Hamilton Cycles in Random Lifts of Graphs

Transcription

1 Hamlto Cycle Radom Lft of Grap K. Burg P. Cebolu C. Cooper A.M. Freze Marc 1, 005 Abtract A -lft of a grap K, a grap wt vertex et V K [] ad for eac edge, EK tere a perfect matcg betwee {} [] ad {} []. If tee matcg are coe depedetly ad uformly at radom te we ay tat we ave a radom -lft. We ow tat tere are cotat 1, uc tat f 1 te a radom -lft of te complete grap K amltoa wp ad f te a radom -lft of te complete bpartte grap K, amltoa wp. 1 Itroducto For a grap K, a -lft G of K a vertex et V K [] were for eac vertex v V K, {v} [] called te pllar above v ad wll be deoted by Π v. Te edge et of a a -lft G cot of a perfect matcg betwee pllar Π u ad Π w for eac edge u, w EK. Te et of -lft wll be deoted L K. I t paper we dcu radom -lft, coe uformly from L K. I t cae, te matcg betwee pllar are coe depedetly ad uformly at radom. Lft of grap were troduced by Amt ad Lal [1] were tey proved tat f K a coected, mple grap wt mmum degree δ 3, ad G coe radomly from L K te G δ coected wp, were te aymptotc are for. Tey cotued te tudy of radom lft [] were tey proved expao properte of lft. Togeter wt Matoušek, tey gave boud o te depedece umber ad cromatc umber of radom lft [3]. Lal ad Rozema [4] gve a tgt aaly for we a radom -lft a a perfect matcg. I t paper we dcu te probablty tat a radom -lft amltoa. I partcular we tudy te cae were K te complete grap K or te complete bpartte grap K,. We ue te otato y r Y for y coe uformly at radom from Y. Teorem 1. Tere ext a cotat 1 uc tat f 1 ad G r L K te G amltoa wp. Teorem. Tere ext a cotat uc tat f ad G r L K, te G amltoa wp. Teorem 1 proved te ext ecto. Teorem proved Secto 3. Departmet of Matematcal Scece, Carege Mello Uverty, Pttburg PA1513, USA Departmet of Matematcal Scece, Carege Mello Uverty, Pttburg PA1513, USA Scool of Matematcal Scece, Uverty of Nort Lodo, Lodo N7 8DB, UK Departmet of Matematcal Scece, Carege Mello Uverty, Pttburg PA1513, USA. Reearc upported part by a NSF grat. 1

2 Proof of Teorem 1.1 Structural Properte of L K Te vertce of L K wll be deoted by V ad t edge wl be deoted E. We wll ue te colorg argumet of Feer ad Freze [7] to ow G amltoa wp. For G L K we cooe a et H 1 = H 1 G EG a follow: Eac vertex of G arbtrarly cooe 1 edge of G cdet wt t. Tu te umber of dtct edge coe betwee 6 ad 1 ad te mmum degree of te grap duced by H 1 at leat 1. Next let P 0 = P 0 G be a pecfc loget pat G. Let F G = P 0 H 1 be te fxed edge of G. Te aaly ue a upecfed, uffcetly mall, potve cotat β < 1. Let B = BG be te et of ubet of EG of ze β. We ay tat a ubet of edge H acceptable f H = B F for ome B BG. Let HG be te collecto of acceptable ubgrap of G. For a lft G, eac B BG defe a colorg of te edge of G wc te edge of H = B F are colored blue ad te edge of R = G\H are colored gree. Let S V be of ze ad let S be te terecto of S V wt pllar Π for []. Te umber of coce for S ad by coderg te umber of coce for te S we ee tat e = = For a grap G = V, E ad S V let NS = {v V \S : u S uc tat u, v EG} be te dot egborood of S. For G r L K ad et S Π ad T Π, S =, T = t, PrNS Π T = tt 1... t Trougout t ecto all tatemet old for ad uffcetly large. Lemma 1. For G r L K, Pr S V : S 10 ad S cota at leat S edge = o1 t. Proof Ug 1 we ee tat te expected umber of et S of ze tat cota at leat edge o more ta φ = = e e e e 3 4 4

3 Te /10 =5 φ = o1. Lemma. If G r L K ad H r HG, te wp H atfe S V, S /4 mple N H S S. 3 Proof Aume frt tat S /10 ad let U = S NS. Let a be te umber of edge cotaed S ad let b be te umber of edge from S to NS. Te degree um of S H 1 at leat 1 S ad o a + b 1 S. But te U cota at leat a + b 6 S edge ad we ca aume by Lemma 1 tat U > 3 S. T complete te argumet for S /10. Let H be defed by cludg a edge of G H depedetly wt probablty β were β < β. Te H a bomal radom varable woe expected value le ta β. Te Ceroff boud mple tat for a mootoe creag property of lft Q, f H Q wp, te H Q wp. For /10 < S /4, let T = NS ad t = T. Ug 1 ad, te expected umber Z of et S wt N H S < S bouded a follow: I te frt le of te followg dplay, te otato deote + t > + t or + t = + t ad >. Z = = /4 1 =/10 t= = t 1+ +t =t /4 1 =/10 t= = /4 1 =/10 t= = /4 =/10 t= = /4 1 =/10 t= = /4 1 =/10 t=0 /4 1 =/10 t=0 /4 =/10 e β/ e e e e t t 1+ +t =t t 1+ +t =t t 1+ +t =t t 1+ +t =t t =1 t =1 t t =1 t t 1 β 1 + t 1 β β +t + t + 1 β β +t + t / + 1 β β + t + 1 β =1 β +t +t / + t β 1+t/ + t β 1+t/ t e exp { β 1 + t } 1 + t/ t 1 3 { } exp β 10 1+t/ 4 3

4 Lemma 3. If G r L K ad H r HG, te wp H coected. Proof If H ot coected, Lemma mple tat wp H te uo of a cotat umber of compoet of ze at leat /4. We wll aga work uder te aumpto tat edge are cluded H depedetly wt probablty β were β < β. Aume wtout lo of geeralty tat S /. Te expected umber of et S of ze S [/4, /] wt o edge betwee S ad t complemet o more ta / =/4 / =/4 / =/4 e β / = e β 1 =1 β + 1 β { } e exp β / β 1/ 5 Let P = v 0,..., v k be a loget pat grap H. A Póa rotato of P [10] wt v 0 fxed gve aoter loget pat P = v 0,... v v k... v +1 created by addg edge v k, v ad deletg edge v, v +1. Let END H v 0, P be te et of edpot obtaed by a equece of Póa rotato tartg wt P, keepg v 0 fxed ad ug a edge v k, v of H. Eac vertex v END H v 0, P ca te be ued a te tal vertex of aoter et of loget pat END H v, P, t tme ug v a te fxed vertex, but aga oly addg edge from H. Let END H P = {v 0 } END H v 0, P. Te Póa codto NENDv, P ENDv, P 1 for v END H P togeter wt Lemma mple te followg. Lemma 4. If G r L K ad H r HG, te wp END H v, P /4 for all v END H P, P = P G. We ay ext tat a ordered par of pllar Π k, Π l good w.r.t. a loget pat P f {u Π k END H P : {v Π l END H u, P : u, v EH} /500} / I word, Π k cota at leat /500 vertce u END H P for wc tere at leat /500 vertce v Π l END H u, P uc tat te edge u, v / EH. Lemma 5. If 3 old te G a at leat /3000 good pllar par. Proof We ow frt tat for u END H tere are at leat /7 1 pllar for wc {v Π l END H u, P : u, v EH} /8 7 4

5 old. Let u END H ad uppoe tat tere are m pllar for wc 7 fal. Te total umber of vertce ENDu, H mut be at leat /4 by Lemma 4 wc gve te equalty o tat m 6/7. cotag u. m/8 + m /4 We get /7 1 good pllar, becaue we ave to dcout te pllar Next, we ay tat a o-edge x, y / EH mut be avoded f x END H ad y ENDu, H. We ave ut ow tat for eac u END H, tere are at leat /57 edge cdet wt u tat mut be avoded. A END H ENDu, H ad eac o-edge couted at mot twce, te total umber of o-edge G tat mut be avoded at leat 1 /4 /57. Aume ow tat tere are δ pllar par tat cota at leat /50 edge tat mut be avoded. We te get te equalty δ + 1 δ /50 /456 wc gve δ > 1/3000. Let Π k, Π l be a pllar par tat cota at leat /50 edge tat mut be avoded. To ow tat Π k, Π l good, let {u END Π k : {v ENDu, H Π l, u, v / EH /500} = γ. 8 We te get te equalty o γ > 1/500. γ + 1 γ /500 /50. Te Proof For a lft G, let DG be te ubet of HG wc H coected ad atfe 3 for S > /10 ad let D = G DG. Let A be te ubet of L K uc tat for G A ad H coe radomly from HG, PrH DG 1 α. were α = e β/400. Let C be te ubet of L K tat ot amltoa ad let F = A C. To ow tat PrC 0, we wll frt ow tat A = 1 o1 L K ad te ue te colorg argumet of Feer ad Freze [7] to ow tat PrF 0. Lemma 6. A = 1 o1 L K Proof If G r L K ad H r HG te PrH D = PrH D GPrG G L K = PrH D GPrG + PrH D GPrG G A G A PrA + 1 α1 PrA = 1 α + αpra 9 5

6 ad 4 ad 5 mply tat Puttg 9 ad 10 togeter, we get o tat PrH D 1 α α + αpra 1 α. PrA 1 α. To get a upper boud o te umber of grap G L K uc tat G F, we cotruct a 0-1 matrx A = a,. Row dex correpod to a grap G L K ad dex rage over all acceptable ubgrap H HG. Subgrap of G wll be deoted by H,. Let a, = 1 f S V, S /4 mple NH, S S H, coected H, P 0 G v G ot Hamltoa v E H, Π k, Π l [1 ± 1/3 β], k l [] 11 Note tat, ad v mply loget pat P of H,, u, v ER, : u END H, P, v END H, u, P 1 Now let be te umber of oe A. Lemma 7. If G F te N 1 = a, 13 a, 1 o1 1 β. 13 r Proof G F ad H, HG mple tat H, atfe,, ad v wp. Now B 1, B BG may gve re to te ame ubgrap H f te edge ot B 1 B are all F. So we cout te umber of way to elect R a a lower boud o HG. We ave H β + 13 ce tere are at mot 13 edge P0 ad H 1. Te te umber of coce for R at leat te umber of way to elect a et of 1 β 13 edge from te 13 ot F. Codto v old troug te Ceroff boud. It follow mmedately from Lemma 7 tat 13 N 1 1 o1 1 β F We ow obta a upper boud o N 1. Let X = {H :, for wc H, = H ad a, = 1} 6

7 Te followg boud follow from te defto ad a cocetrato equalty for amplg wtout replacemet, ee Hoeffdg [9], Teorem 4: X 1 + o1 β! β For a fxed H X let Tu, G H = {G : H, = H ad a, = 1}. N 1 = H X G H. Lemma 8. H X mple G H e c 1 β + O 1/3! 15 for ome abolute cotat c > 0. Proof We beg wt H ad cout te umber of way to add back te edge of R to form a lft G G H. Te umber of edge Rk, l betwee two pllar of G o more ta 1 β + O 1/3. Tu tere are at mot 1 β + O 1/3! poble matcg to add back betwee eac par of pllar. We addg back ew edge to H we mut avod edge u, v were u END H ad v ENDu, H o tat a, = 1 te reultg grap. For a good pllar par Π k, Π l a defed 6, tere are at leat /500 vertce x Π k, eac adacet to at leat /500 vertce y Π l tat gve re to a edge x, y tat mut be avoded. Te probablty tat we avod all uc edge betwee a good pllar par at mot /500 1 =0 1 /500 e /50,000 A tere are at leat /3000 good pllar par, te probablty tat a et of ew edge avod all requred edge G at mot e /50,000 /3000. It follow from 13, 14 ad 15 tat F L K e c 1 β + O 1/3! e c / β = o1 β β 14 e c /+14 l1/β bouded above by 13 1 o1! 13 1 β o1 β! were te ecod le ue a x b x b x a x x a b. β 7

8 3 Proof of Teorem 3.1 Structural Properte of L K, Let V 1, V be te bpartto of K, ad let W 1, W be te bpartto of te lft of K, tat t duce. We ow prove mlar properte to toe Secto.1. Let H 1, P 0 be et of edge defed a Secto.1 ad let F = P 0 H 1. Aga we ue a upecfed, utably mall cotat β < 1, let B be a et of β edge G ad BG te collecto of ubgrap B. A et of edge H G acceptable f H = B F for ome B BG. Let HG be te collecto of acceptable ubgrap of G ad let R = G\H. Trougout t ecto all tatemet old for ad uffcetly large. Te proof mlar to tat for K ad o we wll omt calculato tat are almot detcal to toe of te prevou ecto. Te ma dffculty wt ug a Poá type argumet tat f a loget pat P G eve te t caot be cloed to a cycle, coectvty otwttadg.e. we ga otg from avodg coog edge to o v to ENDv. I t cae, tere are o edge to avod. We terefore ave to modfy te argumet. We follow Bollobá ad Koayakawa [6] wo coderably mplfed te argumet of [8]. Lemma 9. For G r L K, Pr S V : S 0 ad S cota at leat S edge = o1 Lemma 10. If G r L K, ad H r HG, te wp H atfe S W, S /4 mple N H S S. 16 Lemma 11. If G r L K, ad H r HG te wp H coected. Lemma 1. If K a a -factor ad G L K, te G a a -factor. Proof Let C V K be oe of te cycle of a -factor of K ad let G[C] te ubgrap of G duced by te pllar above te vertce of C. Let v 1,..., v k be a orderg of te vertce of C uc tat v, v +1 a edge of C were v 1 = v k+1 ad let Π be te pllar of G above v C. Let σ be te permutato tat defe te matcg from pllar Π to Π +1 for eac Π G[C]. For eac Π 1, defe σ = σ k σ σ 1 to be te permutato o te vertce of Π 1 tat reult from followg te permutato σ 1 troug σ k back to Π 1. Te a cycle of σ a cycle of G o tat te cycle of σ defe a -factor of G[C]. T proce ca be repeated for all cycle of a -factor of K to obta a -factor of G L K. We ow decrbe a exteo-rotato proce wc attempt to traform te -factor F of Lemma 1 to a Hamlto cycle. Geeral Step: Gve te curret -factor tally F cooe a edge e = x, y of G wc o two dtct cycle C, C. T poble becaue G coected wp. Let f be a edge of 8

9 C cdet wt x ad f be a edge of C cdet wt y. Let P be te pat C C {e}\{f, f }. Tere are ow everal poblte. a: Tere a edpot u ay, of P wc a a egbour v a cycle C dot from P. We exted P by replacg P, C by P C {u, v} \ f were f a edge of C cdet wt v. We repeat t operato a log a we ca. We te carry out b or c. b Te edpot u, v of P are coected by a edge H. Addg u, v to P create a -factor wt at leat oe le cycle ta at te tart of te Geeral Step ad complete t. c Carry out rotato o P utl eter we cotruct a pat Q wt a edpot x wc adacet to a vertex y o cycle C outde Q or we atfy te codto of b. I te latter cae we proceed a b above. I te former cae we exted Q by addg te edge x, y ad deletg a edge of C cdet wt y. We cotue te above operato utl we eter obta a Hamlto cycle or obta a pat P 0 = P 0 G = v 0, v 1,..., v p tat caot be exteded or cloed to a cycle va a equece of rotato. Note tat t pat ecearly of odd legt. We terefore let P 0 be a loget pat of odd legt wc caot be exteded by rotato ad for wc tere are a et of vertex dot cycle coverg te vertce ot P. We ue te Póa codto wc tll old ad Lemma 10 to get te followg. Lemma 13. If G r L K, ad H v END H P 0, P 0 = P 0 G. r HG, te wp END H v, P 0 /4 for all We ay ext tat a ordered par of pllar Π k, Π l good w.r.t. a loget pat P f Π k W x, Π l W 3 x, x = 1, ad {u Π k END H P : {v Π l END H u, P : u, v EH} /500} / I word, Π k cota at leat /500 vertce u END H P for wc tere at leat /500 vertce v Π l END H u, P uc tat te edge u, v / EH. Lemma 14. If 16 old te G a at leat /3000 good pllar par. Proof We frt ote tat P 0 ad te pat obtaed by rotato are of odd legt ad o eac a oe edpot eac of W 1, W. Now we ca argue a Lemma 5 tat for eac u W x END H, x = 1, tere are at leat /7 pllar Π W 3 x ENDu, H for wc {v Π k END H u, P : u, v EH} /8. Te ret of te proof detcal to tat of Lemma Te Proof Defe te et A, C, F a te proof of Teorem 1. We ave A 1 o1 L K, ug te argumet Lemma 6 wt te reult from Lemma 10 ad 11. Defe alo te matrx A ad N 1 a te proof of Teorem 1. Te proof of te followg Lemma are mlar to te proof of Lemma 7 ad 8. 9

10 Lemma 15. If G F te 5 a, 1 o1 1 β. 5 It follow mmedately from Lemma 15 tat 5 N 1 1 o1 1 β F We ow obta a upper boud o N 1. Let X = {H :, for wc G, = H ad a, = 1} It follow from te defto tat X 1 + o1 β! β For a fxed H H let Tu, G H = {G, : H, = H ad a, = 1}. N 1 = H X G H. Lemma 16. H X mple G H e c 1 β + O 1/3!. 0 for ome abolute cotat c > 0. It follow from 18, 19 ad 0 tat F L K bouded above by e c 1 β + O 1/3! o1 e c / β β β 4 e c /+4 l1/β = o1. 1 o1! 5 1 β 5 β β! Referece [1] A. Amt ad N. Lal, Radom Grap Coverg I: Geeral Teory ad Grap Coectvty, Combatorca 00, [] A. Amt ad N. Lal, Radom Lft of Grap II: Edge Expao, Combatorc Probablty ad Computg, to appear. 10

11 [3] A. Amt, N. Lal ad J. Matoušek, Radom Lft of Grap III: Idepedece ad Cromatc Number, Radom Structure ad Algortm 0 00, 1-. [4] N. Lal, ad E. Rozema, Radom Lft of Grap: Perfect Matcg, to appear. [5] B. Bollobá, Radom Grap, Cambrdge Uverty Pre 001. [6] B. Bollobá ad Y. Koayakawa, Te ttg tme of Hamlto cycle radom bpartte grap, Grap Teory, Combatorc, Algortm ad Applcato, Y. Alav, F.R.K. Cug, R. Graam ad D.F. Hu, Ed [7] T.I. Feer ad A.M. Freze, O te extece of Hamlto cycle a cla of radom grap, Dcrete Matematc , [8] A.M. Freze, Lmt dtrbuto for te extece of amltoa cycle radom bpartte grap, Europea Joural of Combatorc [9] W. Hoeffdg, Probablty Iequalte for Sum of Bouded Radom Varable, Amerca Stattcal Aocato Joural 1963, [10] L. Póa, Hamlto crcut radom grap, Dcrete Matematc ,