Geometrc drawngs of K n wth few crossngs Bernardo M. Ábrego, Slva Fernández-Merchant Calforna State Unversty Northrdge {bernardo.abrego,slva.fernandez}@csun.edu ver 9 Abstract We gve a new upper bound for the rectlnear crossng number cr(n) of the complete geometrc graph K n.weprovethatcr(n) 0.380559 n + Θ(n 3 ) by means of a new constructon based on an teratve duplcaton strategy startng wth a set havng a certan structure of halvng lnes. Keywords: Rectlnear crossng number, crossng number, geometrc graph, complete graph, convex quadrlateral. AMS Subect Classfcaton: 05C62, 52A10, 52A22, 52C10. 1 Introducton The crossng number cr (G) of a smple graph G s the mnmum number of edge crossngs n any drawng of G n the plane, where each edge s a smple curve. The rectlnear crossng number cr (G) s the mnmum number of edge crossngs n any geometrc drawng of G, thatsadrawngofg n the plane where the vertces are ponts n general poston and the edges are straght segments. The crossng numbers have many applcatons to Dscrete Geometry and Computer Scence, see for example [9] and [10]. In ths paper we contrbute to the problem of determnng cr (K n ),wherek n denotes the complete graph on n vertces. Specfcally, we construct geometrc drawngs of K n wth a small number of crossngs. For smplcty we wrte cr (n) =cr (K n ). We note that a geometrc drawng of K n s determned by the locaton of ts vertces and two edges cross each other f and only f the quadrlateral determned by ther vertces s convex. Thus our drawngs also provde constructons of n-element pont sets wth small number of convex quadrlaterals determned by the n ponts. The problem of fndng the asymptotc behavor of cr (n) s also mportant because of ts close relaton to Sylvester s four pont problem [13]: determne the probablty that four ponts selected unformly at random on a gven doman, form a convex quadrlateral. It s easy to see that {cr (n) / n } s a non-decreasng sequence bounded above by 1. Thus v = lm n cr (n) / n exsts, and Schenerman and Wlf [12] proved that v s the nfmum, over all open sets R wth fnte area, of the probablty that four randomly chosen ponts n R are n convex poston. When ths problem was frst nvestgated, the best lower bounds were obtaned by usng an averagng argument over subsets whose crossng numbers were known. For example, t s well known that cr(5) = 1, so by countng the crossngs generated by every subset of sze 5 t s easy to get cr(n) (1/5) n. Wagner [1] was the frsttouseadfferent approach, he proved v 0.3288. Then the authors [1] and ndependently Lovasz et al. [8], used allowable sequences to prove v 3/8 = 0.375. Lovasz et al. [8] managed to even mprove ths v by 10 5, and Balogh and Salazar [5] refned ths technque even further to obtan the currently best bound of v 0.37533. The hstory of the mprovements on the upper bounds s as follows: In the early seventes Jensen [7] and Snger [11] obtaned v 7/18 < 0.3888, andv 5/13 < 0.3862 respectvely. Much later Brodsky et al. [6] constructed pont sets wth nested non-concentrc trangles yeldng 1
v 667/1688 < 0.3838. Then Achholzer et al. [3] devsed a lense-replacement constructon dependng on a sutable ntal set. They obtaned v 0.380739 and later on Jensen provded a better ntal set [], whch gave the prevously known best bound of v 0.38062. In ths paper we prove the followng, Theorem 1 cr(n) 78750 29969 n + Θ(n 3 ) < (0.380559) n Ths theorem s based on the followng stronger result whch may mprove the upper bound n the future. To accomplsh ths we would only need a pont-set P havng a halvng-lne matchng (defnednext)andwthabetter n coeffcent. Theorem 2 If P s a N-element pont set n general poston, wth N even, and P has a halvng-lne matchng; then µ 2 cr(p )+3N 3 7N 2 +(30/7)N cr(n) N µ n Let P = {p 1,p 2,...,p N } be a general poston pont set n the plane. We defne a halvng lne of P as a lne passng through two ponts n P and leavng the same number of ponts of P on ether sde of the lne. Accordng to ths, N needs to be even for P to have a halvng lne. Consder a bpartte graph G =(P, H) where H s the set of halvng lnes of P and p P s adacent to l H f p s on the lne l. If there s a matchng for P n the graph G we say that ths matchng s a halvng-lne matchng of P. Note that such a matchng nduces a functon on P, p 7 p f(),suchthatp p f() s a halvng lne of P. In general, most even sets P mnmzng the crossng number have a large number of halvng lnes and consequently most of these sets have a halvng-lne matchng. The only excepton seems to be P, the set consstng of a trangle wth a pont nsde. Ths set acheves cr() = 0 but t only has three halvng lnes. The prevously best upper bound, obtaned by Achholzer et al. [3], used a lens-replacement constructon yeldng µ 2 cr(p )+3N 3 7N 2 µ +6N n cr(n) N + Θ(n 3 ) (1) for any ntal pont set P wth N elements, N even. Jensen (reported n []) dscovered a set of 36 ponts wth 21175 crossngs whch happened to be the best ntal set known at the tme. The bound obtaned s cr(n) (0.38062) n Note that our constructon from Theorem 2 always gves a better bound compared to (1) as long as the startng set P has a halvng-lne matchng. 2 The Constructon The constructon s based on the followng lemma, Lemma 3 If P s a N-element pont set, N even, and P has a halvng-lne matchng; then there s a pont set Q = Q(P ) n general poston, Q =2N, Q also has a halvng-lne matchng, and cr(q) =16cr(P )+(N/2)(2N 2 7N +5). Proof. Q s constructed as follows. Each pont p P wll be replaced by a par of ponts q 1 and q 2. Usng the functon f nducedbythehalvng-lnematchngwedefne q 1 = p + ε p f() p p f() p and q 2 = p ε p f() p p f() p, where ε s small enough so that all ponts q 1,q 2 comng from a pont p located to the left (rght) of p p f() ; are also located to the left (rght) of the lnes q 1 q f()1 and q 1 q f()2. In other words, the 2
cone wth vertex q 1 spanned by the segment q f()1 q f()2 does not contan any of the ponts q x wth 6=, f(). One way to fnd such an ε s the followng: suppose that P scontanednadskwth dameter D, sncep s n general poston there s δ>0 so that the strps of wdth δ centered at the lnes p p f() have only the ponts p and p f() wth P n common. The porton contaned n the dsk of any cone wth center n the lne p p f(),axsp p f(), and wth an openng angle of 2arctan(δ/2D); s a subset of the strp of wdth δ around p p f(). Thus we can start wth a small arbtrary value ε 0 <δsuch that all ponts q 1,q 2 arensdethedsk. Obtanε as the dstance from p f() to the lnes subtendng the cone wth center q 1,axsp p f(), and angle 2arctan(δ/2D). And fnally set ε as the mnmum of all the ε. Fgure 1: Constructon of the set Q From the halvng-lne matchng defnton we deduce that no two ponts n P are assocated to the same halvng lne. Ths, together wth the choce of ε, guarantees that Q s n general poston whenever P s n general poston. By constructon the lne q 1 q 2 s a halvng lne of Q. In addton, snce q 1 s n the nteror of the trangle q 2,q f()1,q f()2 the lne q 1 q f()1 (and also the lne q 1 q f()2 ) s a halvng lne of Q. Then { q 1, q 1 q f()1 : =1, 2,...,N} {(q2, q 1 q 2 ): =1, 2,...,N} s a halvng-lne matchng of Q. Now we proceed to calculate cr(q) by countng crossngs accordng to three dfferent types: Type I Type II(1) Type II(2) Type III = f( ) k k Fgure 2: Dfferent types of crossngs Type I. Two ponts n par and two n par 6=. Thereare N 2 ways of choosng pars and and all of them determne a crossng except when = f() or = f(). Snce there are exactly N pars (, f()), the total number of crossngs n ths case s N 2 N. Type II Two ponts n par and the two other n pars and k all pars dstnct. Frst there are N choces for the par. Thenwehavetwocases,when, k 6= f(), and when ether or k 3
equals f(). Inthefrstcase(seeFgure2,TypeII(1)),tohaveacrossng,bothp and p k must be on the same sde of the lne q 1 q 2. Thus there are 2 N/2 1 2 ways of choosng and k and then choces for the second ndces x and y for the ponts q x and q ky. In the second case (see Fgure 2, TypeII(2)) we can assume = f(). Then there are 2 choces for the second ndex x of q x. Agan, to have a crossng, we need q x and p k on the same sde of q 1 q 2. So there are N/2 1 ways of choosng ³ k and 2 choces for the second ndex y of q ky. The total number of Type II crossngs s N 8 N/2 1 2 +(N/2 1). Type III Each pont n a dfferent par. To have a crossng each of the four pars must come from acrossngnp,sotherearecr(p ) possble pars, and there are 2 choces for the second ndex n each par. Thus there are 16cr(P ) number of crossngs of Type III. The lst of crossngs types s complete snce by constructon none of the segments q 1 q 2 partcpate n any crossng. Addng Types I-III yelds the result. n cr(p n ) cr(q(p n )) old bound for cr(2n) new bound for cr(2n) 30 9726 17951 17955 17951 36 21175 381010 381020 381010 8 71028 1239096 1239139 1239096 Table 1: Improved bounds for 60,72,and 96 ponts. The strength of ths Lemma can be easly seen when t s appled to the set P 6 that mnmzes cr(6) = 3. We obtan cr(12) cr(q) = 153 whch happens to be the correct value of cr(12) (a drawng s shown n Fgure 1). It took some effort [3] to fnd ths pont set n the past. In addton, f we use the sets P 30,P 36,P 8 obtaned by Acholzer et al., and reported n web page [2], we obtan sets P 60,P 72,P 96 wth less number of crossngs than those prevously known (see Table 1). th pont = p =(x-coordnate,y-coordnate) p 1 = (9259, 16598) p 9 = (2877, 16613) p 17 = (511, 23755) p 25 = (9075, 15320) p 2 = (9763, 16199) p 10 = (15909, 1615) p 18 = (915, 17055) p 26 = (7921, 1307) p 3 = (9977, 16397) p 11 = (96, 15905) p 19 =(0, 3239) p 27 = (5206, 851) p = (1028, 16225) p 12 = (950, 1651) p 20 = (6820, 20921) p 28 = (9121, 15603) p 5 = (10666, 16385) p 13 = (9262, 16627) p 21 = (999, 1615) p 29 = (80, 0) p 6 = (1289, 16335) p 1 = (9282, 1697) p 22 = (9355, 16177) p 30 = (632, 10637) p 7 = (18577, 1651) p 15 = (8912, 17261) p 23 = (919, 15893) p 8 = (10391, 16281) p 16 = (782, 19232) p 2 = (916, 15771) Table 2: Coordnates of pont set P 30 wth 9776 crossngs 3 Proof of the Theorems Let S 0 = P and S k+1 = Q(S k ) for k 0, whereq(s k ) s the set gven by Lemma 3. By nducton and Lemma 3 t follows that cr(s k )=16 k cr(p )+N 3 8 k 1 2 k 1 7 6 N 2 k 1 k 1 + 5 1 N2k 1 8 k 1. Lettng n = S k =2 k N we get µ 2cr(P )+3N 3 7N 2 +(30/7)N cr(n) cr(s k )= 2N n 1 8 n3 + 7 2 n2 5 28 n.
a, (a, b) means (p a,p a p b ) 1, (1, 3) 9, (9, 13) 17, (17, 18) 25, (25, 26) 2, (2, ) 10, (10, 11) 18, (18, 19) 26, (26, 27) 3, (3, 7) 11, (11, 12) 19, (19, 21) 27, (27, 28), (, 5) 12, (12, 1) 20, (20, 21) 28, (28, 29) 5, (5, 8) 13, (13, 1) 21, (21, 23) 29, (29, 27) 6, (6, 7) 1, (1, 15) 22, (22, 23) 30, (30, 29) 7, (7, 9) 15, (15, 16) 23, (23, 2) 8, (8, 10) 16, (16, 20) 2, (2, 25) Table 3: Halvng-lne matchng of P 30 Ths proves Theorem 2. Now, to prove Theorem 1, consder the set P = P 30 wth coordnates n Table 2 obtaned from [2]. It satsfes that cr(p 30 ) = 9726 and t can be verfed that the set of pont-lne pars n Table 3 represents a halvng-lne matchng of P 30.Wethenget cr(n) 29969 µ n 78750 References [1] Ábrego, B., Fernández-Merchant, S.: A Lower Bound for the Rectlnear Crossng Number, Graphs and Combnatorcs, to appear (2005). [2] Achholzer, O.: Rectlnear Crossng Number Page. http://www.st.tugraz.at/staff/achholzer/crossngs.html. [3] Achholzer, O., Aurenhammer, F., Krasser, H.: On the crossng number of complete graphs. In: Proc. 18th Ann ACMSympCompGeom., pp. 19-2, Barcelona Span, 2002. [] Achholzer, O., Krasser, H.: Abstract order type extensons and new results on the rectlnear crossng number. Submtted. [5] Balogh, J., Salazar G.: On k-sets, quadrlaterals and the rectlnear crossng number of K n.manuscrpt,submtted. [6] Brodsky, A., Durocher, S., Gethner, E.: Toward the rectlnear crossng number of K n : new drawngs, upper bounds, and asymptotcs. Dscrete Mathematcs 262, 59-77 (2003) [7] Jensen, H.F.: An upper bound for the rectlnear crossng number of the complete graph. J. Combn. Theory Ser B 10, 212-216 (1971) [8] Lovász, L., Vesztergomb, K., Wagner, U., Welzl, E.: Convex quadrlaterals and k-sets. In: Pach, J. edtor: Towards a theory of geometrc graphs, (Contemporary Mathematcs Seres, vol. 32, pp.139-18) AMS 200. [9] Matoušek, J.: Lectures on Dscrete Geometry, New York, N.Y.: Sprnger-Verlag 2002. [10] Pach, J., Tóth, G.: Thrteen problems on crossng numbers. Geombnatorcs 9, 19-207 (2000) [11] Snger, D.: Rectlnear crossng numbers. Manuscrpt (1971) [12] Schenerman, E. R., Wlf, H. S., The rectlnear crossng number of a complete graph and Sylvester s four pont problem of geometrc probablty, Amer. Math. Monthly 101 (199), 939-93. [13] Sylvester, J.J., On a specal class of questons n the theory of probabltes, Brmngham Brtsh Assoc., Report 35, 8-9 (1865). [1] Wagner, U.: On the rectlnear crossng number of complete graphs. In: Proc. of the 1th Annual ACM-SIAM Symposum on Dscrete Mathematcs (SODA), pp. 583-588, 2003. [15] Zarankewcz, K.: On a problem of P. Turán concernng graphs. Fund. Math. 1, 137-15 (195) 5