Relaxing planarity for topological graphs

Transcription

1 Rlaing planariy for opological graph Jáno Pach 1, Radoš Radoičić 2, and Géza Tóh 3 1 Ciy Collg, CUNY and Couran Iniu of Mahmaical Scinc, Nw York Univriy, Nw York, NY 10012, USA pach@cim.nyu.du 2 Dparmn of Mahmaic, Maachu Iniu of Tchnology, Cambridg, MA 02139, USA rado@mah.mi.du 3 Rényi Iniu of h Hungarian Acadmy of Scinc, H-1364 Budap, P.O.B. 127, Hungary gza@rnyi.hu Abrac. According o Eulr formula, vry planar graph wih n vric ha a mo O(n) dg. How much can w rla h condiion of planariy wihou violaing h concluion? Afr urvying om claical and rcn rul of hi kind, w prov ha vry graph of n vric, which can b drawn in h plan wihou hr pairwi croing dg, ha a mo O(n) dg. For raigh-lin drawing, hi amn ha bn ablihd by Agarwal al., uing a mor complicad argumn, bu for h gnral ca prviouly no bound br han O(n 3/2 ) wa known. 1 Inroducion A gomric graph i a graph drawn in h plan o ha i vric ar rprnd by poin in gnral poiion (i.., no hr ar collinar) and i dg by raigh-lin gmn conncing h corrponding poin. Topological graph ar dfind imilarly, cp ha now ach dg can b rprnd by any impl (non-lfinrcing) Jordan arc paing hrough no vric ohr han i ndpoin. Throughou hi papr, w aum ha if wo dg of a opological graph G har an inrior poin, hn a hi poin hy proprly cro. W alo aum, for impliciy, ha no hr dg cro a h am poin and ha any wo dg cro only a fini numbr of im. If any wo dg of G hav a mo on poin in common (including hir ndpoin), hn G i aid o b a impl opological graph. Clarly, vry gomric graph i impl. L V (G) and E(G) dno h vr and dg of G, rpcivly. W will mak no noaional diincion bwn h vric (dg) of h undrlying abrac graph, and h poin (arc) rprning hm in h plan. I follow from Eulr Polyhdral Formula ha vry impl planar graph wih n vric ha a mo 3n 6 dg. Equivalnly, vry opological graph wih n vric and mor han 3n 6 dg ha a pair of croing dg. Wha happn if, inad of a croing pair of dg, w wan o guaran h inc of om largr configuraion involving vral croing? Wha kind of unavoidabl ubrucur mu occur in vry gomric (or opological) graph G having n vric and mor han Cn dg, for an appropria larg conan C > 0? In h n four cion, w approach hi quion from four diffrn dircion, ach lading o diffrn anwr. In h la cion, w prov ha any opological graph wih n vric and no hr pairwi croing dg ha a mo O(n) dg. For impl opological graph, hi rul wa fir ablihd by Agarwal-Aronov-Pach-Pollack-Sharir [AAPPS97], uing a mor complicad argumn. Jáno Pach ha bn uppord by NSF Gran CCR , by PSC-CUNY Rarch Award , and by OTKA T Géza Tóh ha bn uppord by OTKA-T and by an award from h Nw York Univriy Rarch Challng Fund.

2 2 Jáno Pach, Radoš Radoičić, and Géza Tóh 2 Ordinary and opological minor A graph H i aid o b a minor of anohr graph G if H can b obaind from a ubgraph of G by a ri of dg conracion. If a ubgraph of G can b obaind from H by rplacing i dg wih indpndn pah bwn hir ndpoin, hn H i calld a opological minor of G. Clarly, a opological minor of G i alo i (ordinary) minor. If a graph G wih n vric ha no minor iomorphic o K 5 or o K 3,3, hn by Kuraowki horm i i planar and i numbr of dg canno cd 3n 6. I follow from an old rul of Wagnr ha h am concluion hold undr h wakr aumpion ha G ha no K 5 minor. A fw yar ago Madr [M98] provd h following famou conjcur of Dirac: Thorm 2.1. (Madr) Evry graph of n vric wih no opological K 5 minor ha a mo 3n 6 dg. If w only aum ha G ha no opological K r minor for om r > 5, w can ill conclud ha G i par, i.., i numbr of dg i a mo linar in n. Thorm 2.2. (Komló-Szmrédi [KSz96], Bollobá-Thomaon [BT98]) For any poiiv ingr r, vry graph of n vric wih no opological K r minor ha a mo cr 2 n dg. Morovr, Komló and Szmrédi howd ha h abov amn i ru wih any poiiv conan c > 1/4, providd ha r i larg nough. Apar from h valu of h conan, hi horm i harp, a i hown by h union of pairwi dijoin copi of a compl bipari graph of iz roughly r 2. W hav a br bound on h numbr of dg, undr h rongr aumpion ha G ha no K r minor. Thorm 2.3. (Koochka [K84], Thomaon [T84]) For any poiiv ingr r, vry graph of n vric wih no K r minor ha a mo cr log rn dg. Th b valu of h conan c for which h horm hold wa aympoically drmind in [T01]. Th horm i harp up o h conan. (Warning! Th lr c and C ud in vral amn will dno unrlad poiiv conan.) Rvring Thorm 2.3, w obain ha vry graph wih n vric and mor han cr log rn dg ha a K r minor. Thi immdialy impli ha if h chromaic numbr χ(g) of G i a la 2cr log r + 1, hn G ha a K r minor. According o Hadwigr nooriou conjcur, for h am concluion i i nough o aum ha χ(g) r. Thi i known o b ru for r 6 ( [RST93]). 3 Quai-planar graph A graph i planar if and only if i can b drawn a a opological graph wih no croing dg. Wha happn if w rla hi condiion and w allow r croing pr dg, for om fid r 0? Thorm 3.1. [PT97] L r b a naural numbr and l G b a impl opological graph of n vric, in which vry dg cro a mo r ohr. Thn, for any r 4, w hav E(G) (r + 3)(n 2). Th ca r = 0 i Eulr horm, which i harp. In h ca r = 1, udid in [PT97] and indpndnly by Gärnr, Thil, and Ziglr (pronal communicaion), h abov bound can b aaind for all n 12. Th rul i alo harp for r = 2, providd ha n 5 (mod 15) i ufficinly larg ( Figur 1).

3 Rlaing planariy for opological graph 3 Figur 1. Howvr, for r = 3, w hav rcnly provd ha E(G) 5.5(n 2), and hi bound i b poibl up o an addiiv conan [PRTT02]. For vry larg valu of r, a much br uppr bound can b dducd from h following horm of Ajai-Chváal-Nwborn-Szmrédi [ACNS82] and Lighon [L84]: any opological graph wih n vric and > 4n dg ha a la conan im 3 /n 2 croing. Corollary 3.2. [PRTT02] Any opological graph wih n vric, who ach dg cro a mo r ohr, ha a mo 4 rn dg. On can alo obain a linar uppr bound for h numbr of dg of a opological graph undr h wakr aumpion ha no dg can cro mor han r ohr dg incidn o h am vr. Thi can b furhr gnralizd, a follow. Thorm 3.3. [PPST02] L G b a opological graph wih n vric which conain no r + dg uch ha h fir r ar incidn o h am vr and ach of hm cro h ohr dg. Thn w hav E(G) C rn, whr C i a conan dpnding only on. In paricular, i follow ha if a opological graph conain no larg gridlik croing parn (wo larg of dg uch ha vry lmn of h fir cro all lmn of h cond), i numbr of dg i a mo linar in n. I i a challnging opn problm o dcid whhr h am arion rmain ru for all opological graph conaining no larg compl croing parn. For any poiiv ingr r, w call a opological graph r-quai-planar if i ha no r pairwi croing dg. A opological graph i -monoon if all of i dg ar -monoon curv, i.., vry vrical lin cro hm a mo onc. Clarly, vry gomric graph i -monoon, bcau i dg ar raigh-lin gmn (ha ar aumd o b non-vrical). If h vric of a gomric graph ar in conv poiion, hn i i aid o b a conv gomric graph r Figur 2. Conrucion howing ha Thorm 3.4 i harp (n = 13, r = 4) 6

4 4 Jáno Pach, Radoš Radoičić, and Géza Tóh Thorm 3.4. [CP92] Th maimum numbr of dg of any r-quai-planar conv gomric graph wih n 2r dg i ( ) 2r 1 2(r 1)n. 2 Thorm 3.5. (Valr [V98]) Evry r-quai-planar -monoon opological graph wih n vric ha a mo C r n log n dg, for a uiabl conan C r dpnding on r. Thorm 3.6. [PSS96] For any r 4, vry r-quai-planar impl opological graph G wih n vric ha a mo C r n(log n) 2(r 3) dg, for a uiabl conan C r dpnding only on r. In Scion 6, w will poin ou ha Thorm 3.6 rmain ru vn if w drop h aumpion ha G i impl, i.., wo dg may cro mor han onc. For 3-quai-planar opological graph w hav a linar uppr bound. Thorm 3.7. [AAPPS97] Evry 3-quai-planar impl opological graph G wih n vric ha a mo Cn dg, for a uiabl conan C. In Scion 7, w giv a hor nw proof of h la horm, howing ha hr, oo, on can drop h aumpion ha no wo dg cro mor han onc (i.., ha G i impl). In hi ca, prviouly no bound br han O(n 3/2 ) wa known. Thorm 3.7 can alo b ndd in anohr dircion: i rmain ru for vry opological graph G wih no r + 2 dg uch ha ach of h fir r dg cro h la wo and h la wo dg cro ach ohr. Of cour, h conan C in h horm now dpnd on r [PRT02]. All horm in hi cion provid (uually linar) uppr bound on h numbr of dg of opological graph aifying crain condiion. In ach ca, on may ak whhr a rongr amn i ru. I i poibl ha h graph in quion can b dcompod ino a mall numbr planar graph? For inanc, h following rongr form of Thorm 3.7 may hold: Conjcur 3.8. Thr i a conan k uch ha h dg of vry 3-quai-planar opological graph G can b colord by k color o ha no wo dg of h am color cro ach ohr. McGuinn [Mc00] provd ha Conjcur 3.8 i ru for impl opological graph, providd ha hr i a clod Jordan curv croing vry dg of G prcily onc. Th amn i alo ru for r-quai-planar conv gomric graph, for any fid r ( [K88], [KK97]). 4 Gnralizd hrackl and hir rlaiv Two dg ar aid o b adjacn if hy har an ndpoin. W ay ha a graph drawn in h plan i a gnralizd hrackl if any wo dg m an odd numbr of im, couning hir common ndpoin, if hy hav any. Tha i, a graph i a gnralizd hrackl if and only if i ha no wo adjacn dg ha cro an odd numbr of im and no wo non-adjacn dg ha cro an vn numbr of im. In paricular, a gnralizd hrackl canno hav wo non-adjacn dg ha ar dijoin. Alhough a fir glanc hi propry may appar o b h ac oppoi of planariy, urpriingly, h wo noion ar no ha diffrn. In paricular, for bipari graph, hy ar quivaln. Thorm 4.1. [LPS97] A bipari graph can b drawn in h plan a a gnralizd hrackl if and only if i i planar. Uing h fac ha vry graph G ha a bipari ubgraph wih a la E(G) /2 dg, w obain ha if a graph G of n vric can b drawn a a gnralizd hrackl, hn E(G) = O(n).

5 Rlaing planariy for opological graph 5 Thorm 4.2. (Cairn-Nikolayvky [CN00]) Evry gnralizd hrackl wih n vric ha a mo 2n 2 dg. Thi bound i harp. Figur 3. A gnralizd hrackl wih n vric and 2n 2 dg A gomric graph G i a gnralizd hrackl if and only if i ha no wo dijoin dg. (Th dg ar uppod o b clod, o ha wo dijoin dg ar ncarily non-adjacn.) On can rla hi condiion by auming ha G ha no r pairwi dijoin dg, for om fid r 2. For r = 2, i wa provd by Hopf-Pannwiz [HP34] ha vry graph aifying hi propry ha a mo n dg, and ha hi bound i harp. For r = 3, h fir linar bound on h numbr of dg of uch graph wa ablihd by Alon-Erdő [AE89], which wa lar improvd o 3n by Goddard- Kachalki-Kliman [GKK96]. For gnral r, h fir linar bound wa ablihd in [PT94]. Th b currnly known ima i h following: Thorm 4.3. (Tóh [T00]) Evry gomric graph wih n vric and no r pairwi dijoin dg ha a mo 2 9 (r 1) 2 n dg. I i likly ha h dpndnc of hi bound on r can b furhr improvd o linar. If w wan o prov h analogu of Thorm 4.3 for opological graph, w hav o mak om addiional aumpion on h rucur of G, ohrwi i i poibl ha any wo dg of G cro ach ohr. Conjcur 4.4. (Conway Thrackl Conjcur) L G b a impl opological graph of n vric. If G ha no wo dijoin dg, hn E(G) n. For many rlad rul, conul [LPS97], [CN00], [W71]. Th n inring opn quion i o dcid whhr h maimum numbr of dg of a impl opological graph wih n vric and no hr pairwi dijoin dg i O(n). 5 Locally planar graph For any r 3, a opological graph G i calld r-locally planar if G ha no lfinrcing pah of lngh a mo r. Roughly paking, hi man ha h mbdding of h graph i planar in a nighborhood of radiu r/2 around any vr. In [PPTT02], w howd ha hr i 3-locally planar gomric graph wih n vric and wih a la conan im n log n dg. Somwha

6 6 Jáno Pach, Radoš Radoičić, and Géza Tóh urpriingly (o u), Tardo [T02] managd o nd hi rul o any fid r 3. H conrucd a qunc of r-locally planar gomric graph wih n vric and a uprlinar numbr of dg (approimaly n im h r/2 im irad logarihm of n). Morovr, h graph ar bipari and all of hir dg can b abbd by h am lin. Th following poiiv rul i probably vry far from bing harp. Thorm 5.1. [PPTT02] Th maimum numbr of dg of a 3-locally planar opological graph wih n vric i O(n 3/2 ). For gomric graph, much rongr rul ar known. Thorm 5.2. [PPTT02] Th maimum numbr of dg of a 3-locally planar -monoon opological graph wih n vric i O(n log n). Thi bound i aympoically harp. For 5-locally planar -monoon opological graph, w hav a lighly br uppr bound on h numbr of dg: O(n log n/ log log n). Thi bound can b furhr improvd undr h addiional aumpion ha all dg of h graph cro h y-ai. Thorm 5.3. [PPTT02] L G b an -monoon r-locally planar opological graph of n vric all of who dg cro h y-ai. Thn, w hav E(G) cn(log n) 1/ r/2 for a uiabl conan c. 6 Srnghning Thorm 3.6 In hi cion, w oulin h proof of Thorm 6.1. Evry r-quai-planar opological graph wih n vric ha a mo f r (n) := C r n(log n) 4(r 3) dg, whr r 2 and C r i a uiabl poiiv conan dpnding on r. L G b a graph wih vr V (G) and dg E(G). Th bicion widh b(g) of G i dfind a h minimum numbr of dg, who rmoval pli h graph ino wo roughly qual ubgraph. Mor prcily, b(g) i h minimum numbr of dg running bwn V 1 and V 2, ovr all pariion of h vr of G ino wo dijoin par V 1 V 2 uch ha V 1, V 2 V (G) /3. Th pair-croing numbr pair-cr(g) of a graph G i h minimum numbr of croing pair of dg in any drawing of G. W nd a rcn rul of Maoušk [M02], who analogu for ordinary croing numbr wa provd in [PSS96] and [SV94]. Lmma 6.2. (Maoušk) L G b a graph of n vric wih dgr d 1, d 2,..., d n. Thn w hav ) n b 2 (G) c(log n) (pair-cr(g) 2 +, whr c i a uiabl conan. W follow h ida of h original proof of Thorm 3.6. W ablih Thorm 6.1 by doubl inducion on r and n. By Thorm 7.1 (in h n cion), h amn i ru for r = 3 and for all n. I i alo ru for any r > 2 and n n r, providd ha C r i ufficinly larg in rm of n r, bcau hn h ad bound cd ( n 2). (Th ingr nr can b pcifid lar o a o aify crain impl chnical condiion.) i=1 d 2 i

7 Rlaing planariy for opological graph 7 Aum ha w hav alrady provd Thorm 6.1 for om r 3 and all n. L n n r+1, and uppo ha h horm hold for r + 1 and for all opological graph having fwr han n vric. L G b an (r + 1)-quai-planar opological graph of n vric. For impliciy, w u h am lr G o dno h undrlying abrac graph. For any dg E(G), l G G dno h opological graph coniing of all dg of G ha cro. Clarly, G i r-quai-planar. Thu, by h inducion hypohi, w hav pair-cr(g) 1 2 E(G) E(G ) 1 2 E(G) f r(n). Uing h fac ha n i=1 d2 i 2 E(G) n hold for vry graph G wih dgr d 1, d 2,..., d n, Lmma 6.2 impli ha b(g) ( c(log n) 2 E(G) f r (n) ) 1/2. Conidr a pariion of V (G) ino wo par of iz n 1, n 2 2n/3 uch ha h numbr of dg running bwn hm i b(g). Obviouly, boh ubgraph inducd by h par ar (r + 1)-quaiplanar. Thu, w can apply h inducion hypohi o obain E(G) f r+1 (n 1 ) + f r+1 (n 2 ) + b(g). Comparing h la wo inqualii, h rul follow by om rouin calculaion. 7 Srnghning Thorm 3.7 Th aim of hi cion i o prov h following rongr vrion of Thorm 3.7. Thorm 7.1. Evry 3-quai-planar opological graph wih n vric ha a mo Cn dg, for a uiabl conan C. L G b a 3-quai-planar opological graph wih n vric. Rdraw G, if ncary, wihou craing 3 pairwi croing dg o ha h numbr of croing in h ruling opological graph G i a mall a poibl. Obviouly, no dg of G cro ilf, ohrwi w could rduc h numbr of croing by rmoving h loop. Suppo ha G ha wo diinc dg ha cro a la wic. A rgion nclod by wo pic of h paricipaing dg i calld a ln. Suppo hr i a ln l ha conain no vr of G. Conidr a minimal ln l l, by conainmn. Noic ha by wapping h wo id of l, w could rduc h numbr of croing wihou craing any nw pair of croing dg. In paricular, G rmain 3-quai-planar. Thrfor, w can conclud ha Claim 1. Each ln of G conain a vr. W may aum wihou lo of gnraliy ha h undrlying abrac graph of G i conncd, bcau ohrwi w can prov Thorm 7.1 by inducion on h numbr of vric. L 1, 2,..., n 1 E(G) b a qunc of dg uch ha 1, 2,..., i form a r T i G for vry 1 i n 1. In paricular, 1, 2,..., n 1 form a panning r of G. Fir, w conruc a qunc of croing-fr opological graph (r), T1, T 2,..., T n 1. L T 1 b dfind a a opological graph of wo vric, coniing of h ingl dg 1 (a wa drawn in G). Suppo ha T i ha alrady bn dfind for om i 1, and l v dno h ndpoin of i+1 ha do no blong o T i. Now add o T i h pic of i+1 bwn v and i fir croing wih T i. Mor prcily, follow h dg i+1 from v up o h poin v whr i hi T i for h fir im, and dno hi pic of i+1 by ẽ i+1. If v i a vr of T i, hn add v and ẽ i+1 o T i and l T i+1 b h ruling opological graph. If v i in h inrior of an dg of T i, hn inroduc a nw vr a v. I divid ino wo dg, and. Add boh of hm o T i, and dl. Alo add v and ẽ i+1, and l T i+1 b h ruling opological graph.

8 8 Jáno Pach, Radoš Radoičić, and Géza Tóh Afr n 2 p, w obain a opological r T := T n 1, which (1) i croing-fr, (2) ha fwr han 2n vric, (3) conain ach vr of G, and (4) ha h propry ha ach of i dg i ihr a full dg, or a pic of an dg of G T T Figur 4. Conrucing T from T L D dno h opn rgion obaind by rmoving from h plan vry poin blonging o T. Dfin a conv gomric graph H, a follow. Travling around h boundary of D in clockwi dircion, w ncounr wo kind of diffrn faur : vric and dg of T. Rprn ach uch faur by a diffrn vr i of H, in clockwi ordr in conv poiion. No ha h am faur will b rprnd by vral i : vry dg will b rprnd wic, bcau w vii boh of i id, and vry vr will b rprnd a many im a i dgr in T. I i no hard o ha h numbr of vric i V (H) do no cd 8n. N, w dfin h dg of H. L E b h of dg of G \ T. Evry dg E may cro T a vral poin. Th croing poin divid ino vral pic, calld gmn. L S dno h of all gmn of all dg E. Wih h cpion of i ndpoin, vry gmn S run in h rgion D. Th ndpoin of blong o wo faur along h boundary of D, rprnd by wo vric i and j of H. Connc i and j by a raigh-lin dg of H. Noic ha H ha no loop, bcau if i = j, hn, uing h fac ha T i conncd, on can aily conclud ha h ln nclod by and by h dg of T corrponding o i ha no vr of G in i inrior. Thi conradic Claim 1. Of cour, vral diffrn gmn may giv ri o h am dg i j E(H). Two uch gmn ar aid o b of h am yp. Obrv ha wo gmn of h am yp canno cro. Indd, a no dg inrc ilf, h wo croing gmn would blong o diinc dg 1, 2 E. Sinc any wo vric of G ar conncd by a mo on dg, a la on of i and j corrpond o an dg (and no o a vr) of T, which oghr wih 1 and 2 would form a pairwi inrcing ripl of dg, conradicing our aumpion ha G i 3-quai-planar. Claim 2. H i a 3-quai-planar conv gomric graph. To ablih hi claim, i i ufficin o obrv ha if wo dg of H cro ach ohr, hn h faur of T corrponding o hir ndpoin alrna in h clockwi ordr around h boundary of D. Thrfor, any hr pairwi croing dg of H would corrpond o hr pairwi croing gmn, which i a conradicion. A gmn i aid o b hildd if hr ar wo ohr gmn, 1 and 2, of h am yp, on on ach id of. Ohrwi, i calld pod. An dg E i aid o b pod if a la on of i gmn i pod. Ohrwi, i calld a hildd dg. In viw of Claim 2, w can apply Thorm 3.4 [CP92] o H. W obain ha E(H) 4 V (H) 10 < 32n, ha i, hr ar fwr han 32n diffrn yp of gmn. Thr ar a mo wo pod gmn of h am yp, o h oal numbr of pod gmn i mallr han 64n, and hi i alo an uppr bound on h numbr of pod dg in E. I rmain o bound h numbr of hildd dg in E.

9 Rlaing planariy for opological graph 9 Claim 3. Thr ar no hildd dg. Suppo, in ordr o obain a conradicion, ha hr i a hildd dg E. Orin arbirarily, and dno i gmn by 1, 2,..., m S, lid according o hi orinaion. For any 1 i m, l i S b h (uniqu) gmn of h am yp a i, running clo o i on i lf id. Sinc hr i no lf-inrcing dg and mpy ln in G, h gmn i and i+1 blong o h am dg f E, for vry i < m ( Fig. 5). Howvr, hi man ha boh ndpoin of and f coincid, which i impoibl. W can conclud ha E ha fwr han 64n lmn, all of which ar pod. Thu, aking ino accoun h n 1 dg of h panning r T, h oal numbr of dg of G i mallr han 65n. i i+1 i i+1 i i+1 i i+1 i i+1 i i+1 Figur 5. i and i+1 blong o h am dg Rfrnc [AAPPS97] P. K. Agarwal, B. Aronov, J. Pach, R. Pollack, and M. Sharir, Quai-planar graph hav a linar numbr of dg, Combinaorica 17 (1997), 1 9. [ACNS82] M. Ajai, V. Chváal, M. Nwborn, and E. Szmrédi, Croing-fr ubgraph, in: Thory and Pracic of Combinaoric, Norh-Holland Mah. Sud. 60, Norh-Holland, Amrdam-Nw York, 1982, [AE89] N. Alon and P. Erdő, Dijoin dg in gomric graph, Dicr Compu. Gom. 4 (1989), [BT98] Bollobá and A. Thomaon, Proof of a conjcur of Madr, Erdő and Hajnal on opological compl ubgraph, Europan J. Combin. 19 (1998), [BKV02] P. Braß, G. Károlyi, and P. Valr, A Turán-yp rmal hory for conv gomric graph, in: Dicr and Compuaional Gomry Th Goodman-Pollack Fchrif (B. Aronov al., d.), Springr Vrlag, Brlin, 2003, o appar. [CN00] G. Cairn and Y. Nikolayvky, Bound for gnralizd hrackl, Dicr Compu. Gom. 23 (2000), [CP92] V. Capoyla and J. Pach, A Turán-yp horm on chord of a conv polygon, Journal of Combinaorial Thory, Sri B 56 (1992), [GKK96] W. Goddard, M. Kachalki, and D. J. Kliman, Forcing dijoin gmn in h plan, Europan J. Combin. 17 (1996), [HP34] H. Hopf and E. Pannwiz, Aufgab Nr. 167, Jahrbrich dr duchn Mahmaikr-Vrinigung 43 (1934), 114. [KSz96] J. Komló and E. Szmrédi, Topological cliqu in graph II, Combin. Probab. Compu. 5 (1996), [K84] A. V. Koochka, Lowr bound of h Hadwigr numbr of graph by hir avrag dgr, Combinaorica 4 (1984), [K88] A. V. Koochka, Uppr bound on h chromaic numbr of graph (in Ruian), Trudy In. Ma. (Novoibirk), Modli i Mody Opim., 10 (1988), [KK97] A. V. Koochka and J. Kraochvíl, Covring and coloring polygon-circl graph, Dicr Mah. 163 (1997), [L84] F. T. Lighon, Nw lowr bound chniqu for VLSI, Mah. Sym Thory 17 (1984), [LPS97] L. Lováz, J. Pach, and M. Szgdy, On Conway hrackl conjcur, Dicr and Compuaional Gomry, 18 (1997), [M98] W. Madr, 3n 5 dg do forc a ubdiviion of K 5, Combinaorica 18 (1998), [M02] J. Maoušk, Lcur a DIMACS Workhop on Gomric Graph Thory, Nw Brunwick, Spmbr 2002.

10 10 Jáno Pach, Radoš Radoičić, and Géza Tóh [Mc00] S. McGuinn, Colouring arcwi conncd in h plan I, Graph & Combin. 16 (2000), [P99] J. Pach, Gomric graph hory, in: Survy in Combinaoric, 1999 (J. D. Lamb and D. A. Prc, d.), London Mahmaical Sociy Lcur No 267, Cambridg Univriy Pr, Cambridg, 1999, [PPST02] J. Pach, R. Pinchai, M. Sharir, and G. Tóh, Topological graph wih no larg grid, o appar. [PPTT02] J. Pach, R. Pinchai, G. Tardo, and G. Tóh, Gomric graph wih no lf-inrcing pah of lngh hr, in: Graph Drawing (M. T. Goodrich, S. G. Kobourov, d.), Lcur No in Compur Scinc 2528, Springr-Vrlag, Brlin, 2002, [PRT02] J. Pach, R. Radoičić, and G. Tóh, On quai-planar graph, in prparaion. [PRTT02] J. Pach, R. Radoičić, G. Tardo, and G. Tóh, Graph drawn wih a mo 3 croing pr dg, o appar. [PSS96] J. Pach, F. Shahrokhi, and M. Szgdy, Applicaion of h croing numbr, Algorihmica 16 (1996), [PT97] J. Pach and G. Tóh, Graph drawn wih fw croing pr dg, Combinaorica 17 (1997), [PT94] J. Pach and J. Törőcik, Som gomric applicaion of Dilworh horm, Dicr Compu. Gom. 12 (1994), 1 7. [PR02] R. Pinchai and R. Radoičić, On h numbr of dg in gomric graph wih no lf-inrcing cycl of lngh 4, Proc. 19h Annual Sympoium on Compuaional Gomry, ubmid. [RST93] N. Robron, P. Symour, and R. Thoma, Hadwigr conjcur for K 6-fr graph, Combinaorica 13 (1993), [SV94] O. Sýkora and I. Vr o, On VLSI layou of h ar graph and rlad nwork, Ingraion, Th VLSI Journal 17 (1994), [T02] G. Tardo, On h numbr of dg in a gomric graph wih no hor lf-inrcing pah, in prparaion. [T84] A. Thomaon, An rmal funcion for conracion of graph, Mah. Proc. Cambridg Philo. Soc. 95 (1984), [T00] G. Tóh, No on gomric graph, J. Combin. Thory, Sr. A 89 (2000), [T01] A. Thomaon, Th rmal funcion for compl minor, J. Combin. Thory Sr. B 81 (2001), [V98] P. Valr, On gomric graph wih no k pairwi paralll dg, Dicr and Compuaional Gomry 19 (1998), [W71] D. R. Woodall, Thrackl and dadlock, in: Combinaorial Mahmaic and i Applicaion (Proc. Conf., Oford, 1969), Acadmic Pr, London, 1971,