any two curves intersect

in at most one point. Assume furthermore that every curve is incident to at least one

2=3 2

point. Then i = O pl + p + l if p =i l g and i = O l g +1 + p if

p =i l g i l =64.

Theorem 7. Suppose G is a multigraph with n nodes, e edges and maximum edge

3 2

multiplicity m. Then either e<5nm or cr G ce =n m.

Takeany simple graph H for which Theorem 1 is tight with a drawing which shows it,

and substitute each edge with m closely drawn parallel edges. For the new graph m H

Theorem 7 is tight. For the pro of of Theorem 7 we need the following simple fact:

Prop osition 1. cr k H =k cr H .

Pro of. Takeany drawing of H with cr H crossings. Substitute the edges of H with

k parallel edges closely drawn to the original edge. In this waywe obtain a drawing of

k H with k cr H edges. On the other hand, takeany drawing of k H . Picking one

jE H j

representative of parallel edges in k ways, we obtain a drawing of H , which exhibits

at least cr H crossings. Any pair of crossing edges in k H that arose ab ove is counted

jE H j2

k times.

Pro of of Theorem 6.For 0 i log m, let G denote the subgraph of G in which

twovertices a; b are joined with t>0 edges if and only if in G they are joined by exactly

4 L aszl oSz ekely

i i+1

t edges and 2 t<2 . Set

i+3

A = fi 2 [0; log m]: jEG j2 ng and B =[0;log m] n A:

2 2

Wemay assume e 5nm. Therefore, since jE G j4nm,wehave

i2A

X X

jEGjjEGj jEG jjEGj4mn jEGj=5:

i i

i2B i2A

Let G denote the simple graph obtained from G by identifying parallel edges. Wehave

log m log m

2 2

X X X

jE G j

2i 2i i

2 cr G cr G 2 cr G c

i=0 i=0

i2B

X X

jE G j c jE G j

i i

c = ;

2 i 2

i=3

n 2 n

i2B i2B

with the second inequality following from Prop osition 1. Hence, by applying Holder's

inequality with p = 3 and q =3=2, we nd that

3 3 2

X X X

c jE G j ce c

i=3 i=3 3=2

cr G 2 jE G j : 2

2 2 2

i=3

n n m n m

i2B i2B i2B

Theorem 7 has the following immediate consequence concerning incidences between

p oints and curves.

Theorem 8. Given p points and l simple curves in the plane, such that any two curves

intersect in at most t points and any two points belong to at most m curves, the number

of incidences is at most

2=3 1=3

clp tm + l +5mp:

Although Theorem 2 has b een given several generalizations in the vein of Theorem 8,

they seem to admit algebraic curves only [19,6,11].

In 1946 Erd}os [8] conjectured that n p oints in the plane determine at least cn= log n

distinct distances and showed that the minimum is at least n. Over the years, this

2=3

lower b ound has b een improved several times. In 1952 Moser [17] proved n , then

2=3 5=7

in 1984 Chung [5] improved n to n and in an unpublished manuscript Beck [3]

58=81"

proved n . The b est lower b ound to date has b een proved by Chung, Szemer edi

and Trotter [7].

4=5 c

Theorem 9. Chung, Szemer edi and Trotter [7] At least n =log n distinct

distances are determinedbynpoints in the plane.

They proved this for a large c and claimed that a much smaller c, p erhaps even c =0,

can be obtained with their metho d. However, as they remarked, they did not prove

the existence of a single p oint from which determined many distances. Given p oints

x ;:::;x ,wesay that x determines at least t distances from the others if there are t

1 n i

are distinct ;:::;dx ;x ;dx ;x such that the distances dx ;x ;:::;x p oints x

i j i j i j j j

t 2 1 t 1

and p ositive. Here is the b est published result for this mo di ed problem.

Crossing Numbers and HardErd}os Problems in Discrete Geometry 5

Theorem 10. Clarkson, Edelsbrunner, Guibas, Sharir and Welzl [6] Given

3=4

n points in the plane, one of them determines at least cn distinct distances from the

others.

4=5

Theorem 11. Given n points in the plane, one of them determines at least cn

distinct distances from the others.

Pro of. Let t b e the maximum numb er of distinct distances measured from any p oint.

Wemay assume t = on= log n, otherwise there is nothing to prove. Draw circles around

every p oint with every distance as radius, which can b e measured from that p oint. We

have drawn at most t concentric circles around each p oint. De ne amultigraph whose

vertices are the p oints, and whose edges are arcs connecting consecutive p oints on the

circles wehave drawn. Delete edges lying in circles containing at most two p oints. Since

t = on, the resulting multigraph G still has cn edges. Theorem 7 cannot b e applied

immediately, since very high edge multiplicities may o ccur.

Prop osition 2. The number of pairs f; a, where f is a line with at least k points,

a is an arc representing an edge of G, and f is the symmetry axis of a, is at most

2 2

ctn =k + ctn log n.

i 2 3i

Pro of. By Theorem C, the numb er of lines with at least 2 p oints is at most cn =2 ,as

far as 2 n.For each such line, the numb er of bisected edges is at most 2t times the

numberofpoints on the line. Therefore, the numb er of pairs f; a with k jfj4 n

is at most

2 2

n ctn

ct 2 :

3i 2

2 k

i:k 2 n

A simple and well-known inclusion-exclusion argument see [20] shows that the number

of lines with numb er of p oints b etween a and 2a 4 n a is at most cn=a. Hence the

contribution of such big lines to the numb er of pairs is at most

ct 2 < ctn log n:

i: n<2

Pro of of Theorem 11.Now the numb er of pairs in Prop osition 2 is just the number of

edges joining pairs of p oints which are joined by at least k edges. Thus, deleting all such

t for a suitable constant K ,we arriveatamultigraph G still having edges with k = K

2 2 2

cn edges. The crossing number of G is at most 2n t .Thus using Theorem 7 wehave

6 3

cn cjE G j

2 2

p p

2n t cr G

2 2

n K t n K t

and the theorem follows.

I am indebted to the anonymous referee, whose suggestions substantially improved the

presentation of the pap er.

6 L aszl oSz ekely

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