On Grids in Topological Graphs

Transcription

1 On Grids in Topological Graphs Eyal Ackrman Dpartmnt of Computr Scinc Fri Univrsität Brlin Takustr. 9, Brlin, Grmany Jacob Fox Dpartmnt of Mathmatics Princton Univrsity Princton, NJ Andrw Suk Dpartmnt of Mathmatics Courant Institut, NYU 251 Mrcr St., Nw York, NY János Pach City Collg, CUNY and Courant Institut NYU,NwYork,NY ABSTRACT A topological graph is a graph drawn in th plan with vrtics rprsntd by points and dgs as arcs conncting its vrtics. A k-grid in a topological graph is a pair of substs of th dg st, ach of siz k, such that vry dg in on subst crosss vry dg in th othr subst. It is known that for a fixd constant k, vry n-vrtx topological graph with no k-grid has On dgs. W conjctur that this rmains tru vn whn: 1 considring grids with distinct vrtics; or 2 all dgs ar straight-lin sgmnts and th dgs within ach subst of th grid ar rquird to b pairwis disjoint. Ths conjcturs ar shown to b tru apart from log n and log 2 n factors, rspctivly. W also sttl th conjcturs for som spcial cass. Catgoris and Subjct Dscriptors G.2.1 [Discrt Mathmatics]: Combinatorics Gnral Trms Thory Kywords gomtric graphs, topological graphs, grids, Turán-typ problms Supportd by a fllowship from th Alxandr von Humboldt Foundation. Rsarch supportd by an NSF Graduat Rsarch Fllowship and a Princton Cntnnial Fllowship. Supportd by NSF Grant CCF , and by grants from NSA, PSC-CUNY, th Hungarian Rsarch Foundation OTKA, and BSF. Prmission to mak digital or hard copis of all or part of this work for prsonal or classroom us is grantd without f providd that copis ar not mad or distributd for profit or commrcial advantag and that copis bar this notic and th full citation on th first pag. To copy othrwis, to rpublish, to post on srvrs or to rdistribut to lists, rquirs prior spcific prmission and/or a f. SCG 09, Jun 8 10, 2009, Aarhus, Dnmark. Copyright 2009 ACM /09/06...$ INTRODUCTION Th intrsction graph of a st C of gomtric objcts has vrtx st C and an dg btwn vry pair of objcts with a nonmpty intrsction. Th problms of finding maximum indpndnt st and maximum cliqu in th intrsction graph of gomtric objcts hav rcivd a considrabl amount of attntion in th litratur du to thir applications in VLSI dsign [9], map labling [1], frquncy assignmnt in cllular ntworks [12], and lswhr. Hr w study th intrsction graph of th dg st of graphs that ar drawn in th plan. It is known that if this intrsction graph dos not contain a larg complt bipartit subgraph, thn th numbr of dgs in th original graph is small. W show that this rmains tru vn undr som vry rstrictiv conditions. A topological graph is a graph drawn in th plan with points as vrtics and dgs as arcs conncting its vrtics. Th arcs ar allowd to cross, but thy may not pass through vrtics xcpt for thir ndpoints. W only considr graphs without paralll dgs or slf-loops. A topological graph is simpl if vry pair of its dgs intrsct at most onc. If th dgs ar drawn as straight-lin sgmnts, thn th graph is gomtric. Givn a topological graph G, th intrsction graph of EG has th dg st of G as its vrtx st, and an dg btwn vry pair of crossing dgs. Not that w considr th dgs of G as opn curvs, thrfor, dgs that intrsct only at a common vrtx ar not adjacnt in th intrsction graph. A complt bipartit subgraph in th intrsction graph of EG corrsponds to a grid structur in G. Dfinition 1.1. A k, l-grid in a topological graph is a pair of dg substs E 1,E 2 such that E 1 = k, E 2 = l, and vry dg in E 1 crosss vry dg in E 2. A k-grid is an abbrviation for a k, k-grid. Thorm 1.2 [15]. Givn fixd constants k, l 1 thr xists anothr constant c k,l, such that any topological graph on n vrtics with no k, l-grid has at most c k,l n dgs. Th proof of Thorm 1.2 in [15] actually guarants a grid in which all th dgs of on of th substs ar adjacnt to a common vrtx. For two rcnt and diffrnt proofs of

2 obtain a linar bound for th numbr of dgs in an ordrd graph that dos not contain a crtain ordrd matching s [10] for mor dtails. Sinc crossings in convx gomtric graphs ar dtrmind by th ordr of th vrtics, this also sttls Conjctur 1.3 for convx gomtric graphs. Figur 1: a natural grid Thorm 1.2 s [8] and [7]. Tardos and Tóth [19] xtndd th rsult in [15] by showing that thr is a constant c k such that a topological graph on n vrtics and at last c k n dgs must contain thr substs of k dgs ach, such that vry pair of dgs from diffrnt substs cross, and for two of th substs all th dgs within th subst ar adjacnt to a common vrtx. Not that according to Dfinition 1.1 th dgs within ach subst of th grid ar allowd to cross or shar a common vrtx, as is indd rquird in th proofs of [15] and [19]. Howvr, a drawing similar to Figur 1 usually coms to mind whn on thinks of a grid. That is, w would lik vry pair of dgs within ach subst of th grid to b disjoint, i.., nithr to shar a common vrtx nor to cross. W say that a k, l-grid formd by dg substs E 1 and E 2 is natural if th dgs within E 1 ar pairwis disjoint, and th dgs within E 2 ar pairwis disjoint. Conjctur 1.3. Givn fixd constants k, l 1 thr xists anothr constant c k,l, such that any simpl topological graph G on n vrtics with no natural k, l-grid has at most c k,l n dgs. Not that it is alrady not trivial to show that an n-vrtx gomtric graph with no k pairwis disjoint dgs has On dgs s [17] and [20]. Morovr, it is an opn qustion whthr a simpl topological graph on n vrtics and no k disjoint dgs has On dgs th bst uppr bound, du to Pach and Tóth [16], is On log 4k 8 n. Thrfor, a proof of Conjctur 1.3 is probably hard to obtain. Hr w prov th following bounds for gomtric and simpl topological graphs with no natural k-grids. Thorm 1.4. i An n-vrtx gomtric graph with no natural k-grid has Ok 2 n log 2 n dgs. ii An n-vrtx simpl topological graph with no natural k-grid has On log 4k 6 n dgs. An n-vrtx topological graph with no 1, 1-grid is planar and hnc has at most 3n 6dgs,forn>2. W sttl Conjctur 1.3 for th first nontrivial cas. Thorm 1.5. An n-vrtx simpl topological graph with no natural 2, 1-grid has On dgs. Many xtrmal problms on gomtric graphs bcom asir for convx gomtric graphs gomtric graphs whos vrtics ar in convx position. Indd, it was alrady pointd out by Klazar and Marcus [10] that it is not hard to modify th proof of th Marcus-Tardos Thorm [13] and Corollary 1.6. Givn a fixd constant k 1 thr xists anothr constant c k, such that any convx gomtric graph on n vrtics with no natural k-grid has at most c k n dgs. Th constant c k in Corollary 1.6 is hug. Using diffrnt tchniqus, w prov tightr uppr bounds for th numbr of dgs in convx gomtric graphs avoiding natural 2, 1-, 2, 2-, or k, 1-grids. Conjctur 1.3 is clarly fals for not ncssarily simpl topological graphs: th complt graph can b drawn as a topological graph in which vry pair of dgs intrsct at most twic [16]. Thrfor, for topological graphs w hav to sttl for only on of th componnts of disjointnss. Conjctur 1.7. Givn fixd constants k, l 1 thr xists anothr constant c k,l, such that any topological graph on n vrtics with no k, l-grid with distinct vrtics has at most c k,l n dgs. This conjctur is shown to b tru for l =1. Thorm 1.8. An n-vrtx topological graph with no k, 1-grid with distinct vrtics has On dgs. For th gnral cas w provid a slightly suprlinar uppr bound. Thorm 1.9. Evry n-vrtx topological graph with no k-grid with distinct vrtics has at most c k n log n vrtics, whr c k = k Olog log k and log is th itratd logarithm function. Not that c k is just barly suprpolynomial in k. Organization. Th rst of this papr is organizd as follows. W discuss topological graphs with no grids with distinct vrtics in Sction 2. In Sction 3 w prov th bounds for th numbr of dgs in simpl topological graphs with no natural grids. Convx gomtric graphs ar considrd in Sction 4. W systmatically omit floor and ciling signs whnvr thy ar not crucial for th sak of clarity of prsntation. W also do not mak any srious attmpt to optimiz absolut constants in our statmnts and proofs. All logarithms in this papr ar bas GRIDS ON DISTINCT VERTICES In this sction w prov Thorms 1.9 and Topological graphs with no k-grid with distinct vrtics Hr w prov Thorm 1.9. W us th following thr rsults from diffrnt paprs. A graph is a string graph if it is an intrsction graph of a collction of curvs in th plan.

3 Lmma 2.1 [5]. Evry string graph with m vrtics and ɛm 2 dgs contains th complt bipartit graph K t,t as a subgraph with t ɛ c 1 m, whr c1 is an absolut log m constant. Th pair-crossing numbr pair-crg of a graph G is th minimum possibl numbr of unordrd pairs of crossing dgs in a drawing of G. Thbisction width, dnotd by bg, is dfind for vry simpl graph G with at last two vrtics. It is th smallst nonngativ intgr such that thr is a partition of th vrtx st V = V 1 V 2 with 1 V 3 Vi 2 V for i =1, 2, and EV1,V2 = bg. 3 W will us th following rsult of Kolman and Matoušk [11] which rlats th pair-crossing numbr and th bisction width of a graph. Lmma 2.2 [11]. Thr is an absolut constant c 2 such that if G is a graph with n vrtics of dgrs d 1,...,d n, thn bg c 2 log n pair-crg+ n. Lt G b a topological graph with n vrtics and mor than nlog n c 3 log h dgs. It is shown in [6] that G has h pairwis crossing dgs. In [7], it is shown that G has h pairwis crossing dgs with distinct vrtics. This strongr vrsion was ndd in th proof of an uppr bound on th numbr of dgs in a string graph without a forbiddn bipartit subgraph. Hr w nd an vn strongr vrsion for th proof of Thorm 1.9. Thorm 2.3. Lt G b a topological graph with n vrtics such that th dgs of G ar colord with ach color class forming a matching. If G dos not contain h pairwis crossing dgs of diffrnt colors and with distinct vrtics, thn th numbr m of dgs of G is at most nlog n c 3 log h. Th proof of Thorm 2.3 is so similar to th proof of th prvious wakr vrsions that w only outlin th proof ida, showing dtails only whr thy diffr from th prvious vrsions in [6] and [7]. Th proof is by induction on n and h. If th intrsction graph of th m dgs is spars, i.., thr ar at most cm 2 /log n 4 pairs of intrscting dgs for som small absolut constant c>0, thn w apply Lmma 2.2 and find a partition of th vrtics into two substs with fw dgs btwn thm. In this cas, w ar don by th induction hypothsis applid to ach of ths vrtx substs. If th intrsction graph of th dgs is dns, i.., thr ar mor than cm 2 /log n 4 pairs of dgs that intrsct, thn using Lmma 2.1 w find two larg dg substs E 1,E 2 with E 1 = E 2 such that vry dg in E 1 intrscts vry dg in E 2. In [7], it is shown that on can pick E 1 E 1 and E 2 E 2 with E 1 = E 2 E 2 /8 such that th vrtics that ar in dgs in E 1 ar diffrnt from th vrtics that ar in dgs in E 2. With th nxt lmma, with A i = E i for i =1, 2, w find substs E 1 E 1 and E 2 E 2 with E 1 = E 2 such that vry dg in E 1 has diffrnt color from vry dg in E 2. Lmma 2.4. Lt A 1,A 2 b two disjoint sts such that A 1 = A 2 2n. Suppos th lmnts of A 1 A 2 ar colord such that no color class has mor than n/2 lmnts. Thn thr ar A 1 A 1 and A 2 A 2 with A 1, A 2 A 1 /4 such that vry lmnt of A 1 has a diffrnt color from vry lmnt of A 2. d 2 i Proof. Lt c 1,...,c t b th colors. In incrasing ordr of j, atstpj, if thr ar at last as many lmnts in A 1 of color c j as thr ar in A 2 of color c j,thnwplacth lmnts of color c j which ar in A 1 in A 1.Ifthnumbrof lmnts of color c j which ar in A 2 is mor than th numbr of lmnts of color c j in A 2,thnwplacalllmntsof color c j which ar in A 2 in A 2. W stop this procss aftr j 0 colors if thr is i such that A i A i /2. This procss stops at som stp j 0, sinc at last half of th lmnts considrd ar placd in A 1 or A 2. Suppos without loss of gnrality that A 1 A i /2. For j 0 <j t, w also plac all lmnts of color c j in A 2.Sincatmost A 1 A 1 /2+n/2 lmnts of A 2 ar not in A 2, thn A 2 A 2 A 1 /2 n/2 = A 2 /2 n/2 A 2 /4. By construction, A 1, A 2 A 1 /4, and no lmnt of A 1 has th sam color as an lmnt in A 2. Not both E 1 and E 2 contain h/2 pairwis crossing dgs of distinct colors and distinct vrtics, sinc othrwis togthr w would hav h pairwis crossing dgs of distinct colors and distinct vrtics. Th induction hypothsis thrfor givs an uppr bound on th siz of E 1,whichcomplts th proof of Thorm 2.3. Lt hk b th minimum h such that if a collction C of h pairwis intrscting curvs is such that ach of th curvs is partitiond into on or two subcurvs, thn thr ar k subcurvs intrscting k othr subcurvs, and ths 2k subcurvs com from distinct curvs in C. Not that h1 = 2. Lmma 2.5. For k 2, w hav hk c 4k log k for som absolut constant c 4. Proof. Lt h = c 4k log k, whrc 4 =16 c1+1,whrc 1 is th absolut constant in Lmma 2.1. For ach curv γ C, randomly pick on of th at most two subcurvs to kp. For ach pair γ,γ C, thr is a probability at last 1/4 that th subcurv of γ w pick intrscts th subcurv of γ w pick. So th xpctd numbr of intrscting pairs of curvs is at last 1 h 4 2. So thr is a collction C consisting of on subcurv of th at most two subcurvs for ach curv such that th numbr of intrscting pairs of curvs in C is at last 1 h 4 2. Sinc C has cardinality h and at last 1 h h2 intrscting subcurvs, thn by Lmma 2.1, th intrsction graph of C contains a complt bipartit graph with parts of siz c1 1 h 16 log h k, sincwpickdc 4 sufficintly larg. Lt f k n dnot th maximum numbr of dgs of a topological graph with n vrtics and no k-grid with distinct vrtics. Th rmaindr of this subsction is dvotd toward proving Thorm 1.9, which says that f k n c k n log n.it will b hlpful to considr a rlatd function. Lt f k n, Δ dnot th maximum numbr of dgs of a topological graph with n vrtics, maximum dgr at most Δ, and no k-grid with distinct vrtics. W collct svral usful lmmas bfor proving Thorm 1.9. For a graph G and vrtx sts A and B, lt GA dnot th numbr of dgs with both vrtics in A and GA, B dnot th numbr of pairs a, b A B that ar dgs of G.

4 Lmma 2.6. Thr is an absolut constant c such that if Δ=logn c log k k c log log k,thn f k n f k n, Δ + k c log log k n. Proof. Lt G =V,E b a topological graph with n vrtics, f k n dgs, and no k-gridwithdistinctvrtics. Partition V = A B, whra consists of thos vrtics with dgr mor than Δ. W construct a squnc of topological graphs G i with vrtx st A. LtG 0 simply b th inducd subgraph of G with vrtx st A. Suppos w alrady hav topological graph G i. If thr is a vrtx v B adjacnt to two vrtics a 1,a 2 A which ar not adjacnt, thn w rplac th path of lngth two with dgs a 1,vandv, a 2 by an dg from a 1 to a 2, and lt G i+1 b th rsulting topological graph. W vntually stop at som stp j and w hav a topological graph G j on A. Notic that ach at stp, w dlt two dgs from B to A and rplac it by on dg btwn two vrtics in A. For ach vrtx v B, th st A v of vrtics in A adjacnt to v aftr constructing G j form a cliqu in G j,othrwisv is adjacnt to two vrtics a 1,a 2 A that ar not adjacnt in G j, which contradicts that w stoppd at stp j. Not that th subgraph of G j inducd by A has j mor dgs than th subgraph of G inducd by A. W first provid an uppr bound on th numbr of dgs of G j. Each dg in G j corrsponds to ithr an dg or a path of lngth two in G. W assign ach dg of G j a color, whrachdgofg j that is an dg of G gts its own color, and w color th dgs of G j that form a path of lngth two in G by th middl vrtx v B. Not that by construction this coloring of th dgs of G j has th proprty that ach color class is a matching. So if thr ar hk pairwis intrscting dgs in th subgraph of G j inducd by A with distinct vrtics and distinct colors, thn G contains a k-grid with distinct vrtics, a contradiction. By Thorm 2.3 and Lmma 2.5, w hav GA+j = Gj A A log A c 3 log hk A log n c 3 logc 4 k log k A log n c 6 log k for som absolut constant c 6. As discussd abov, for ach vrtx v B, th st A v of vrtics in A adjacnt to v aftr constructing G j form a cliqu in G j. This cliqu can not hav hk pairwisintrscting dgs with distinct vrtics and distinct colors, othrwis it contains a k-gridwithdistinctvrtics. ByThorm 2.3, w hav A v 2 A v log A v c 3 log hk, so dividing both sids by A v w gt and finally A v 2log A v c 3 log hk +1 A v hk c 7 log log hk for som absolut constant c 7. Also using Lmma 2.5, w hav A v k c 8 log log k for som absolut constant c 8. Th numbr GA, B of dgs of G with on vrtx in A and th othr vrtx in B is 2j + A v 2j + B k c 8 log log k. v B Sinc ach vrtx in A has dgr at last Δ in G, th numbr GA+ GA, B ofdgsing containing at last on vrtx in A is at last A Δ/2. So A Δ/2 GA+ GA, B GA+2j + B k c 8 log log k 2 A log n c 6 log k + B k c 8 log log k 2 A log n c 6 log k + nk c 8 log log k If nk c 8 log log k 2 A log n c 6 log k,thnwgt Δ 8log n c 6 log k k c 8 log log k, which contradicts Δ = log n c log k k c log log k with c asufficintly larg constant. So nk c 8 log log k > 2 A log n c 6 log k, and th numbr of dgs in G containing a vrtx in A is at most 2k c 8 log log k n k c log log k n. Not that vry vrtx in B in G hasdgratmostδ,so GB f k B, Δ f k n, Δ, whr th last inquality follows by adding isolatd vrtics to B to gt a st of n vrtics. Thrfor, th numbr f k n ofdgsofg is at most f k n, Δ+k c log log k n. Lt d k n = max n n f k n /n and d k n, Δ = max n n f k n, Δ/n. Lmma 2.6 dmonstrats that d k n d k n, Δ + k c log log k 1 whr Δ = k c log log k log n c log k. Not that a triangulatd planar graph with n vrtics has 3n 6dgs,sod 1n = 3 6 for n 3, so d n kn 1forn 3. By Thorm 2.3, w hav d k n log n c 3 log 2k 2 sinc a st of 2k pairwis crossing dgs with distinct vrtics in a topological graph contains a k-gridwithdistinct vrtics. W will improv this bound significantly. Lmma 2.7. Thr ar absolut constants c 9 and c 10 > 0 such that for ach k, n and Δ with Δ k and n Δ c 9, thr is n 1 2n/3 such that d k n 1, Δ d k n, Δ 1 n c 10. Proof. Lt G b a topological graph with at most n vrtics,maximumdgratmostδ,andnok-grid with distinct vrtics which has maximum possibl avrag dgr among all such topological graphs. Without loss of gnrality, w may suppos that th numbr of vrtics of G is actually n, andltm = f k n, Δ. Sinc ach vrtx has dgr at most Δ, thn G dos not contain a 4kΔ-grid. Lt th numbr of crossing pairs of dgs of G b ɛm 2, so th undrlying graph of G has pair-crossing numbr at most ɛm 2.By Lmma 2.1, G has an l-grid with l ɛ c 1 m. Thrfor, w log m hav th inquality ɛ c 2 1 m 3c 4kΔ, and w gt ɛ m 1, log m whr w us 4kΔ m 1/6 and log m m 1/6. By Lmma 2.2, thr is an absolut constant c 2 such that if d 1,...,d n

5 is th dgr squnc of G, thn bg c 2 log n pair-crg+ n c 2 log n ɛ 1/2 m +Δ n c 2 log n m 1 3c 1 1 +Δ n d 2 i m 1 c 10 for som constant c 10 > 0. Thrfor, thr is a partition V G = V 1 V 2 such that V 1, V 2 2 n and GV1,V2 3 m1 c 10. Sinc G has m dgs in total, thr is i {1, 2} such that GV i V i n m m 1 c 10. Thrfor, th subgraph of G inducd by V i has avrag dgr at last a fraction 1 m c 10 1 n c 10 of th avrag dgr of G. Ltting n 1 = V i, whavn 1 2n/3 and th subgraph of G inducd by V i also has maximum dgr at most Δ and dos not contain a k-grid with distinct vrtics, complting th proof. Rpatdly applying Lmma 2.7, w obtain th following lmma. Lmma 2.8. Lt Δ=logn c log k k c log log k as in Lmma 2.6. Thr is a constant c such that d k Δ c 1 1 d Δ kn, Δ d k n, Δ 1. Proof. Lt n 0 = n. Aftr on application of Lmma 2.7, w gt d k n 1, Δ d k n, Δ 1 n c 10 for som n 1 2n/3. Aftr j applications of Lmma 2.7, w gt d k n j, Δ d k n, Δ j 1 n c 10 i 1 for som nj 2/3 j n.lti 0 b th first valu such that n i0 Δ c,whrc is a sufficintly larg constant. W hav d k Δ c d k Δ c, Δ d k n i0, Δ d k n, Δ i 0 d k n, Δ 1 1 n c 10 i 1 i 0 n c 10 i 1 d k n, Δ 1 n c 10 i 0 1 2/3 c 10i i=0 d k n, Δ 1 n c 1 10 i /3 c 10 d k n, Δ 1 Δ c c /3 c 10 d k n, Δ1 1 Δ. By 2, w hav d k n log n c 3 log 2k.Sincc was chosn sufficintly larg in Lmma 2.6, w hav d k n, Δ d k n Δ. Summarizing, d k Δ c 1 1 Δ d kn, Δ d k n, Δ 1. Th last inquality in Lmma 2.8 follows from 2 and th fact that th constant c is chosn sufficintly larg. Combining Lmma 2.6, which givs us inquality 1, and Lmma 2.8 w thrfor gt that thr is an absolut constant C such that d k log n C log k d k n k C log log k if k log n, and 3 d k k C log log k d k n k C log log k othrwis. 4 Itrating 3 until log n<k, applying 4 if k C log log k <n< 2 k, and finally applying th trivial inquality d k n n/2 if n k C log log k,wgtthatd k n =Ok C log log k log n, and hnc f k n =Ok C log log k n log n, complting th proof of Thorm Topological graphs with no k, 1-grid with distinct vrtics Lt G =V,E b a topological graph. For vry dg E dfin X to b st of dgs in E that cross and shar no common vrtx with it. Givn a st of dgs E E, thvrtx covr numbr of E is th minimum siz of a st of vrtics V V such that vry dg in E has at last on of its ndpoints in V. Thorm 1.8 will follow from th nxt lmma, whos proof is du to Rom Pinchasi [18]. Lmma 2.9. Lt k b a fixd intgr and lt G =V,E b a topological graph on n vrtics, such that for vry E th vrtx covr numbr of X is at most k. Thn thr is a constant c k, such that G has at most c k n dgs. Proof. W us a standard sampling argumnt. Lt m b th numbr of dgs in G, andlt0<q<1baconstant. Lt G b th graph obtaind from G by taking vry vrtx of G indpndntly with probability q. Callandg in G good if thr is no dg f in G that crosss and shars no vrtx with it. Dnot by n and m th xpctd numbr of vrtics and good dgs in G, rspctivly. Clarly, n = qn. Th probability that an dg is good is at last q 2 1 q k, thus m q 2 1 q k m. Obsrv that two good dgs may cross only if thy shar a vrtx. Thus, th good dgs form a planar graph by th Hanani-Tutt Thorm s,.g., [21]. Thrfor, q 2 1 q k m m 3n =3qn, and 3 thus, m n. q1 q k Now lt G b an n-vrtx topological graph with no k, 1- grid with distinct vrtics. W claim that for vry EG th vrtx covr numbr of X is at most 2k. Assum not. Thn thr is an dg EG such that th vrtx covr numbr of X is at last 2k+1. Pick an dg u, v X and rmov all th othr dgs in X that ar covrd by v or u. This can b rpatd k tims, for othrwis X can b covrd by at most 2k vrtics. Th dgs w pickd along with th dg form a k, 1-grid with distinct vrtics. This provs Thorm NATURAL GRIDS IN GEOMET- RIC AND SIMPLE TOPOLOGICAL GRAPHS In this sction w considr natural grids in gomtric and simpl topological graphs and prov Thorms 1.4 and 1.5.

6 3.1 Proof of Thorm 1.4 In this sction w prov Thorm 1.4, which givs an uppr bound on th numbr of dgs of a gomtric graph or a simpl topological graph without a natural k-grid. W us th following thr rsults from thr diffrnt paprs. Pach t al. [14] provd th following rlationship btwn th crossing numbr and th bisction width of a graph. Lmma 3.1 [14]. If G is a graph with n vrtics of dgrs d 1,...,d n,thn bg 7crG 1/2 +2 n d 2 i. Lt m b th numbr of dgs in G. Sinc n di =2m and d i n for vry i, whav bg 7crG 1/2 +3 mn. 5 Th following lmma is tight apart from th constant factor. Lmma 3.2 [7]. For ach p thr is a constant c p such that if H is a graph with n vrtics, at last c ptn dgs, and is an intrsction graph of curvs in th plan in which ach pair of curvs intrsct in at most p points, thn H contains th complt bipartit graph K t,t as a subgraph. W will only nd to us th cas p =1. Thlasttoolw us is an uppr bound on th numbr of dgs of a gomtric graph with no k pairwis disjoint dgs. Lmma 3.3 [20]. Any gomtric graph with n vrtics and no k pairwis disjoint dgs has at most 2 9 k 1 2 n dgs. W now prov Thorm 1.4i. As th proofs of i and ii ar so similar, w only giv th dtails for i and discuss how to modify th proof to obtain ii. Proof of Thorm 1.4i: Lt g k n bthmaximum numbr of dgs of a gomtric graph with n vrtics and no natural k-grid. Lt G b a gomtric graph on n vrtics and m = g k n dgs with no natural k-grid. Lt c = max2 20 c 1, 144, whr c 1 is th constant with p = 1 from Lmma 3.2. W prov by induction on n that g k n ck 2 n log 2 n. Suppos for contradiction that g k n >ck 2 n log 2 n. Lt ɛ =10 3 log 2 n. Th proof splits into two cass, dpnding on whthr or not th numbr of pairs of crossing dgs of G is lss than ɛm 2. Cas 1: Th numbr of pairs of crossing dgs is lss than ɛm 2. Thn th crossing numbr of G is lss than ɛm 2. By 5, thr is a partition V G =V 1 V 2 with V 1, V 2 2n/3 and th numbr of dgs with on vrtx in V 1 and on vrtx in V 2 is at most bg 7crG 1/2 +3 mn 7ɛ 1/2 m +3 mn =7ɛ 1/2 +3 n/mm. Lt n 1 = V 1 and n 2 = V 2, son = n 1 + n 2.Thnwhav m = g k n g k V 1 +g k V 2 +bg g k n 1+g k n 2+7ɛ 1/2 +3 n/mm ck 2 n 1 log n 1 + ck 2 n 2 log n 2 +7ɛ 1/2 +3 n/mm ck 2 n log 2n/3+7ɛ 1/2 +3 n/mm ck 2 n log n ck 2 n log 3/2+7ɛ 1/2 +3 n/mm. This implis g k n =m ck 2 log n log 3/2 n 1 7ɛ 1/2 3 n/m <ck 2 1 log 3/2log n 1 n log n 1 log 1 n/4 3c 1/2 k 1 log 1 n <ck 2 n log n, whrwus3c 1/2 k 1 1/4. This complts th proof in this cas. Cas 2: Th numbr of pairs of crossing dgs is at last ɛm 2. Considr th intrsction graph of th dgs whr two dgs ar adjacnt if thy cross. Sinc this intrsction graph has m vrtics and at last ɛm 2 dgs and ach pair of dgs intrsct at most onc, Lmma 3.2 implis it contains a complt bipartit graph with parts of siz t ɛm log 2 nm c > k 2 n>2 9 k 2 n, c c c 1 whr c 1 is th constant with p = 1 from Lmma 3.2. Thrfor, G contains dg substs E 1,E 2 with E 1 = E 2 = t and vry dg in E 1 crosss vry dg of E 2, i.., G contains a t-grid. Sinc t>2 9 k 2 n, Lmma 3.3 implis that E i contains k disjoint dgs for i =1, 2. Ths two substs of k disjoint dgs cross ach othr and hnc form a natural k-grid, complting th proof. To prov Thorm 1.4ii, ssntially th sam proof works as abov, xcpt rplacing th bound Ok 2 noftóth [20] on th numbr of dgs of a gomtric graph with no k disjoint dgs by th bound On log 4k 8 nofpachandtóth [16] on th numbr of dgs of a simpl topological graph with no k disjoint dgs. 3.2 Natural 2, 1-grids: proof of Thorm 1.5 Lt G =V,E b a simpl topological graph on n vrtics without a natural 2, 1-grid. For vry E assign th color rd if X has vrtx covr numbr at most 3, othrwis assign th color blu. It follows from Lmma 2.9 that G has at most 29n rd dgs by picking q =1/4. Th nxt lmma is crucial for bounding th numbr of blu dgs. For F E dnot by V F thstofvrtics inducd by F. Lmma 3.4. Lt =u, v b a blu dg, and lt f 1 X. Thn if thr is an dg =u, w such that w / V X and crosss f 1,thn crosss vry dg f X. Proof. Assum not. Thn thr is an dg f X such that and f do not cross. Not that and f must b disjoint sinc w / V X. If f and f 1 ar disjoint, thn,f, andf 1 form a natural 2, 1-grid. If f and f 1 cross, thn,f, andf 1 form a natural 2, 1-grid. Thus, f and f 1 must shar a vrtx, and V {f} {f 1} =3. Sinc is blu

7 u f 2 f 1 f w v u f 1 w f f 2 v a f 2 crosss b f 2 and ar disjoint Figur 2: Illustrations for th proof of Lmma 3.4 a Cas 1: 2 and ar disjoint b Cas 2: 2 and cross thrmustbandgf 2 X not sharing an ndpoint with f or f 1 and also not sharing an ndpoint with sinc w / V X. Thrfor f 2 must cross both f and f 1. If f 2 crosss thn f 2,,andf form a natural 2, 1-grid s Figur 2a. Othrwis, if f 2 and ar disjoint, thn f 2,, and f 1 form a natural 2, 1-grid s Figur 2b. Nxt w rmov all th rd dgs and procss th blu dgs in som arbitrary ordr. Lt B b th st of th currntly unmarkd and undltd blu dgs. Initially all th blu dgs ar in B. Lt =u, v b th nxt dg in B. Dlt all th dgs that hav on ndpoint in V X B and th othr ndpoint in {u, v}. Lt E u b th dgs u, x B such that x/ V X B and thr is an dg X B that crosss u, x. Similarly, lt E v b th dgs v, x B such that x / V X B andthris an dg X B that crosss v, x. Assum, w.l.o.g., that E u E v and rmov th dgs E v. Rcall that according to Lmma 3.4, if thr is an dg u, x such that x / V X, and u, x crosss som dg in X, thn u, x crosss vry dg in X. A thrackl is a simpl topological graph in which vry pair of dgs mt xactly onc, ithr at a vrtx or at a crossing point. It is known that a thrackl on n vrtics has at most 3n 1/2 dgs [3] and it is a famous opn problm known as Conway s Thrackl Conjctur to show that th tight bound is n. St thrackl = B {} X {u, x X thatcrosssu, v}. Mark all th blu dgs in thrackl, and continu to crat thrackls as long as thr ar unmarkd blu dgs. Lmma 3.5. thrackl is a thrackl. Proof. By dfinition mts vry othr dg in thrackl. A pair of dgs in X cannotbdisjoint,for othrwis thy will form a natural 2, 1-grid with. Finally, by Lmma 3.4 vry dg in thrackl ofthformu, x such that x/ Xv mustcrossallthdgsinxv. Lmma 3.6. If 1 thrackl and 2 / thrackl thn 1 and 2 do not cross. Proof. Assum not and lt 1 and 2 b th first such pair along th procss of crating th thrackls. Thn, w.l.o.g. 2 is unmarkd whn thrackl is cratd. Clarly 1 for othrwis 2 X. If 1 X thn 2 dos not shar a vrtx with, for othrwis it would hav bn addd to thrackl or rmovd. Thus,, 1,and 2 form a natural 2, 1-grid. Othrwis, 1 shars a vrtx with 1 2 c Cas 3a: 2 and shar a vrtx and 2 crosss 1 2 d Cas 3b: 2 and shar a vrtx and 2 and ar disjoint Figur 3: Illustrations for th proof of Lmma 3.6 and thr is an dg X thatcrosss 1. Not that 2 cannot shar a vrtx with, sinc if it shars th sam vrtx as 1 thn thy cannot cross, and othrwis it would hav bn rmovd. Thr ar thr possibl cass to considr s Figur 3: Cas 1: 2 and ar disjoint. Thn 2,,and 1 form a natural 2, 1-grid. Cas 2: 2 and cross. Thn 2,,and form a natural 2, 1-grid. Cas 3: 2 and shar a vrtx. Sinc is blu thr must an dg X that do not shar a vrtx with or 2. By Lmma 3.4 must cross 1. If a 2 crosss, thn 2,,and form a natural 2, 1-grid. Othrwis, if b 2 and ar disjoint thn 2,,and 1 form a natural 2, 1-grid. Sinc any nwly cratd thrackl contains no dgs of a prvious thrackl, w obtain a partition of th dgs that wr not dltd into thrackls t 1,t 2,...,t j. Lt t i = thrackl u i,v i and dnot by V i th vrtx st of t i.rcall that whn t i was cratd at most 2 V i dgs of th form x i,y i x i {u i,v i} y i V X u i,v i and at most V i dgs of th form x i,y i x i {u i,v i} y i / V X u i,v i wr dltd. Th numbr of dgs in t i is at most 3 V i /2, thus, it rmains to show that j Vi = On. TothisndwdrawanwgraphG with th sam vrtx st V. For vry thrackl t i =thracklx i,y i w draw a crossing-fr tr T i with V i 1 dgs as follows. First, draw th dg from x i from y i. Nxt, for vry vrtx v V i \ T i pick on dg t i that has v as on of its ndpoints. Follow from v until it ithr hits a vrtx ncssarily x i or y ior crosss an alrady drawn dg. In th first cas draw an

8 v 1 v 3 x i yi v 4 v 2 a Constructing G c 3. It follows from Lmma 3.7 that thr is a path on dgs of t 3 btwn p and w. This path must cross c 1 or c 2 at a point diffrnt from u and v, hnc thr ar dgs from diffrnt thrackls that cross, contradicting Lmma 3.6. It follows that thr ar no two adjacnt 2-facs that is, sharing an dg in G. Considr th paralll dgs btwn two vrtics in G according to thir ordr around on of th vrtics, and rmov vry othr dg. Th rmaining graph has at last half of th dgs of G and no 2-facs, thus it has at most 3n dgs. Thrfor, G has at most 6n dgs, and thus th numbr of dgs in all th thrackls is at most 9n and th total numbr of blu dgs is at most 36n. W conclud that th original graph G hasatmost65n dgs. b G dgs might contain paralll 4. NATURAL GRIDS IN CONVEX GEO- METRIC GRAPHS For spcific valus of k or l w ar abl to provid tightr bounds in trms of th constant c k,l for th numbr of dgs in convx gomtric graphs avoiding natural k, l-grids, than th ons guarantd by Thorms 1.5 and Corollary1.6. Thorm 4.1. An n-vrtx convx gomtric graph with no natural 2, 1-grid has lss than 5n dgs. Figur 4: Th graph G dg idntical to. In th scond cas draw th sgmnt of from v almost until th crossing point, thn continu th dg vry clos to in on of th dirctions until a vrtx is rachd. S Figur 4a for an xampl. It follows from Lmma 3.6 and th construction of G that G is planar. Not that it is possibl for G to contain paralll dgs s Figur 4b for an xampl. Howvr, it can b shown that thy can b liminatd by rmoving at most half of th dgs in G. Rcall that th standard proof using Eulr s polyhdral formula that a planar graph on n vrtics has at most 3n 6dgsforn 3 uss th fact that th graph has no fac of siz 2 a 2-fac. Th nxt lmma will b usful in showing that G has not too many 2-facs. Lmma 3.7. Lt t i = thrackl b a thrackl and lt p and q b two points on dgs of t i. Thn thr is a path on dgs of t i btwn p and q that dos not go through any vrtx. Proof. It is nough to show that thr is a path from p to. Lt p bthdgthatcontainsp. If p = thn w ar don. If p crosss thn th sgmnt of p btwn p and th crossing point is th rquird path. Finally, if p dos not cross, thn thr is an dg t i that crosss both and p. Th sgmnt of p from p to th crossing point of p and along with th sgmnt of from that crossing point to th crossing point of and crat th rquird path. Lt t 1,t 2,t 3 b thr diffrnt thrackls that yild thr paralll dgs c 1,c 2,c 3 in G btwn two vrtics u, v. Th closd curv c 1 c 2 splits th plan into two rgions, on containing th intrior of c 3. Thn this rgion must contain vry vrtx in V 3 \{u, v}. For othrwis, lt w V 3 \{u, v} b a vrtx outsid that rgion and lt p b som point on Thorm 4.2. An n-vrtx convx gomtric graph with no natural 2, 2-grid has lss than 8n dgs. Thorm 4.3. A convx gomtric graph with n 3k vrtics and no natural k, 1-grid has at most 6kn 12k 2 dgs. W mntion first som basic notions and facts bfor moving to th proofs. Lt G b a convx gomtric graph. W dnot by d Gv th dgr of a vrtx v in G, andbyδg th minimum dgr in G. For u, v V G, w say that v and u ar conscutiv vrtics if thy appar nxt to ach othr on th convx hull of th vrtics of G. Foru, v V G w dnot by Ru, v V G thstofvrticsfromu to v in clockwis ordr, not including u and v. A convx gomtric graph G is a gomtric minor of G if G can b obtaind from G by prforming a finit numbr of th following two oprations: 1. Vrtx dltion. 2. Conscutiv vrtx contraction, i.., only conscutiv vrtics can contract. Rcall that th contraction of two vrtics x and y, rplacsx and y in G with a vrtx v, such that v is adjacnt to all th nighbors of x and y. Notic that if two dgs 1 and 2 cross in G, thn thy cross in G. Likwis, if 1 and 2 ar disjoint in G, thn thy ar disjoint in G. Assum that G is a convx gomtric graph with n vrtics and at last cn dgs. Lt G b a minimal gomtric-minor of G such that EG / V G c. Thn w can conclud that: 1. δg c othrwis vrtx dltion can b applid; and 2. vry conscutiv pair of vrtics v and u must hav at last c 1 common nighbors othrwis conscutiv vrtx contraction can by applid.

9 b v 3 v 2 a v k 1 a b An illustration for th proof of Tho- Figur 5: rm 4.1 v 1 v v k An illustration for th proof of Tho- Figur 6: rm 4.2 y x v u y x Proof of Thorm 4.1: Suppos that EG 5n. Lt G b a minimal gomtric-minor of G such that EG / V G 5. Not that V G 11. For a vrtx u V G dnotbyu 1,u 2,... th nighbors of u in clockwis ordr. Not that u 1 immdiatly follows u in clockwis ordr, sinc a straight-lin sgmnt conncting two conscutiv vrtics in G cannot b crossd by any dg of G and hnc w can assum w.l.o.g. that it is an dg of G. Lt v V G bthvrtxsuchthat: Rv 3,v = min Ru 3,u u V G Sinc δg 5, u 3 xists for vry u. Sinc v 1 and v ar conscutiv vrtics thy shar at last 4 common nighbors. Hnc v 1 and v ar both adjacnt to a vrtx a V G, such that a/ {v 2,v k 1,v k },whrk = d G v. By minimality of Rv 3,v, v k has at last thr nighbors in Rv k,v 3, S Figur 5. Thus v k has a nighbor b Rv k,v 3 othr than v and v 1. Hnc w hav a natural 2, 1-gridwithdgs v, v k 1, v 1,a, and v k,bing, and hnc in G. Proof of Thorm 4.2: Assum that EG 8n. LtG b a minimal gomtric-minor of G with EG / V G 8. Not that V G 17, δg 8, and vry pair of conscutiv vrtics in G shar at last 7 common nighbors. Lt x, x andy,y b a pair of disjoint dgs such that: 1. x and y ar conscutiv vrtics with x following y in clockwis ordr; 2. Rx, x, Ry,y 2; and 3. Ry,y is maximizd subjct to 1 and 2 abov. This is possibl sinc conscutiv vrtics shar at last 7 common nighbors. Now lt u, v b th nxt two vrtics aftr x in clockwis ordr. Sinc u and v ar conscutiv, w know that thy shar at last 7 common nighbors. Now u and v can hav at most 3 common nighbors in Rv, y {y }, sinc othrwis w would contradict th maximality of Ry,y. Hnc u and v must hav two common nighbors a, b Ry,y. S Figur 6. Hnc x, x, y,y, u, a, v, b forms a natural 2, 2-grid in G, and hnc in G. Proof of Thorm 4.3: Our proof uss th tchniqu from [4]. Lt k 1 b fixd. W will prov th thorm by induction on th numbr of vrtics n. For n =3k w nd to show that EG 6k 2, howvr, thr ar at most 3k 2 9k 2 dgs. Assum now that th claim is tru whn 2 q i q i q 2 q n2 q 1 a b p n1 p 1 a q ip j q i p j p 2 p j p j q us q is 1 q u b Th scond cas in th proof of Proposition 4.4 Figur 7: Illustrations for th proof of Thorm 4.3 th numbr of vrtics is smallr than n and lt G b an n- vrtx convx gomtric graph with no natural k, 1-grid. If thr is no dg whos ndpoints ar sparatd by at last 2k vrtics along both arcs of th boundary of th n-gon, thn EG 2kn 6kn 12k 2 sinc n 3k. Sow may assum that thr xists such an dg = ab. Assum w.l.o.g. that is vrtical. Lt p n1,...,p 1 dnot th vrtics on th right-hand sid of a, b andltq 1,...,q n2 dnot th vrtics on its lft-hand sid, both in clockwis ordr. Dfin a partial ordr on th st of dgs that cross a, b as follows: q ip j q i p j i<i and j<j s Figur 7a. W dnot by rankq ip j th largst intgr r such that thr is a squnc of dgs q i1 p j1 q i2 p j2 q ir p jr = q ip j. Sinc G has no natural k, 1-grid, vry dg that crosss ab has rank at most k 1. Nxt w dfin a convx gomtric graph G 1 with n 2 + k +1 vrtics {a, p k 1,...,p 1,b,q 1,...,q n2 } in clockwis ordr. Lt G 1 b th sam as G whn rstrictd to th vrtics {a, b, q 1,...,q n2 }. Thn lt q ip r b in EG 1ifandonly if thr is an dg q ip j EG whos rank is r. First w will show that if thr ar t pairwis disjoint dgs in G 1 with thir lft ndpoints insid an intrval q i,q j, thn thr ar t pairwis disjoint dgs in G with thir lft ndpoints insid th intrval q i,q j. Proposition 4.4. Lt q i1 p r 1,...,q it p r t b t pairwis disjoint dgs in G 1 that cross ab. Thn thr ar t pairwis disjoint dgs q u1 p v1,...,q ut p vt such that 1. u t = i t. 2. u x i x for x =1,...,t rankq ux p vx =r x,forx =1,...,t. a b p vs p v

10 Proof. By rvrs induction on x. InG w can pick th dg q it p vt that has rank r t. W know on xists sinc q it p r t xists in G 1. Assum that w hav alrady found th dgs q ux p vx for x = t, t 1,...,s > 1 that satisfy th abov. Lt q up v b an dg of rank r s 1 such that q up v q us p vs. If u i s 1, thn w can pick q up v as th nxt dg. Othrwis, lt b an dg of rank r s 1 with q is 1 as an ndpoint. Sinc and q up v hav th sam rank, thy must cross, which implis that q us p vs and so w can pick as th nxt dg. S Figur 7b. Proposition 4.5. G 1 dos not contain a natural k, 1- grid. Proof. Assum that G 1 contains a natural k, 1-grid. Thn by considring th possibl dgs involvd in such a grid and using Proposition 4.4 abov, on concluds that thr is a natural k, 1-grid in G, which is a contradiction. W also dfin a convx gomtric graph G 2 with n 1+k+1 vrtics {a, p n1,...,p 1,b,q1,...,q k 1} in clockwis ordr. Lt G 2 b th sam as G whn rstrictd to th vrtics {a, p n1,...,p 1,b}. Thnltqr,p jbineg 2ifandonly ifthrisandgq i,p j EG whos rank is r. By th sam argumnts G 2 dos not contain a natural k, 1-grid. Lt E r dnot th dgs in G with rank r, 1 r k 1. Proposition 4.6. E r d G1 p r+d G2 q r 1. Proof. Th dgs in E r cannot form a cycl. Indd, considr a path q i1 p j1,q i2 p j1,q i2 p j2,... and assum w.l.o.g. that i 1 <i 2. Thn j 2 <j 1 for othrwis q i1 p j1 and q i2 p j2 ar disjoint. Similarly, w hav i l >i l 1 and j l <j l 1,for any l>1, thrfor th path can not form a cycl. Sinc thr ar d G1 p r+d G2 qr vrtics that ar ndpoints of dgs in E r, th claim follows. Dnot by E 1 th dgs in G 1 that do not cross ab and by E 2 th dgs in G 2 that do not cross ab not that ab E i, i =1, 2. Rcall that ab has at last 2k vrtics on ach of its sids, thrfor, V G 1, V G 2 3k. Thn: k 1 EG = E 1 + E 2 1+ E r r=1 k 1 = E 1 + E 2 1+ d G1 p r+d G2 qr 1 r=1 = EG 1 + EG 2 k ind hyp 6kn 1 + k +1 12k 2 +6kn 2 + k +1 12k 2 k = 6kn 12k 2 k 6kn 12k 2 This complts th proof of Thorm 4.3. Acknowldgmnts. W thank Rom Pinchasi for hlpful discussions and for his prmission to includ his proof for Lmma 2.9 in this papr. W also thank anonymous rfrs for svral hlpful rmarks. 5. REFERENCES [1] P. K. Agarwal, M. van Krvld, and S. Suri, Labl placmnt by maximum indpndnt st in rctangls, Comput. Gom. Thory Appl , [2] P. Brass, G. Károlyi, and P. Valtr, A Turán-typ xtrmal thory for convx gomtric graphs, Discrt and Computational Gomtry th Goodman-Pollack Fstschrift B. Aronov t al., ds., Springr 2003, [3] G. Cairns and Y. Nikolayvsky, Bounds for gnralizd thrackls, Discrt and Computational Gomtry , [4] V. Capoylas and J. Pach, A Turán-typ problm on chords of a convx polygon, J. Comb. Thory Sr. B , [5] J. Fox and J. Pach, String graphs and incomparability graphs, manuscript, [6] J. Fox and J. Pach, Coloring K k -fr intrsction graphs of gomtric objcts in th plan, 24th ACM Symp. on Computational Gomtry, Collg Park, MD, USA, Jun 2008, [7] J. Fox and J. Pach, A sparator thorm for string graphs and its applications, submittd, [8] J. Fox, J. Pach, and Cs. D. Tóth, A bipartit strngthning of th Crossing Lmma, submittd, [9] D. S. Hochbaum and W. Maass, Approximation schms for covring and packing problms in imag procssing and VLSI, J. ACM , [10] M. Klazar and A. Marcus, Extnsions of th linar bound in th Fürdi-Hajnal conjctur, Adv. in Appl. Math , [11] P. Kolman and J. Matoušk, Crossing numbr, pair-crossing numbr, and xpansion, J. Combin. Thory Sr. B , [12] E. Malsinska, Graph-thortical modls for frquncy assignmnt problms, Ph.D. Thsis, Tchnisch Univrsität Brlin, [13] A. Marcus and G. Tardos, Excludd prmutation matrics and th Stanly-Wilf conjctur, J. Comb. Thory Sr. A , [14] J. Pach, F. Shahrokhi, and M. Szgdy, Applications of th crossing numbr, Algorithmica , [15] J. Pach, R. Pinchasi, M. Sharir, and G. Tóth, Topological graphs with no larg grids, Graphs and Combinatorics , [16] J. Pach and G. Tóth, Disjoint dgs in topological graphs, in: Combinatorial Gomtry and Graph Thory J. Akiyama t al., ds., Lctur Nots in Computr Scinc 3330, Springr-Vrlag, Brlin, 2005, 133-Ű140. [17] J. Pach and J. Törőcsik, Som gomtric applications of Dilworth s thorm, Discrt and Computational Gomtry , 1 7. [18] R. Pinchasi, prsonal communication, Octobr [19] G. Tardos and G. Tóth, Crossing stars in topological graphs, SIAM Journal on Discrt Mathmatics , [20] G. Tóth, Not on gomtric graphs, J. Comb. Thory Sr. A , [21] W. T. Tutt, Toward a thory of crossing numbrs, J. Comb. Thory ,