FORBIDDING HAMILTON CYCLES IN UNIFORM HYPERGRAPHS

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1 FORBIDDING HAMILTON CYCLES IN UNIFORM HYPERGRAPHS JIE HAN AND YI ZHAO Absrac For d l <, we give a ew lower boud for he miimum d-degree hreshold ha guaraees a Hamilo l-cycle i -uiform hypergraphs Whe 4 ad d < l =, his boud is larger ha he cojecured miimum d-degree hreshold for perfec machigs ad hus disproves a wellow cojecure of Rödl ad Rucińsi Our simple) cosrucio geeralizes a cosrucio of Kaoa ad Kiersead ad he space barrier for Hamilo cycles Iroducio The sudy of Hamilo cycles is a impora opic i graph heory A classical resul of Dirac [4] saes ha every graph o 3 verices wih miimum degree /2 coais a Hamilo cycle I rece years, researchers have wored o exedig his heorem o hypergraphs see rece surveys [6, 8, 26] To defie Hamilo cycles i hypergraphs, we eed he followig defiiios Give 2, a -uiform hypergraph i shor, -graph) cosiss of a verex se V ad a edge se E V ), where every edge is a -eleme subse of V Give a -graph H wih a se S of d verices where d ) we defie deg H S) o be he umber of edges coaiig S he subscrip H is omied if i is clear from he coex) The miimum d-degree δ d H) of H is he miimum of deg H S) over all d-verex ses S i H For l, a -graph is a called a l-cycle if is verices ca be ordered cyclically such ha each of is edges cosiss of cosecuive verices ad every wo cosecuive edges i he aural order of he edges) share exacly l verices I -graphs, a )-cycle is ofe called a igh cycle We say ha a -graph coais a Hamilo l-cycle if i coais a l-cycle as a spaig subhypergraph Noe ha a Hamilo l-cycle of a -graph o verices coais exacly / l) edges, implyig ha l divides Le d, l For l)n, we defie h l d, ) o be he smalles ieger h such ha every -verex -graph H saisfyig δ d H) h coais a Hamilo l-cycle Noe ha wheever we wrie h l d, ), we always assume ha d Moreover, we ofe wrie h d, ) isead of h d, ) for simpliciy Similarly, for N, we defie m d, ) o be he smalles ieger m such ha every -verex -graph H saisfyig δ d H) m coais a perfec machig The problem of deermiig m d, ) has araced much aeio recely ad he asympoic value of m d, ) is cojecured as follows Noe ha he o) erm refers o a fucio ha eds o 0 as hroughou he paper Cojecure [6, 5] For d ad, { m d, ) = max 2, ) } d ) ) d + o) d Cojecure has bee cofirmed [, 7] for mi{ 4, /2} d he exac values of m d, ) are also ow i some cases, eg, [23, 25]) O he oher had, h l d, ) has also bee exesively sudied [2, 3, 5, 7, 8, 9, 0,, 2, 3, 4, 9, 20, 22, 24] I paricular, Rödl, Rucińsi ad Szemerédi [20, 22] showed ha h, ) = /2 + o)) The same auhors proved i [2] ha m, ) = /2 + o)) laer hey deermied m, ) exacly [23]) This suggess ha he values of h d, ) ad m d, ) are closely relaed ad ispires Rödl ad Rucińsi o mae he followig cojecure Cojecure 2 [8, Cojecure 28] Le 3 ad d 2 The h d, ) = m d, ) + o d ) Dae: Jue 4, Mahemaics Subjec Classificaio Primary 05C45, 05C65 Key words ad phrases Hamilo cycles, hypergraphs The firs auhor is suppored by FAPESP Proc 204/864-5, 205/ ) The secod auhor is parially suppored by NSF gra DMS

2 2 JIE HAN AND YI ZHAO By usig he value of m d, ) from Cojecure, Küh ad Oshus saed his cojecure explicily for he case d = Cojecure 3 [6, Cojecure 53] Le 3 The h, ) = ) ) ) + o) I his oe we provide ew lower bouds for h l d, ) whe d l Theorem 4 Le d ad = d, he ) /2 /2 /2 + ) /2 h d, ) /2 + ) + o) Theorem 5 Le d l ad = d The h l d, ) b, l 2 + o) ) ), where b, l equals o he larges sum of l cosecuive biomial coefficies ae from { 0),, ) } Theorem 4 disproves boh Cojecures 2 ad 3 Corollary 6 For all, ) ) 5 h 2, ) 9 + o), h 3, ) o) ) ) ), h 4, ) 3 ad i geeral, for ay d, ) ) ) h d, ) > 3 d)/2 + d ) o) ) ) 4 These bouds imply ha Cojecure 2 is false whe 4 ad mi{ 4, /2} d 2, ad Cojecure 3 is false wheever 4 We will prove Theorem 4, Theorem 5, ad Corollary 6 i he ex secio We believe ha Cojecure 2 is false wheever 4 bu due o our limied owledge o m d, ), we ca oly disprove Cojecure 2 for he cases whe m d, ) is ow This boud h 2, ) o)) coicides wih he value of m 3, ) i was show i [6] ha m 3, ) = 5/9 + o)) ) 2, ad i was widely believed ha h 3, ) = 5/9 + o)) ), eg, see [9] O, which is smaller ha 5 he oher had, i is ow [7] ha m 2 4, ) = 2 + o)) 2 9 Therefore = 4 ad d = 2 is he smalles case whe Theorem 4 disproves Cojecure 2 More imporaly, ) shows ha h d, )/ d) eds o oe as d eds o For example, as becomes sufficiely large, h l, ) is close o d d), he rivial upper boud I coras, Cojecure suggess ha here exiss c > 0 idepede of ad d c = /e, where e = 278, if Cojecure is rue) such ha m d, ) c) d d) Similarly, by Theorem 5, if l = o ), h l d, )/ ) eds o oe as eds o because b, l 2 l 2 /2 ) o ) π/2 Theorem 5 also implies he followig special case: suppose is odd ad l = d = 2 The = 2 ad b, l = b 2,2 = 3, ad cosequely h 2 2, ) 4 + o)) Previously i was oly ow ha h 2 2, ) + )2 + o)) by wih a = / l) = + )/2 from Secio 2 Whe is large, he boud provided by Theorem 5 is much beer Fially, we do o ow if Theorems 4 ad 5 are bes possible Glebov, Perso, ad Weps [5] gave a geeral upper boud far away from our lower bouds) h l d, ) where c is a cosa idepede of d, l,, ) ) d c 3 3, d

3 FORBIDDING HAMILTON CYCLES IN HYPERGRAPHS 3 2 The proofs Before provig our resuls, i is isrucive o recall he so-called space barrier Throughou he paper, we wrie X Y for X Y whe ses X, Y are disjoi Proposiio 2 [3] Le H = V, E) be a -verex -graph such ha V = X Y ad E = {e V ) : e X } Suppose X < a l), where a := / l), he H does o coai a Hamilo l-cycle A proof of Proposiio 2 ca be foud i [3, Proposiio 22] ad is acually icluded i our proof of Proposiio 22 below I is o hard o see ha Proposiio 2 shows ha ) ) d ) h l d d, ) + o) a l) d Now we sae our cosrucio for Hamilo cycles i geeralizes he oe give by Kaoa ad Kiersead [, Theorem 3] where j = /2 ) ad he space barrier where j = l + ) simulaeously The special case wih = 3, l = 2, j =, ad X = /3 appears i [9, Cosrucio 2] Proposiio 22 Give a ieger j such ha l+ j, le H = V, E) be a -verex -graph such ha V = X Y ad E = {e ) V : e X / {j, j +,, j + l } Suppose j j+ l a l) < X < a l), where a := / l) ad a := / l), he H does o coai a Hamilo l-cycle Proof Suppose isead, ha H coais a Hamilo l-cycle C The all edges e of C saisfy e X / {j, j +,, j + l } We claim ha eiher all edges e of C saisfy e X j or all edges e of C saisfy e X j + l Oherwise, here mus be wo cosecuive edges e, e 2 i C such ha e X j ad e 2 X j + l However, sice e e 2 = l, we have e X e 2 X l, a coradicio Observe ha every verex of H is coaied i eiher a or a edges of C ad C coais implies ha l edges This a X e X a X e C O he oher had, we have e C e X < j ) l or e C e X > j + l) l I eiher case, j j+ l we ge a coradicio wih he assumpio a l) < X < a l) Noe ha by reducig he lower ad upper bouds for X by small cosas, we ca coclude ha H acually coais o Hamilo l-pah I he proofs of Theorems 4 ad 5, we will cosider biomial coefficies p q) wih q < 0 i his case p ) q = 0 We will coveiely wrie X = x, where 0 < x <, isead of X = x his does o affec our calculaios as is sufficiely large I he proofs of Theorem 4 we apply Proposiio 22 wih j ad x such ha j < x < j+ Noe ha whe j, d, ad = d) j < x < j+ implies ha j d + < x < j+ +, i paricular, 2 j d x + ) j Proof of Theorem 4 Le x = /2 / + ) Sice j= j, j+ /2 ) = 0, ) ad /3 + /2, here exiss a ieger j [ ] such ha j < /2 Le H = V, E) be a -verex -graph such ha + < j+ V = X Y, X = x ad E = {e ) V : e X j} Sice j Hamilo cycle by Proposiio 22 wih l = Now le us compue δ d H) For 0 i d, le S i be ay d-verex subse of V ha coais exacly i verices i X By he defiiio of H, ) ) ) d X i Y d i) deg H S i ) = j i j + i j+ < X <, H coais o igh I ca be show ha amog all 0 < x < ad 0 j such ha j < x < j+, our choice of x ad j miimizes max j d i j i) x i x) i, ad hus maximizes δ d H) i 23)

4 4 JIE HAN AND YI ZHAO Noe ha his holds for i > j or i < j rivially So we have { ) ) )} d X i Y d i) δ d H) = mi 0 i d j i j + i ) { ) )} X Y = max j d i j i + o ) Wrie X = x ad Y = y Whe 0 i, we have ) ) X Y i i = x)i y) i i! i + o ) = )! Whe i < 0 or i >, we have X i ) Y 23) δ d H) = i ) ) i x i y i + o ) ) i = 0 = ) i x i y i ) I all cases, we have ) { ) } ) max j d i j i x i y i + o ) Le a i := i) x i y i Sice x = /2 / + ) ad y = x, i is easy o see ha max 0 i a i = a /2 eg, by observig ai a i+ = y x i+ i for 0 i < ) Moreover, by 2, we have j d /2 j Therefore, ) /2 /2 max {a /2 + ) /2 i} = a /2 = j d i j /2 + ), ad we complee he proof by subsiuig i io 23) Now we ur o he proof of Theorem 5, i which we assume ha X = /2, hough a furher improveme of he lower boud may be possible by cosiderig oher values of X Proof of Theorem 5 The proof is similar o he oe of Theorem 4 Le H = V, E) be a -verex -graph such ha V = X Y, X = /2 ad E = {e V ) : e X / { l/2,, l/2 + l }} Noe ha a l) = l) l ) = l + > 2 l/2 ), ad l a l) = l) + l ) < 2 l/2 ) = 2 l/2 + l) l So we have l/2 a l) < X = 2 < l/2 + l a l) Thus, H coais o Hamilo l-cycle by Proposiio 22 Fix d ad le = d Now we compue δ d H) For 0 i d, le S i be ay d-verex subse of V ha coais exacly i verices i X I is easy o see ha deg H S i ) = ) i + l p=i X p ) ) Y p + o ), where i = l/2 i Usig X = Y = /2 ad he similar calculaios i he proof of Theorem 4, we ge By he defiiio of b, l, we have deg H S i ) = ) δ d H) = mi 0 i d deg HS i ) i + l p=i ) p 2 Corollary 6 follows from Theorem 4 via simple calculaios ) + o ) ) ) b, l 2 + o )

5 FORBIDDING HAMILTON CYCLES IN HYPERGRAPHS 5 Proof of Corollary 6 Le = d ad ) /2 /2 /2 + ) /2 f) := /2 + ) Theorem 4 saes ha h, ) f) + o)) ) for ay Sice f = 4 9, f3) = 3 26, ad f4) = 8 625, he bouds for h, ), = 2, 3, 4, are immediae To see ), i suffices o show ha for, 24) f) > 3/2 + Whe is odd, /2 /2 /2 +) /2 +) = /2 ; whe is eve, /2 /2 /2 + ) /2 < + 2 ) Thus, for all, we have ) f) /2 2, where a sric iequaliy holds for all eve Now we use he fac ) 2m m 2 2m / 3m +, which holds for all iegers m Thus, for all eve, we have f) / 3/2 + ; for all odd, ) f) /2 2 = ) + 2 /2 + 2 < 3 + )/2 + 3/2 + Hece f) / 3/2 + for all Moreover, by he compuaio above, regardless of he pariy of, he sric iequaliy always holds ad hus 24) is proved We ex show ha wheever 4 ad 2, f) > max { 2, ) } This implies ha Cojecure 3 fails for 4, ad Cojecure 2 fails for 4 ad mi{ 4, /2} d 2 because m d, )/ ) { d = max 2, ) } d + o) i his case) I suffices o show ha for 4 ad 2, f) < /2 ad f) < ) The firs iequaliy immediaely follows from 24) ad / 3/2 + /2 For he secod iequaliy, oe ha f) < < 3/2 + e < ) ) for all 5 For = 2, 3 ad all 4, oe ca verify f) < 3/4) ) easily Also, for = 4 ad all 5, we have f4) < 4/5) 4 )4 Acowledgeme We ha wo referees for heir valuable commes Refereces [] N Alo, P Fral, H Huag, V Rödl, A Rucińsi, ad B Sudaov Large machigs i uiform hypergraphs ad he cojecure of Erdős ad Samuels J Combi Theory Ser A, 96):200 25, 202 [2] E Buß, H Hà, ad M Schach Miimum verex degree codiios for loose Hamilo cycles i 3-uiform hypergraphs J Combi Theory Ser B, 036): , 203 [3] A Czygriow ad T Molla Tigh codegree codiio for he exisece of loose Hamilo cycles i 3-graphs SIAM J Discree Mah, 28):67 76, 204 [4] G A Dirac Some heorems o absrac graphs Proc Lodo Mah Soc 3), 2:69 8, 952 [5] R Glebov, Y Perso, ad W Weps O exremal hypergraphs for Hamiloia cycles Europea J Combi, 334): , 202 [6] H Hà, Y Perso, ad M Schach O perfec machigs i uiform hypergraphs wih large miimum verex degree SIAM J Discree Mah, 23: , 2009

6 6 JIE HAN AND YI ZHAO [7] H Hà ad M Schach Dirac-ype resuls for loose Hamilo cycles i uiform hypergraphs Joural of Combiaorial Theory Series B, 00: , 200 [8] J Ha ad Y Zhao Miimum degree codiios for Hamilo /-cycles i -uiform hypergraphs mauscrip [9] J Ha ad Y Zhao Miimum codegree hreshold for Hamilo l-cycles i -uiform hypergraphs Joural of Combiaorial Theory, Series A, 32:94 223, 205 [0] J Ha ad Y Zhao Miimum verex degree hreshold for loose hamilo cycles i 3-uiform hypergraphs Joural of Combiaorial Theory, Series B, 4:70 96, 205 [] G Kaoa ad H Kiersead Hamiloia chais i hypergraphs Joural of Graph Theory, 30:205 22, 999 [2] P Keevash, D Küh, R Mycrof, ad D Oshus Loose Hamilo cycles i hypergraphs Discree Mahemaics, 37): , 20 [3] D Küh, R Mycrof, ad D Oshus Hamilo l-cycles i uiform hypergraphs Joural of Combiaorial Theory Series A, 77):90 927, 200 [4] D Küh ad D Oshus Loose Hamilo cycles i 3-uiform hypergraphs of high miimum degree Joural of Combiaorial Theory Series B, 966):767 82, 2006 [5] D Küh ad D Oshus Embeddig large subgraphs io dese graphs I Surveys i combiaorics 2009, volume 365 of Lodo Mah Soc Lecure Noe Ser, pages Cambridge Uiv Press, Cambridge, 2009 [6] D Küh ad D Oshus Hamilo cycles i graphs ad hypergraphs: a exremal perspecive Proceedigs of he Ieraioal Cogress of Mahemaicias 204, Seoul, Korea, Vol 4:38 406, 204 [7] O Pihuro Perfec machigs ad K4 3 -iligs i hypergraphs of large codegree Graphs Combi, 244):39 404, 2008 [8] V Rödl ad A Rucińsi Dirac-ype quesios for hypergraphs a survey or more problems for edre o solve) A Irregular Mid, Bolyai Soc Mah Sudies 2:56 590, 200 [9] V Rödl ad A Rucińsi Families of riples wih high miimum degree are Hamiloia Discuss Mah Graph Theory, 34:36 38, 204 [20] V Rödl, A Rucińsi, ad E Szemerédi A Dirac-ype heorem for 3-uiform hypergraphs Combiaorics, Probabiliy ad Compuig, 5-:229 25, 2006 [2] V Rödl, A Rucińsi, ad E Szemerédi Perfec machigs i uiform hypergraphs wih large miimum degree Europea J Combi, 278): , 2006 [22] V Rödl, A Rucińsi, ad E Szemerédi A approximae Dirac-ype heorem for -uiform hypergraphs Combiaorica, 28: , 2008 [23] V Rödl, A Rucińsi, ad E Szemerédi Perfec machigs i large uiform hypergraphs wih large miimum collecive degree J Combi Theory Ser A, 63):63 636, 2009 [24] V Rödl, A Rucińsi, ad E Szemerédi Dirac-ype codiios for Hamiloia pahs ad cycles i 3-uiform hypergraphs Advaces i Mahemaics, 2273): , 20 [25] A Treglow ad Y Zhao Exac miimum degree hresholds for perfec machigs i uiform hypergraphs II J Combi Theory Ser A, 207): , 203 [26] Y Zhao Rece advaces o dirac-ype problems for hypergraphs I Rece Treds i Combiaorics, volume 59 of he IMA Volumes i Mahemaics ad is Applicaios Spriger, New Yor, 206 Isiuo de Maemáica e Esaísica, Uiversidade de São Paulo, Rua do Maão 00, , São Paulo, Brazil address, Jie Ha: jha@imeuspbr Deparme of Mahemaics ad Saisics, Georgia Sae Uiversiy, Alaa, GA 30303, USA address, Yi Zhao: yzhao6@gsuedu