Many Touchings Force Many Crossings

Transcription

1 May Touchigs Force May Crossigs Jáos Pach 1, ad Géza Tóth 1 École Polytechique Fédérale de Lausae, St. 8, Lausae 1015, Switzerlad pach@cims.yu.edu Réyi Istitute, Hugaria Academy of Scieces 1364 Budapest, POB 17, Hugary geza@reyi.hu Abstract. Give cotiuous ope curves i the plae, we say that a pair is touchig if they have oly oe iterior poit i commo ad at this poit the first curve does ot get from oe side of the secod curve to its other side. Otherwise, if the two curves itersect, they are said to form a crossig pair. Let t ad c deote the umber of touchig pairs ad crossig pairs, respectively. We prove that c 1 t, provided 10 5 that t 10. Apart from the values of the costats, this result is best possible. 1 Itroductio I the cotext of the theory of topological graphs ad graph drawig, may iterestig questios have bee raised cocerig the adjacecy structure of a family of curves i the plae or i aother surface [FP10]. I particular, durig the past four decades, various importat properties of strig graphs (i.e., itersectio graphs of curves i the plae) have bee discovered, ad the study of differet crossig umbers of graphs ad their relatios to oe aother has become a vast area of research. A useful tool i these ivestigatios is the so-called crossig lemma of Ajtai, Chvátal, Newbor, Szemerédi ad Leighto [ACNS8], [Le83]. It states the followig: Give a graph of vertices ad e > 4 edges, o matter how we draw it i the plae by ot ecessarily straight-lie edges, there are at least costat times e 3 / crossig pairs of edges. This lemma has ispired a umber of results establishig the existece of may crossig subcofiguratios of a give type i sufficietly rich geometric or topological structures [D98], [Sh03], [SoT01], [GNT00]. I this ote, we will be cocered with families of curves i the plae. By a curve, we mea a o-selfitersectig cotiuous arc i the plae, that is, a homeomorphic image of the ope iterval (0,1). Two curves are said to touch each other if they have precisely oe poit i commo ad at this poit the first curve does ot pass from oe side of the secod curve to the other. Ay other pair of curves with oempty itersectio is called crossig. A family of curves is i geeral positio if ay two of them itersect i a fiite umber of poits ad o three pass through the same poit.

2 Let be eve, t be a multiple of, ad suppose that t < 4. Cosider a collectio A of t > pairwise disjoit curves, ad aother collectio B curves such that of t (i) A B is i geeral positio, (ii) each elemet of B touches precisely elemets of A, ad (iii) o two elemets of B touch each other. The family A B cosists of curves such that the umber of touchig pairs amog them is t. The oly pairs of curves that may cross each other belog to B. Thus, the umber of crossig pairs is at most ( t/ ) t. See Figure 1. A B t/ / Fig.1: A set of curves with t touchig pairs ad at most t crossig pairs. The aim of the preset ote is to prove that this costructio is optimal up to a costat factor, that is, ay family of curves ad t touchigs has at least costat times t crossig pairs. Theorem. Cosider a family of curves i geeral positio i the plae which determies t touchig pairs ad c crossig pairs. If t 10, the we have c 1. This boud is best possible up to a costat factor t We make o attempt to optimize the costats i the Theorem. Pach, Rubi, ad Tardos [PRT16] established a similar relatioship betwee t, the umber of touchig pairs, ad C, the umber of crossig poits betwee thecurves.theyprovedthatc t(loglog(t/)) δ,foraabsolutecostatδ > 0. Obviously, we have C c. There is a arragemet of red curves ad blue curves i the plae such that every red curve touches every blue curve, ad the total umber of crossig poits is C = Θ( log); cf. [FFPP10]. Of course, the umber of crossig pairs, c, ca ever exceed ( ). Betwee arbitrary curves, the umber of touchigs t ca be as large as ( 3 4 +o(1))( ) ; cf. [PT06]. However, if we restrict our attetio to algebraic plae curves of bouded degree, the we have t = O( 3/ ), where the costat hidde i the otatio depeds o the degree [ESZ16].

3 May Touchigs Force May Crossigs 3 Proof of Theorem We start with a easy observatio. Lemma. Give a family of 3 curves i geeral positio i the plae, o two of which cross, the umber of touchigs, t, caot exceed 3 6. Proof. Pick a differet poit o each curve. Wheever two curves touch each other at a poit p, coect them by a edge (arc) passig through p. I the resultig drawig, ay two edges that do ot share a edpoit are represeted by disjoit arcs. Accordig to the Haai-Tutte theorem [Tu70], this meas that the uderlyig graph is plaar, so that its umber of edges, t, satisfies t 3 6. Proof of Theorem. We proceed by iductio o. For 0, the statemet is void. Suppose that > 0 ad that the statemet has already bee proved for all values smaller tha. We distiguish two cases. CASE A: t 10 3/. I this case, we wat to establish the stroger statemet By the assumptio, we have c t. 1 t 100. (1) LetG t (resp.,g c )deotethetouchig graph(resp.,crossig graph)associated with the curves. That is, the vertices of both graphs correspod to the curves, ad two vertices are coected by a edge if ad oly if the correspodig curves are touchig (resp., crossig). Let T be a miimal vertex cover i G c, that is, a smallest set of vertices of G c such that every edge of G c has at least oe edpoit i T. Let τ = T. Let U deote the complemet of T. Obviously, U is a idepedet set i G c. Accordig to the Lemma, the umber of edges i G t [U], the touchig graph iduced by U, satisfies E(G t [U]) < 3 U 3. () By the miimality of T, G c has at least T = τ edges. That is, we have c τ, so we are doe if τ 1 t. From ow o, we ca ad shall assume that τ < 1 t. By (1), we have 1 t 100. Hece, T 100 ad U = T (3)

4 4 G c T (cover) T = τ U 0 (isolated) U Fig.: Graph G c. Let U U deote the set of all vertices i U that are ot isolated i the graph G c. By the defiitio of T, all eighbors of a vertex v U i G c belog to T. If U 1 t, the we are doe, because c U. Therefore, we ca assume that U < t 100, (4) where the secod iequality follows agai by (1). Lettig U 0 = U\U, by (3) ad (4) we obtai U 0 = U U Clearly, all vertices i U 0 are isolated i G c. Suppose that G t [T U t ] has at least 10 edges. Cosider the set of curves T U. We have 0 = T U 100 ad, the umber of touchigs, t 0 = E(G t [T U ]) t 10. Therefore, by the iductio hypothesis, for the umber of crossigs we have c 0 = E(G c [T U ]) 1 t t 0 ad we are doe. Hece, we assume i the sequel that G t [T U ] has fewer tha t 10 edges. Cosequetly, for the umber of edges i G t ruig betwee T ad U 0, we have E(G t [T,U 0 ]) t E(G t [T U ]) E(G t [U 0 U ]) t t 10 3 > t. (5) Here we used the assumptio that t 10. Let χ = χ(g c [T]) deote the chromatic umber of G c [T]. I ay colorig of a graph with the smallest possible umber of colors, there is at least oe edge betwee ay two color classes. Hece, G c [T] has at least ( ) χ 1 t edges, ad we are doe, provided that χ > 1 70 t. Thus, we ca suppose that χ = χ(g c [T]) 1 70 t. (6)

5 May Touchigs Force May Crossigs 5 Cosider a colorig of G c [T] with χ colors, ad deote the color classes by I 1,I,...,I χ. Obviously, for every j, I j U 0 is a idepedet set i G c. Therefore, by the Lemma, G t [I j U 0 ] has at most 3 edges. Summig up for all j ad takig (6) ito accout, we obtai E(G t [T,U 0 ]) χ j=1 E(G t [I j U 0 ]) 1 70 t 3 t 0, cotradictig (5). This completes the proof i CASE A. CASE B: t 10 3/. Set p = 103 t Select each curve idepedetly with probability p. Let, t, ad c deote the umber of selected curves, the umber of touchig pairs, ad the umber of crossig pairs betwee them, respectively. Clearly, E[ ] = p, E[t ] = p t, E[c ] = p c. (7) The umber of selected curves,, has biomial distributio, therefore, By Markov s iequality, Prob[ p > 1 4 p] < 1 3. (8) Prob[c > 3p c] < 1 3. (9) Cosider the touchig graph G t. Let d 1,...,d deote the degrees of the vertices of G t, ad let e 1,...,e t deote its edges, listed i ay order. We say that a edge e i is selected (or belogs to the radom sample) if both of its edpoits were selected. Let X i be the idicator variable for e i, that is, { 1 if ei was selected, X i = 0 otherwise. We have E[X i ] = p. Let t = t i=1 X i. It follows by straightforward computatio that for every i, var[x i ] = E[(X i E[X i ]) ] = p p 4, If e i ad e j have a commo edpoit for some i j, the cov[x i,x j ] = E[X i X j ] E[X i ]E[X j ] = p 3 p 4. If e i ad e j do ot have a commo vertex, the X i ad X j are idepedet radom variables ad cov[x i,x j ] = 0. Therefore, we obtai σ = var[t ] = t var[x i ]+ i=1 1 i j t cov[x i,x j ]

6 6 t = var[x i ]+ cov[x i,x j ] i=1 1 i j t = (p p 4 )t+(p 3 p 4 ) d i (d i 1) < p t+p 3 t. i=1 From here, we get σ < p t+ p 3 t < p t = E[t ]. By Chebyshev s iequality, Settig λ = 1 4, Prob[ t p t λσ] 1 λ. Prob[ t p t p t 4 ] 1 4 < 1 3. (10) It follows from (8), (9), ad (10) that, with positive probability, we have p 1 4 p, c 3p c, t p t 1 4 p t. (11) Cosider a fixed selectio of curves with t touchig pairs ad c crossig pairs for which the above three iequalities are satisfied. The we have ad, hece, O the other had, so that t 3 4 p t = t 3, 5 4 p = t, t 10. (1) t 5 4 p t = t 3, 3 4 p = t, 10( ) 3/ / 4 t 3 > t. (13) Accordig to (1) ad (13), the selected family meets the requiremets of the Theorem i CASE A. Thus, we ca apply the Theorem i this case to obtai that c 1 t. I view of (11), we have 3/ 6 3p c c, t 3 4 p t, 5 4 p. Thus, 3p c c t (3p t/4) (5p/4) = ( ) 3 p t 5.

7 May Touchigs Force May Crossigs 7 Comparig the left-had side ad the right-had side, we coclude that c t, as required. This completes the proof of the Theorem. Ackowledgmet. The work of Jáos Pach was partially supported by Swiss Natioal Sciece Foudatio Grats ad Géza Tóth s work was partially supported by the Hugaria Natioal Research, Developmet ad Iovatio Office, NKFIH, Grat K Refereces ACNS8. M. Ajtai, V. Chvátal, M. Newbor, E. Szemerédi, Crossig-free subgraphs. I: Theory ad Practice of Combiatorics, North-Hollad Mathematics Studies 60, North-Hollad, Amsterdam, 198, 9 1. D98. T. K. Dey, Improved bouds o plaar k-sets ad related problems, Discrete Comput. Geom. 19 (1998), ESZ16. J. S. Elleberg, J. Solymosi, ad J. Zahl, New bouds o curve tagecies ad orthogoalities, Discrete Aal. (016), Paper No. 18, pp. FFPP10. J. Fox, F. Frati, J. Pach, ad R. Pichasi, Crossigs betwee curves with may tagecies, i: WALCOM: Algorithms ad Computatio, Lecture Notes i Comput. Sci. 594, Spriger-Verlag, Berli, 010, 1 8. Also i: A Irregular Mid, Bolyai Soc. Math. Stud. 1, Jáos Bolyai Math. Soc., Budapest, 010, FP10. J. Fox ad J. Pach, A separator theorem for strig graphs ad its applicatios, Combi. Probab. Comput. 19 (010), GNT00. A. García, M. Noy, ad J. Tejel, Lower bouds o the umber of crossig-free subgraphs of K, Comput. Geom. 16 (000), o. 4, Le83. T. Leighto, Complexity Issues i VLSI, Foudatios of Computig Series, MIT Press, Cambridge, PRT16. J. Pach, N. Rubi, ad G. Tardos, Beyod the Richter-Thomasse Cojecture, i: Proc. 7th Aual ACM-SIAM Symposium o Discrete Algorithms(SODA 016, Arligto), SIAM, 016, PT06. J. Pach ad G. Tóth, How may ways ca oe draw a graph? Combiatorica 6 (006), o. 5, Sh03. M. Sharir, The Clarkso-Shor techique revisited ad exteded, Combi. Probab. Comput. 1 (003), o., SoT01. J. Solymosi ad C. D. Tóth, Distict distaces i the plae, Discrete Comput. Geom. 5 (001), o. 4, Sz97. L. A. Székely, Crossig umbers ad hard Erdős problems i discrete geometry, Combi. Probab. Comput. 6 (1997), o. 3, Tu70. W. T. Tutte, Toward a theory of crossig umbers, J. Combiatorial Theory 8 (1970),