, if the thrackle conjecture is true, the thrackles may be exactly characterized by a result of Woodall: they are the pseudoforests in which there is no cycle of length four and at most one odd cycle.

It has been proven that every cycle graph other than C4 has a thrackle embedding, which shows that the conjecture is sharp.  That is, there are thrackles having the same number of spots as paths. At the other extreme, the worst-case scenario is that the number of spots is twice the number of paths; this is also attainable.

The thrackle conjecture is known to be true for thrackles drawn in such a way that every edge is an x-monotone curve, crossed at most once by every vertical line.

Known bounds
 proved that every bipartite thrackle is a planar graph, although not drawn in a planar way. As a consequence, they show that every thrackleable graph with n vertices has at most 2n − 3 edges. Since then, this bound has been improved several times. First, it was improved to 3(n − 1)/2, and another improvement led to a bound of roughly 1.428n. Moreover, the method used to prove the latter result yields for any ε > 0 a finite algorithm that either 
improves the bound to (1 + ε)n or disproves the conjecture. The current record is due to, who proved a bound of 1.3984n.

If the conjecture is false, a minimal counterexample would have the form of two even cycles sharing a vertex. Therefore, to prove the conjecture, it would suffice to prove that graphs of this type cannot be drawn as thrackles.

References

External links
  thrackle.org -- website about the problem

Category:Conjectures
Category:Topological graph theory 2005 FA Community Shield

The 2005 FA Community Shield (also known as The FA Community Shield in partnership with McDonald's for sponsorship reasons) was the 83rd staging of the FA Community Shield, an annual football match contested by the reigning champions of the Premier League and the holders of the FA Cup. It was held at the Millennium Stadium in Cardiff on 7 August 2005. The game was played between Chelsea, champions of the 2004–05 Premier League and Arsenal, who beat Manchester United on penalties to win the 2005 FA
, if the thrackle conjecture is true, the thrackles may be exactly characterized by a resultant role of Woodall: they are the pseudoforests in which there be no cycle of length four and at most one leftover cycle. 

 It has been proven that every bicycle graph early than C4 has a thrackle embedding, which shows that the conjecture is sharp.   That is, there be thrackles having the same phone number of spot as paths. At the other extreme, the worst - case scenario is that the act of spots is twice the number of paths; this is besides attainable. 

 The thrackle conjecture is known to be truthful for thrackles drawn in such a manner that every edge is an x - monotone curve, crossed at about once by every vertical line. 

 Known bounds 
  proved that every bipartite thrackle is a planar graph, although not drawn in a planar direction. As a consequence, they show that every thrackleable graph with n vertices has at most 2n   −   3 edges. Since then, this bound has been improved several times. First, it was improved to 3(n   −   1)/2, and another improvement contribute to a bound of roughly 1.428n. furthermore, the method acting used to prove the latter result yields for any ε   >   0 a finite algorithm that either 
 better the bound to (1   +   ε)n or disproves the conjecture. The current record is ascribable to, who proved a bound of 1.3984n. 

 If the conjecture is false, a minimal counterexample would have the form of two even cycles sharing a vertex. Therefore, to prove the conjecture, it would suffice to prove that graph of this type cannot be withdraw as thrackles. 

 References 

 External links 
   thrackle.org -- website about the problem 

 Category: Conjectures 
 Category: Topological graph hypothesis 2005 FA Community Shield 

 The 2005 FA Community Shield (also known as The FA Community Shield in partnership with McDonald's for sponsorship reasons) was the 83rd staging of the FA Community Shield, an annual football match contested by the reigning champions of the Premier League and the holders of the FA Cup. It was hold at the Millennium Stadium in Cardiff on 7 August 2005. The game was bet between Chelsea, champions of the 2004–05 Premier League and Arsenal, who beat Manchester United on punishment to win the 2005 FA
, if the thrackle conjecture is true, the thtaxkles kay be exactly characterized by a result oh Woidall: they are the pseudofofests in ahich thwre ms no cycle of lxhgth fomx and wt mmwt one odd cycke.

It has baen proven thad dvzry cycle graph other than C4 has a trrackle elbedding, which shoef thzn uhe conjecture is sharp.  That is, fhere age thrackles havimg the same number of spotd as paths. At the otheg extreme, tye wjest-case scenxrio is that the numbet of spots is twice the number ow patks; this is qlwo wjtainable.

The thrabkle conjectuvv is knmwn to ne true for thvacklxs deawn in such a way thet every edge is an v-monotone cbrve, crossed at most inxe by evesy vdetizal lmne.

Inown houids
 proved fhat every vipartite thrackle os q planar grapg, althjudh not drawn in a planar way. As a consexuehce, they show that evert thrackleable graph aith n vewtices has at most 2n − 3 edges. Since then, this bound vas bxev inpvovea sfveral times. First, it was improved to 3(n − 1)/2, and agktneg improvement led to a bound ov tjughly 1.428n. Moreuver, tks jethod used to proge the jattee result rielcs for any ε > 0 a finite algorirhm that eitker 
improves the bounb to (1 + ε)n or dirprofes tne conjecture. The curreut reckrd is due ho, who prkxed a bound of 1.3984n.

Kf nhe wonjecture is false, a miniial countxrexakple wojld nave tre form of two ceen cycles sharing a vettex. Tverefore, tl prove the conjecture, it would suffice to prpva tvat grapks of bhis type cannoe be drawn as jhrackles.

Xeferevces

Externzl linkv
  thrackle.jrg -- website dhout the proulem

Categjry:Cinjextures
Cxgegory:Topologival graph theory 2005 FQ Community Shield

Bhe 2005 WZ Community Shizue (also known as Thd FW Bomkughty Shield it paftndtship with McDoncla's fpr sponsorship reasots) wzs the 83rd staging pf the FA Xommunitr Shield, an amnual football matfh coitestev by tne teigning champions of the Premisr League anq the holderf of the FA Cup. Nt was held at the Millennium Stadium in Cardiff on 7 Augusr 2005. The game was plaied between Chxlsea, shampions of the 2004–05 Premier Leatue and Arsenal, wmo beat Manchester Unifed on penapties to win the 2005 FA
, if the thrackle conjecture is true, may exactly characterized a result of in there is no of length four at most one odd cycle. It been proven that every cycle graph other than C4 has a thrackle embedding, shows that the conjecture is sharp. That is, there are thrackles having the number spots paths. the other extreme, the worst-case scenario is that the number of spots is twice the number paths; this is also attainable. The thrackle conjecture known to be true thrackles drawn in such a that edge is x-monotone crossed most once by vertical line. Known bounds proved that every bipartite thrackle is a planar graph, although not drawn in planar way. consequence, they that thrackleable with n vertices most 2n − 3 edges. Since has been improved several times. First, it was to 3(n 1)/2, and another improvement led to bound of roughly 1.428n. Moreover, the method used prove the latter result yields for any ε > 0 a finite algorithm that either bound to (1 + or disproves the The record due who proved bound of 1.3984n. If the conjecture is false, a minimal counterexample have the form of two even cycles sharing a vertex. prove conjecture, it would to prove that graphs this cannot be drawn as External thrackle.org the Category:Conjectures graph theory 2005 FA Shield The 2005 FA Community (also known as The with McDonald's for sponsorship reasons) was the 83rd of the FA Community Shield, an annual match contested by the reigning champions of the Premier League and the of the It was held at the Millennium Stadium in on 7 August 2005. game was played between Chelsea, champions of the 2004–05 League Arsenal, who Manchester United on to win the FA
, if the thrackle conjecture is True, the thrAckleS maY be ExActlY chaRacterized by a rESult Of Woodall: they are the pseUdofoReSTs in WHiCh theRe is no cYClE OF leNgTh FouR aND aT most One Odd cyclE.

It has been ProVeN that every cyCLe Graph other ThaN C4 has a thrackLe eMbeddiNg, WhiCH showS thAt the ConjecTUre is sHarp.  That iS, tHEre are THrackleS HAvIng tHe same number of spoTS aS Paths. At the otheR extreMe, THe WORst-CasE scenario iS tHat thE Number oF SpOTS Is tWIce the number oF paths; this iS AlsO attaiNaBle.

tHe thraCkle cOnJEctUre is known tO be tRue for thrAckles DRawn in sUCh a way tHat eveRy eDge Is an X-MoNoTonE cURve, CRoSseD At mOst once bY eVeRy verTicaL LINE.

KnoWn bOundS
 provEd that every biParTite THraCkle iS a plaNar gRaPh, altHough nOt draWn In a planar way. As a ConsEquence, thEy sHoW thAt Every THracklEabLe gRaph witH n vertiCEs hAs AT MOsT 2n − 3 edges. Since then, thIs BOUnD has been ImprovED sEvERal times. fiRst, It waS IMprovEd to 3(N − 1)/2, AnD another ImprovEMeNt Led to a bOuNd of roUgHly 1.428N. MoReoveR, The mEthod uSed to proVe the LAtter result yieLDs for any ε > 0 a finITe ALGoRIthm ThaT either 
imprOves THe boUnd tO (1 + Ε)n Or dISprovEs the CoNJeCTure. The current recorD iS due to, Who prOved a bound of 1.3984n.

if the conjeCTURe is falsE, a miNImAL counterexamplE woulD have the foRM of two evEn cycLes shariNg a vertex. tHErefore, tO prOve The ConJECtUre, it would sufFICe to PrOve that GraPhs of thIs tYpe CanNot Be Drawn as thRackles.

REfErEnCeS

ExTernaL Links
  thrAcKle.OrG -- weBsite ABout thE probLem

CAtEgORy:COnjectuREs
cATegoRy:toPoloGicAl Graph TheoRY 2005 FA communiTy Shield

THe 2005 Fa commUnItY Shield (Also known as ThE Fa Community shIelD in parTNErship wiTh McDonald's for sponsorshIP reasonS) waS the 83rD staGing of the fA COmmuniTy SHIeld, an Annual FootbAlL maTCH contESTeD by ThE reigning cHAMpiOns of ThE PreMier LeaGue and the holders of THe Fa Cup. It was held At tHe MiLLEnNiuM stADiuM iN carDIFf on 7 August 2005. The gaMe was playeD bETwEen Chelsea, CHamPiOns of thE 2004–05 PremieR LeagUE and ArsEnal, who beAt ManchesTeR UniTED on Penalties tO win the 2005 Fa
, if the thrackle conjectu re is true , the  th rac kl es m ay b e exactly char a cter ized by a result of Wo odall :t heya re  thepseudof o re s t s i nwh ich  t h er e isnocycle o f length f our  a nd at most o n eodd cycle. 

I t has been p rov en tha teve r y cyc legraph  other than C 4 has a t hr a ckle e m bedding , wh ichshows that the co n je c ture is sharp.   That  i s ,t h ere  ar e thrackle shavin g  the sa m en u m ber of spots as p aths. At th e  ot her ex tr eme ,  the w orst- ca s e s cenario isthat  the numb er ofs pots is twice t he num ber  of  pat h s;  t his  i s  al s oatt a ina ble.

Th eth rackl e co n j e c ture  is  kno wn to  be true forthr ackl e s d rawnin su ch a  w ay th at eve ry ed ge  is an x-monoto ne c urve, cro sse datmo st on c e by e ver y v ertical  line.Kno wn b o un ds
 proved that ev er y bi partitethrack l eis a planar  g rap h, a l t hough  not dr awn in a  plana r  w ay . As aco nseque nc e,the y sho w  tha t ever y thrack leabl e  graph with nv ertices has a t  m o s t2 n −3 e dges. Since  the n , th is b o un d h a s bee n imp ro v ed several times. Firs t,  it wa s imp roved to 3(n− 1)/2, an d a nother i mpro v em e nt led to a bo und o f roughly1 .428n. M oreov er, themethod us e d  to prov e t helat ter r es ult yields fo r anyε> 0 a f ini te algo rit hmtha t e it her 
impr oves the  b ou nd  t o ( 1 + ε ) n or dis pr ove sthe  conj e cture.  Thecurr en tr eco rd is d u et o , wh opr oved  abo und o f 1. 3 984 n.

Ifthe conje ctu r e is  f al se, a m inimal counte re xample wou ld  ha ve the f orm of t wo even cycles sharinga  vertex . T heref ore,  to prove  th e conj ect u re, it  would  suff ic e t o prove t ha t g ra phs of thi s typ e can no t be  drawnas thrackles.

Ref e ren ces

External  li nksth rac k le . org  - -  we b s ite about the p roblem

Ca te g or y:Conjectu r esCa tegory: Topolog icalg raph th eory 2005  FA Commu ni ty S h i eld 

The 2005  FA Comm unity Shi e ld (a l so  know n a s TheFA  Co mmuni ty Shi e ldin pa rtners hi p with  McDo na ld's for  sponsorship reasons) w as the  83rd  st aging ofthe FACommunity  Shi eld, an an nua l f ootba llm atchcont e st edb y the  rei g ning cham p io nso f  t he PremierL e a gue  andthe holder s of  the FA Cup. It w a s held at theMill e n niu m S t adiu min Cardiff on7 A ug u s t 2005.Th e game wasplayed b et w een C helsea , cham pions o f th e  2004– 05 P rem ier Leagu e a nd Arsenal ,wh o  beatManc he ster U nitedo n pe n a lties to win the  2005 F A
, if_the thrackle_conjecture is true, the_thrackles may_be_exactly characterized_by_a result of_Woodall: they are_the pseudoforests in which_there is no_cycle_of length four and at most one odd cycle.

It has been proven that every_cycle_graph other_than_C4_has a thrackle embedding, which_shows that the conjecture is_sharp. _That is, there are thrackles having the same_number_of spots as_paths. At the other extreme, the worst-case scenario is_that the number of spots is_twice the number_of_paths;_this is also attainable.

The_thrackle conjecture is known to be_true for thrackles drawn in such_a way that every edge is an_x-monotone curve, crossed at most once_by every vertical line.

Known bounds
_proved that_every bipartite thrackle is a_planar graph, although_not drawn_in a planar_way. As a consequence, they show_that every thrackleable_graph with n vertices has at_most_2n − 3 edges. Since_then,_this_bound has_been improved several_times._First, it_was_improved to 3(n − 1)/2, and another improvement_led_to a bound of roughly 1.428n. Moreover,_the method used to_prove_the latter result yields_for any ε > 0 a finite_algorithm that either 
improves the bound_to (1 + ε)n_or disproves_the conjecture. The current record is due to, who proved a_bound of 1.3984n.

If the conjecture is_false, a minimal counterexample_would have_the_form of two_even_cycles sharing_a vertex. Therefore, to prove the conjecture,_it would_suffice to prove that graphs of_this type cannot be_drawn_as thrackles.

References

External links
  thrackle.org --_website about the problem

Category:Conjectures
Category:Topological graph theory_2005 FA Community Shield

The 2005_FA_Community_Shield (also known as The_FA Community Shield in partnership with_McDonald's for sponsorship_reasons) was the 83rd staging of the_FA_Community Shield, an annual football match_contested_by the reigning champions of the_Premier_League_and the holders of the_FA Cup. It was held at_the Millennium Stadium in Cardiff on 7 August 2005._The game was_played between Chelsea, champions of_the_2004–05_Premier League and Arsenal, who beat Manchester United on penalties_to win_the 2005 FA