Multicolourd Hamilton cycls and prfct matchings in psudo-random graphs Danila Kühn Dryk Osthus Abstract Givn 0 < p < 1, w prov that a psudo-random graph G with dg dnsity p and sufficintly larg ordr has th following proprty: Considr any rd/blu-colouring of th dgs of G and lt r dnot th proportion of dgs which hav colour rd. Thn thr is a Hamilton cycl C so that th proportion of rd dgs of C is clos to r. Th analogu also holds for prfct matchings instad of Hamilton cycls. W also prov a bipartit vrsion which is usd lswhr to giv a minimum-dgr condition for th xistnc of a Hamilton cycl in a 3-uniform hyprgraph. 1 Introduction 1.1 Ovrviw It is wll known that random graphs, psudo-random graphs and ε-suprrgular graphs hav som strong Hamiltonicity proprtis in common. For instanc, a rcnt rsult of Friz and Krivlvich [10] stats that, for vry constant 0 < p < 1, with high probability almost all dgs of th random graph G n,p can b packd into dg-disjoint Hamilton cycls. (Thy driv this from a similar rsult about ε-suprrgular graphs.) Hamiltonicity has also bn invstigatd from th viwpoint of (anti-)ramsy thory. For xampl, Albrt, Friz and Rd [1] provd that thr is a linar function k = k(n) such that for vry dg-colouring of th complt graph K n on n vrtics which uss ach colour at most k tims thr is a Hamilton cycl whr ach dg has a diffrnt colour. This improvs bounds by prvious authors. A rlatd problm for random graphs was also considrd by Coopr and Friz [6]. Hr, w prov a rlatd rsult about colourings of bipartit ε-suprrgular graphs (which will imply analogous statmnts for psudo-random and random graphs). Roughly spaking, w prov that givn a k-colouring of a sufficintly larg ε-suprrgular graph G (whr ε is sufficintly small) thr is a Hamilton cycl C in G which is strongly multicolourd (or wll balancd) in th following sns: for all colours i, th proportion of dgs in C of colour i is clos to th proportion of dgs in G which hav colour i. W driv this from a rlatd rsult about random prfct matchings (Thorm 1.1) which is also a crucial tool in [12], s Sction 1.3. This papr is organizd as follows. In Sctions 2 and 3.1 w collct som tools which w will nd in our proofs. In Sction 3.2 w thn us ths tools to 1
dduc som simpl proprtis of random prfct matchings in ε-suprrgular graphs. Th cor rsult of this papr is Lmma 3.8 in Sction 3.3, which provs Thorm 1.1 for spcial graphs H. In th final sction, th rmaining rsults in this papr ar asily dducd from Lmma 3.8 and th rsults in Sction 3.2. 1.2 Statmnt of rsults Givn a bipartit graph G = (A, B) with vrtx classs A and B, w dnot th dg st of G by E(A, B) and lt (G) = (A, B) = E(A, B). Th dnsity of a bipartit graph G = (A, B) is dfind to b d(a, B) := (A, B) A B. Givn 0 < ε < 1 and d [0, 1], w say that G is (d, ε)-rgular if for all sts X A and Y B with X ε A and Y ε B w hav (1 ε)d < d(x, Y ) < (1 + ε)d. W say that G is (d, ε)-suprrgular if it is (d, ε)-rgular and, furthrmor, if (1 ε)d B < d G (a) < (1+ε)d B for all vrtics a A and (1 ε)d A < d G (b) < (1 + ε)d A for all b B. This is mor or lss quivalnt to th traditional notions of ε-rgularity and ε-suprrgularity s Sction 2. Thorm 1.1 For all positiv constants d, ν 0, η 1 thr is a positiv ε = ε(d, ν 0, η) and an intgr N 0 = N 0 (d, ν 0, η) such that th following holds for all n N 0 and all ν ν 0. Lt G = (A, B) b a (d, ε)-suprrgular bipartit graph whos vrtx classs both hav siz n and lt H b a subgraph of G with (H) = ν(g). Choos a prfct matching M uniformly at random in G. Thn with probability at last 1 εn w hav (1 η)νn M E(H) (1 + η)νn. At first sight it may sm surprising that th only paramtr of H that is rlvant hr is th numbr of its dgs. Howvr, this is quit natural in viw of th fact that th assrtion would b trivial if instad of a prfct matching on would choos n dgs indpndntly and uniformly at random. Th cas whn H is a sufficintly larg inducd subgraph of G was provd arlir by Rödl and Ruciński [13] as a tool in thir altrnativ proof of th Blow-up Lmma of Komlós, Sárközy and Szmrédi. From Thorm 1.1 w will also dduc a (wakr) analogu for Hamilton cycls: Thorm 1.2 For all intgrs k and all positiv constants d, ν, η 1 thr is a positiv ε = ε(d, ν, η) and an intgr N 1 = N 1 (k, d, ν, η) such that th following holds for all n N 1. Lt G = (A, B) b a (d, ε)-suprrgular bipartit graph whos vrtx classs both hav siz n. For ach 1 i k lt H i b a subgraph of G with (H i ) = ν i (G), whr ν i ν. Thn G contains a Hamilton cycl C such that for all 1 i k (1 η)2ν i n C E(H i ) (1 + η)2ν i n. 2
Thorms 1.1 and 1.2 can in turn b usd to dduc analogus for non-bipartit graphs (s th final sction for dtails). For this, w nd to modify th notion of (d, ε)-suprrgularity as follows. Givn 0 < ε < 1 and d [0, 1], w say that a graph G with n vrtics is (d, ε)-rgular if for all disjoint sts X, Y V (G) with X, Y εn w hav (1 ε)d < d(x, Y ) < (1 + ε)d. W say that G is (d, ε)-suprrgular if it is (d, ε)-rgular and, furthrmor, if (1 ε)dn < d G (x) < (1 + ε)dn for all vrtics x of G. Thorm 1.3 For all intgrs k and all positiv constants d, ν, η 1 thr is a positiv ε = ε(d, ν, η) and an intgr N 2 = N 2 (k, d, ν, η) such that th following holds for all n N 2. Lt G b a (d, ε)-rgular graph with n vrtics. For ach 1 i k, lt H i b a subgraph of G with (H) = ν i (G), whr ν i ν for all i k. Thn (i) G contains a Hamilton cycl C such that for all i (1 η)ν i n C E(H i ) (1 + η)ν i n; (ii) if n is vn thn G contains a prfct matching M such that for all i (1 η)ν i n/2 M E(H i ) (1 + η)ν i n/2. Not that th assrtion is not vn trivial (but much asir to prov) in th spcial cas whr G is th complt graph K n. Morovr, Lt G n,p b a random graph on n vrtics obtaind by conncting ach pair of vrtics with probability p (indpndntly of all th othr pairs). For givn 0 < p < 1 and n sufficintly larg, G n,p is (p, ε)-suprrgular with high probability (in fact th probability that this is not th cas is asily sn to dcras xponntially in n). Thus th assrtion of Thorm 1.3 holds with high probability in this cas. Also, if G is dn-rgular and th scond ignvalu of th adjacncy matrix is at most λdn for sufficintly small λ, thn G is (d, ε)-suprrgular (s.g. Chung [7], Thorm 5.1) so th rsult applis in this cas, too (such graphs ar oftn calld psudo-random graphs). 1.3 Application: Loos Hamilton cycls in 3-uniform hyprgraphs A fundamntal thorm of Dirac stats that vry graph on n vrtics with minimum dgr at last n/2 contains a Hamilton cycl. In [12], w prov an analogu of this for 3-uniform hyprgraphs, which w dscrib blow. All th rsults provd in this papr xcpt Thorms 1.2 and 1.3 and Lmma 3.8 ar usd as a tool in [12]. On way to xtnd th notion of th minimum dgr of a graph to that of a 3-uniform hyprgraph H is as follows. Givn two distinct vrtics x and y of H, th nighbourhood N(x, y) of (x, y) in H is th st of all thos vrtics z which form a hyprdg togthr with x and y. Th minimum dgr δ(h) is dfind to b th minimum N(x, y) ovr all pairs of vrtics of H. W say that a 3-uniform hyprgraph C is a cycl of ordr n if thr xists a cyclic ordring v 1,..., v n of its vrtics such that vry conscutiv pair v i v i+1 lis in a hyprdg of C and such that vry hyprdg of C consists of 3 conscutiv vrtics. A cycl is tight if vry thr conscutiv vrtics form a 3
hyprdg. A cycl of ordr n is loos if it has th minimum possibl numbr of hyprdgs among all cycls on n vrtics. A Hamilton cycl of a 3-uniform hyprgraph H is a subhyprgraph of H which is a cycl containing all its vrtics. Th following rsult is provd in [12]. Thorm 1.4 For ach ε > 0 thr is an n 0 = n 0 (ε) such that vry 3-uniform hyprgraph H with n n 0 vrtics and minimum dgr at last n/4 + εn contains a loos Hamilton cycl. Th bound on th minimum dgr is ssntially bst possibl in th sns that thr ar hyrgraphs with minimum dgr n/4 1 which do not vn contain som (not ncssarily loos) Hamilton cycl. Rcntly, Rödl, Ruciński and Szmrédi [14] provd that if th minimum dgr is at last n/2 + εn and n is sufficintly larg, thn on can vn guarant a tight Hamilton cycl. This is also bst possibl up to th rror trm (thy announcd in [14] that th rror trm εn can in fact b omittd). 2 Notation and a probabilistic stimat Givn a graph G, w writ N G (x) for th nighbourhood of a vrtx x in G and lt d G (x) := N G (x). Givn ε > 0, w say that G is ε-rgular if for all sts X A and Y B with X ε A and Y ε B w hav d(a, B) d(x, Y ) < ε. This (mor traditional) notion of rgularity is mor or lss quivalnt to th on dfind in th introduction. Indd, clarly vry (d, ε)-rgular graph is also 2εd-rgular (and thus 2ε-rgular). Convrsly, if d = d(a, B) ε thn vry ε-rgular bipartit graph (A, B) is (d, ε)-rgular. Givn a positiv numbr ε and sts A, Q T, w say that A is split ε-fairly by Q if A Q A Q T ε. Thus, if ε is small and A is split ε-fairly by Q, thn th proportion of all thos lmnts of T which li in A is almost qual to th proportion of all thos lmnts of Q which li in A. W will us th following vrsion of th wllknown fact that if Q is random thn it tnds to split larg sts ε-fairly. It is an asy consqunc of standard larg dviation bounds for th hyprgomtric distribution, s.g. [12] for a proof. Proposition 2.1 For ach 0 < ε < 1 thr xists an intgr q 0 = q 0 (ε) such that th following holds. Givn t q q 0 and a st T of siz t, lt Q b a subst of T which is obtaind by succssivly slcting q lmnts uniformly at random without rptitions. Lt A b a family of at most q 10 substs of T such that A εt for ach A A. Thn with probability at last 1/2 vry st in A is split ε-fairly by Q. 3 Prfct matchings in suprrgular graphs In this sction, w collct and prov svral rsults about (random) prfct matchings in bipartit suprrgular graphs G which w will all nd to prov 4
Thorms 1.1 and 1.2. Morovr, Lmmas 3.6 and 3.7 will also b usd in [12]. Th main rsult of this sction is Lmma 3.8. Givn a rasonably rgular small subgraph H of G, it givs prcis bounds on th likly numbr of all thos dgs of H that ar containd in a random prfct matching M of G. This is provd in th third subsction. In th first subsction, w collct som tools which w will nd in th othr two subsctions. In th scond subsction, w giv likly uppr bounds on th numbr of all thos dgs of an arbitrary spars subgraph H of G that ar containd in a random prfct matching and on th numbr of cycls in th union of two random prfct matchings in G. 3.1 Known rsults on counting prfct matchings W us th following vrsion of Stirling s inquality (th bound is a wak form of a rsult of Robbins, s.g. [4]): Proposition 3.1 For all intgrs n 1 w hav ( n ) n ( n ) n n! 3 n. (1) W will frquntly us th following immdiat consqunc of th lowr bound in Stirling s inquality: ( ) n ( n ) k. (2) k k W will also us that 1 x x x2 for all 0 < x < 0.45 (3) (s.g. [4, Sction 1.1]). W also nd th following rsult of Brégman [5] which sttls a conjctur of Minc on th prmannt of a 0-1 matrix. (A short proof of it was givn by Schrijvr [15], s also [3]). W stat this rsult in trms of an uppr bound on th numbr of prfct matchings of a bipartit graph. Thorm 3.2 Th numbr of prfct matchings in a bipartit graph G = (A, B) is at most (d G (a)!) 1/dG(a). a A An application of Stirling s inquality (Proposition 3.1) to Thorm 3.2 immdiatly yilds th following. Corollary 3.3 For all ε > 0 thr is an intgr d = d 0 (ε) so that th following holds: Lt G = (A, B) b a bipartit graph with A = B = n and lt m(g) dnot th numbr of prfct matchings in G. Thn m(g) (1 + ε) n a A max{d G (a), d 0 }. 5
A vry usful lowr bound on th numbr of prfct matchings in a k-rgular bipartit graph is providd by th following rsult by Egorichv [8] and Falikman [9], which was formrly known as th van dr Wardn conjctur. Thorm 3.4 Lt G b a k-rgular bipartit graph whos vrtx classs hav siz n. Thn th numbr of prfct matchings in G is at last (k/n) n n!. To bound th numbr of prfct matchings in suprrgular graphs, w will us th following thorm of Alon, Rödl and Ruciński [2]. (Actually, w will only apply th lowr bound, which is basd on Thorm 3.4. Th uppr bound in Thorm 3.5 is an asy consqunc of Corollary 3.3.) Not that thir rsult is statd slightly diffrntly in [2] as th dfinition of (d, ε)-suprrgularity in [2] is slightly diffrnt. Thorm 3.5 For vry 0 < ε < 1/4 thr xists an intgr n 1 = n 1 (ε) such that whnvr d > 0 and G is a (d, ε)-suprrgular bipartit graph whos vrtx classs both hav siz n n 1, thn th numbr m(g) of prfct matchings in G satisfis (d(1 4ε)) n n! m(g) (d(1 + 4ε)) n n!. 3.2 Simpl proprtis of random prfct matchings Basd on th rsults in Sction 3.1, w can asily dduc th nxt lmma, which implis that if w ar givn a (supr-)rgular graph G and a bad subgraph F of G which is comparativly spars, thn a random prfct matching of G will probably only contain a fw bad dgs. Th morovr part will only b usd in [12], th assrtion about (d, ε)-rgular graphs will b usd in [12] and th proof of Thorm 1.1. Lmma 3.6 For all positiv constants ε and d with d 1 and ε 1/6 thr xists an intgr n 0 = n 0 (ε, d) such that th following holds. Lt G b a (d, ε)- suprrgular graph whos vrtx classs A and B satisfy A = B =: n n 0. Lt M b a prfct matching chosn uniformly at random from th st of all prfct matchings of G. Lt F b a subgraph of G such that all but at most n vrtics in F hav dgr most dn, whr 1/2 18ε. Thn th probability that M contains at last 9 n dgs of F is at most 2εn. Morovr, th statmnt also holds if w assum that G is dn-rgular, whr dn N. Proof. First suppos that F has maximum dgr at most dn. Lt F F b a subgraph of G such that d F (a) = dn for ach vrtx a A. (Such an F xists sinc d G (a) (1 ε)dn dn as G is (d, ε)-suprrgular.) Givn a st A A, w dnot by F A th bipartit graph with vrtx classs A and B in which vry vrtx a A is joind to all th vrtics b N F (a) whil vry vrtx a A \ A is joind to all th vrtics b N G (a) \ N F (a). For an intgr q 2 n, lt m(q) dnot th numbr of prfct matchings in G which contain prcisly q dgs from F. Evry such matching M can b obtaind by first fixing a q-lmnt st A A and thn choosing a prfct matching in th 6
graph F A. (So th lmnts of A corrspond to th q ndvrtics of th dgs in M E(F ).) If w apply Corollary 3.3 to F A w now obtain ( ) ( n m(q) (1 + ε) n ) dn q ( ) (1 + ε)dn n q q ( ) (2) n q ( ) dn n ( ) q (1 + ε) 2n. q Lt m(g) dnot th numbr of prfct matchings in G. Thn th lowr bound in Thorm 3.5 implis that ( ) m(g) (d(1 4ε)) n n! (1) (1 4ε)dn n. Thus th probability m(q)/m(g) that M contains xactly q dgs from F is at most ( ) n q (1 + 5ε) 3n q (1 + 5ε) 3n (15ε )n 3εn. q (To s th first inquality, us that q 2 n.) By summing this bound ovr all q 2 n, w find that th probability that M contains at last 2 n dgs of F is at most n 3εn 2εn. Sinc F F, this implis that with probability at most 2εn th matching M contains at last 2 n dgs of F. If F is now allowd to hav up to n vrtics whos dgr is largr than dn, this can incras th numbr of dgs of F in M by at most n, which implis th rsult. Th sam proof also works in th cas wr G is dn-rgular. W now us th lowr bound m(g) d n n! (dn/) n which follows from Thorm 3.4 and inquality (1). In th following lmma w will us Thorms 3.4 and 3.5 to show that a randomly chosn 2-factor in a (supr-)rgular graph G will typically only contain fw cycls. W will nd this fact in th proof of Thorm 1.2 (and in [12] again, as mntiond arlir). A similar obsrvation was also usd in Friz and Krivlvich [10]. Th morovr part will only b usd in [12]. Lmma 3.7 For all positiv constants ε < 1/64 and d 1 thr xists an intgr n 0 = n 0 (ε, d) such that th following holds. Lt G b a (d, ε)-suprrgular graph whos vrtx classs A and B satisfy A = B =: n n 0. Lt M 1 b any prfct matching in G. Lt M 2 b a prfct matching chosn uniformly at random from th st of all prfct matchings in G M 1. Lt R = M 1 M 2 b th rsulting 2-factor. Thn th probability that R contains mor than n/(log n) 1/5 cycls is at most n. Morovr, th statmnt also holds if w assum that G is dn-rgular, whr dn N, and that G and M 1 ar disjoint. Proof. Lt G := G M 1. Lt m(g ) dnot th numbr of prfct matchings in G. Sinc th dltion of a prfct matching from G still lavs a (d, 2ε)- suprrgular graph, Thorm 3.5 implis that m(g ) ((1 8ε)d) n n! (1),(3) 9εn ( dn ) n. 7
Lt k := n/(log n) 1/2 and l := (log n) 1/4. Givn an intgr l l, lt f k,l dnot th numbr of 2-factors of G which contain M 1 and hav at last k cycls of lngth 2l. W will now find an uppr bound on f k,l. For this, not that th numbr of possibilitis for choosing a st C k,l of k disjoint cycls of lngth 2l in G whr vry scond dg is containd in M 1 is at most 1 (1) ( k! nlk k nl) k =: ck,l. (Indd, ach such cycl of lngth 2l is dtrmind by an ordrd choic of l dgs in M 1.) By Corollary 3.3, givn som C k,l as abov, th numbr of matchings on th rmaining vrtics of G M 1 is at most ( ) dn n kl ( ) dn n kl (1 + ε) 2n 2εn =: d k,l. Hnc w hav that f k,l c k,l d k,l. Altogthr, this implis that th probability f k,l /m(g ) that a random 2-factor R (chosn as in th statmnt of th lmma) contains at last k cycls of lngth 2l can b boundd as follows. f ( k,l m(g ) 11εn k nl) k ( ) ( ) kl = 11εn l+1 k dn kd l 11εn k k/2 2n. To driv th third inquality, w usd th fact that (/d) l (and thus (/d) l ) is small compard to k. For th final on, w usd that k log k is larg compard to n. Hnc th probability that thr is an l l such that th random 2-factor R contains at last k cycls of lngth 2l is at most l 2n n. Not that th numbr of cycls of lngth at last 2l in R is at most 2n/(2l ). Thus with probability at last 1 n th numbr of cycls in R is at most kl + n/l = 2n/(log n) 1/4, which implis th first part of th lmma. Th proof of th morovr part of Lmma 3.7 is almost th sam, xcpt that w us th lowr bound m(g) (dn/) n on th numbr of prfct matchings in G which follows from Thorm 3.4 by an application of (1). 3.3 Counting prfct matchings which contain a givn numbr of dgs of an almost rgular subgraph Lmma 3.8 For ach positiv constant β 1 thr is a constant f(β) with 0 < f(β) 1 such that th following holds. Suppos that α, ε, ξ, c and d ar positiv constants with ε α, c, d 1 and α, c ξ f(β) 1. Thr xists an intgr n 0 = n 0 (α, ε, ξ, c, d) for which th following is tru. Lt G b a bipartit (d, ε)-suprrgular graph whos vrtx classs V and W satisfy V = W =: n n 0. Lt H b a subgraph of G with vrtx classs C V and D W whr c n C = cn 2c n and αdn d H (v) (1 + ξ)αdn for all vrtics v C. Lt M b a prfct matching chosn uniformly at random from th st of all prfct matchings in G. Thn 8
(i) P ( M E(H) βαcn) f(β)αcn if β < 1, (ii) P ( M E(H) βαcn) f(β)αcn if β > 1. Th intuition bhind this rsult is th following (s also th rmark aftr Thorm 1.1): If th inclusion of th dgs of G into th random prfct matching M would b mutually indpndnt and qually likly, thn th probability that a givn dg is containd in M would b clos to M /(G). Thus th xpctd valu of M E(H) would b clos to n(h)/(g) which in turn is clos to n(αdn)(cn)/(dn 2 ) = αcn. Th abov rsult would thus immdiatly follow by an application of som larg dviation bound on th tail of th binomial distribution. Th basic stratgy of th proof is similar to that of [13], whr th authors assum that H is a sufficintly larg inducd subgraph of G. Th main difficulty of our proof is du to th fact that H is assumd to b rathr small compard to G. Proof. Lt m(g) dnot th total numbr of prfct matchings in G. If w apply Stirling s formula (1) to th lowr bound in Thorm 3.5, w obtain ( ) (1 4ε)dn n (3) ( ) dn n m(g) 5εn. (4) Givn a cn, lt m(a) b th numbr of prfct matchings in G which mt E(H) in prcisly a dgs. Our aim is to show that m(a) is much smallr than m(g) if a is significantly smallr or largr than αcn. Lt J dnot th summation ovr all matchings J in H of cardinality a. Givn such a matching J, lt m(j) dnot th numbr of prfct matchings M in G(J) := G V (J) E(H). Thus M togthr with J forms a prfct matching of G which intrscts H in xactly a dgs and so m(a) = J m(j). W claim that for all matchings J as abov, w hav ( ) dn n a m(j) αcn a ξa+5εn. (5) Th first trm is th roughly th bound w would gt if w would just us th fact that G(J) has maximum dgr (1 + ε)dn. Th scond trm is a small but crucial improvmnt on this stimat. Th third trm is an insignificant rror trm. W now prov (5). By Corollary 3.3, w hav m(j) (1 + ε) n a v V \V (J) max{d G(J) (v), d 0 (ε)}, (6) whr d 0 (ε) is th intgr dfind in Corollary 3.3. Thus w hav rducd th problm of bounding m(j) to that of finding accurat uppr bounds on th dgrs of th vrtics in G(J). Rcall that th vrtx classs of H ar C and D and that (G) (1 + ε)dn sinc G is (d, ε)-suprrgular. For a vrtx v C \ V (J) w hav d G(J) (v) dn(1 + ε α) =: q H. 9
W say that a vrtx v V \ V (H) is avrag for J if in th graph G it has at last (1 ε)d(a εn) nighbours in W V (J). Lt V av b th st of such vrtics. For v V av, w hav d G(J) (v) dn(1 + ε (1 ε)(a/n ε)) =: q J. Sinc G is (d, ε)-suprrgular, w hav that V av n cn εn if a εn. If a εn, thn trivially vry vrtx in v V \V (H) is avrag for J, so th abov bound on V av holds in this cas, too. Morovr, not that both q H d 0 (ε) and q J d 0 (ε) sinc n is sufficintly larg compard to ε. Thus, insrting all ths bounds into (6) givs m(j) (1 + ε) n a n (q H ) C\V (J) (q J ) V av ((1 + 2ε)dn) n a C\V (J) V av. Now not that q J (1 + 2ε)dn to dduc that th right hand sid is maximizd if V av is minimizd. Thus m(j) εn a n (q H ) cn a (q J ) (1 c ε)n ((1 + 2ε)dn) εn ( ) dn n a xp Q, (7) whr Q := εn + Q H + Q J + 2ε(εn) and Q H :=(ε α)(cn a), Q J :=[ε (1 ε)(a/n ε)][(1 c ε)n]. Not that, w mad us of th fact that 1 + x x thr tims in ordr to obtain (7). Now obsrv that Q H αcn + αa + εn, Q J εn a(1 c ε)(1 ε) + εn a(1 2c) + 2εn. Altogthr, w thus hav Q εn αcn + αa + εn a + 2ac + 2εn + εn αcn a + ξa + 5εn, which provs (5). Lt p a dnot th probability that a prfct matching which is chosn uniformly at random in th st of all prfct matchings in G contains xactly a dgs of H. Thus p a = m(a)/m(g) = J m(j)/m(g). Lt J dnot th numbr of summands, i.. th numbr of matchings in H of cardinality a. Each matching of cardinality a in H can b obtaind by first choosing a subst of a vrtics in C and thn choosing on nighbour in H for ach vrtx in this subst. Thus, writing (x/0) 0 := 1 for all x > 0, it follows that J ( ) cn ((1 + ξ)αdn) a (2) a ( 1+ξ αcdn 2 ) a. (8) a 10
Sinc th bound (5) on m(j) is indpndnt of J, w can now combin (4) and (5) to obtain p a = m(j) m(g) ( ) a 5εn αcn a ξa+5εn dn J J (8) ( αcn ) a αcn 2ξa+10εn. a Now dfin β by a = β αcn and lt g(β ) := log{(/β ) β /}. Thn p a ( ( ) ) β αcn β 1 2ξa+10εn xp { αcn(g(β ) + 2ξβ + ξ) }. Now st µ := αcn to obtain p a xp{µ(g(β ) + ξ(1 + 2β ))}. (Not that if ξ = 0 and β < 1, this would b xactly th standard Chrnoff bound on th probability that X β µ, whr X has a binomial distribution with man µ, s.g. Thorm A.12 in [3].) It is asy to chck that g(β ) < 0 if β 1. Th assrtion (i) (i.. th cas β < 1) of th lmma now follows with f(β) := g(β)/4 by summing ovr all valus of a btwn 1 and βµ. Indd, as g(β ) is ngativ and incrasing for β < 1, w hav P ( M E(H) βαcn) βµ xp{µg(β) + 3ξ} βµ xp{µg(β)/2}, as rquird. To prov th assrtion (ii) of th lmma, w first considr th cas 1 < β β 2. As g(β ) is ngativ and dcrasing for β > 1, it follows that p a xp{µ(g(β) + 17ξ)} xp{µg(β)/2}. Nxt considr th cas that β 2. It is asy to chck that g(β ) β. Thus p a xp{µ( β + ξ(1 + 2β ))} xp{ µβ /2}. Similarly to th cas (i), th assrtion of th lmma in cas (ii) now follows by summing th bounds on p a ovr all valus of a btwn βµ and cn. 4 Proof of Thorms 1.1 1.3 W will prov Thorm 1.1 by dcomposing H into small almost rgular subgraphs H ij and a small rmaindr F. W will apply Lmma 3.8 to ach of th H ij sparatly and thn us Lmma 3.6 to show that a random prfct matching contains only a ngligibl numbr of dgs of F. Proof of Thorm 1.1. By adding all th vrtics in V (G) \ V (H) to H, w may assum that H is a spanning subgraph of G. St β := 1 + η/4, dfin 11
f(β) as in th statmnt of Lmma 3.8 and choos paramtrs α, ε, ξ, c so that 0 < ε α, c, d 1 and c α ξ ν, η, f(β). Thus th rstrictions in th statmnt of Lmma 3.8 ar satisfid. Choos N 0 to b sufficintly larg compard to both 1/ε and th intgr n 0 (α, ε, ξ, c, d) dfind in Lmma 3.8. Finally, fix a constant c such that cn N and c c 2c. First, w prov th uppr bound in Thorm 1.1. Lt l b th smallst intgr so that ξl/2 α > 1 + ε. Thus l 2 ξ log(2/α) 1/ c. (9) Lt A 0 b th st of vrtics in A with d H (a) < αdn. For all i 1, lt α i := ξ(i 1)/2 α. Thus α i+1 (1 + ξ)α i (10) sinc ξ/2 1 + ξ (s.g. [4, Sction 1.1]). Morovr, 1 + ε < α l+1 2. (11) For all i with 1 i l, lt A i b th st of vrtics in a A with α i dn d H (a) < α i+1 dn. Sinc G is (d, ε)-suprrgular and thus d H (a) d G (a) (1 + ε)dn for ach a A, it follows that th A i with 0 i l giv a partition of A. W now dfin a partition of th dg st of H into graphs H ij. Givn 1 i l, dfin q i by A i = q i cn and lt q(i) := q i. W partition th vrtics in A i into q(i) + 1 parts A ij with 0 j q(i) as follows: th partition is arbitrary xcpt that w rquir that A ij = cn for all j 1. Thus A i0 < cn and so A i0 lcn cn αn. (12) Lt H ij b th subgraph of H inducd by A ij and B. Thn for all a A ij, w hav α i dn d Hij (a) < α i+1 dn (10) (1 + ξ)α i dn. (13) Lt H 00 b th subgraph of H which is inducd by A 0 and B. Givn 1 i l, lt H i0 b th subgraph of H which is inducd by A i0 and B. Lt F dnot th union of all th H i0 with 0 i l. Thn (F ) αdn A 0 + A i0 α i+1 dn (11),(12) αdn 2 + 2αdn 2 4α(G) η(h)/4. (14) Lt M b a prfct matching chosn uniformly at random from th st of all prfct matchings in G. Lt X ij := M E(H ij ) and µ i := α i cn. (Not that µ i can b thought of as roughly th xpctd valu of X ij ). Thn for all i, j with i, j 1 w can apply Lmma 3.8(ii) to H ij to s that with probability at last 1 f(β)µ i w hav X ij βµ i (apply th lmma with α i taking on th rol of th paramtr α thr). Morovr, w can apply Lmma 3.6 to F as follows: Lt := α. Thn (12) implis that at most n vrtics of F hav 12
dgr mor than dn. Thus Lmma 3.6 implis that with probability at last 1 2εn w hav M E(F ) 9αn ηνn/2. But F and th sts E(H ij ) with i, j 1 form a partition of E(H) and so with probability at last 1 2εn l q(i) f(β)µ i 1 εn w hav M E(H) ηνn/2 + β q(i)µ i ηνn/2 + β A i α i. Now us th fact that l A i α i dn (H) (1 + ε)νdn 2 to s that M E(H) ηνn/2 + β(1 + ε)νn (1 + η)νn, as rquird. Th proof of th lowr bound is almost xactly th sam: in this cas, w lt β = 1 η/4. Th graphs H ij ar dfind as bfor. W now apply Lmma 3.8(i) to H ij to s that with probability at last 1 l q(i) f(β)µ i 1 εn w hav X ij βµ i for all i, j with i 1. Thus with probability at last 1 εn, w hav But M E(H) β (9) β q(i)µ i β (11) ( A i cn)α i β A i α i 4 cn β A i α i dn (10) (1 2ξ) A i α i 2βlcn A i α i ηνdn/2. (15) A i dnα i+1 (1 2ξ)((H) (F )) (14) (1 2ξ)(1 η/4)(h) (1 η/2)νdn 2, which implis th rsult togthr with (15). W can now asily dduc Thorm 1.2 from Thorm 1.1 and Lmma 3.7. Proof of Thorm 1.2. Put ε := min{1/64, d/5, ε(d, ν, η/2)/2} whr ε(d, ν, η/2) is as dfind in Thorm 1.1. Lt N 1 b sufficintly larg compard to 1/η, 1/ν and k as wll as largr than n 0 (ε, d) and N 0 (d, ν, η/2) dfind in Lmma 3.7 and Thorm 1.1 rspctivly. Choos a prfct matching M 1 uniformly at random in G and thn choos a prfct matching M 2 uniformly at random in G M 1. Lmma 3.7 implis that with probability at last 1 n th rsulting 2-factor R = M 1 M 2 contains at most (n/ log n) 1/5 cycls. Morovr, Thorm 1.1 implis that w may assum that (1 η/2)2ν i n R E(H i ) (1 + η/2)2ν i n (16) for all i k. Thus it suffics to prov that thr is a Hamilton cycl C in G which has sufficintly many dgs in common with R. This is achivd using a standard argumnt basd on xpansion proprtis of G. 13
Lt C b any cycl in R with th proprty that thr ar adjacnt vrtics x and y on C such that x has a nighbour z outsid C. (Using that G is (d, ε)-suprrgular, it is asy to s that such a cycl always xists unlss R is alrady a Hamilton cycl. Indd, sinc δ(g) (1 ε)dn, ach cycl in R of lngth at most dn will hav a nighbour outsid and thus can b takn to b C. On th othr hand, N G (X) (1 ε)n for any st X of siz at last dn/2 εn which lis in on of th vrtx classs of G. This implis that if all th cycls in R hav lngth at last dn and R is not a Hamilton cycl thn w can tak for C any cycl of R.) Lt C dnot th cycl in R which contains z. Lt P dnot th path obtaind from C C by adding th dg xz and dlting xy as wll as on of th dgs on C adjacnt to z. Not that th lngth of P is odd. If on of th ndpoints of P has a nighbour outsid P, w can furthr nlarg P in a similar way. So suppos w can no longr nlarg P in this way and viw P as a dirctd path whos first vrtx is dnotd by x and whos final vrtx is dnotd by y. Thus all th nighbours of x and y li on P. Morovr, sinc P is odd, x and y li in diffrnt vrtx classs of G. W claim that thr is a cycl C which has th sam vrtx st as P. Lt X 1 b th st consisting of th first d G (x)/2 nighbours of x on P and lt X 2 consist of all othr nighbours. Dfin Y 1 and Y 2 similarly. It is asily sn that ithr (i) all vrtics in Y 1 com bfor all thos in X 2 or (ii) all vrtics in X 1 com bfor thos in Y 2. Suppos first that (i) holds. Not that X i, Y j δ(g)/4 (1 ε)dn/4 εn and so th (d, ε)-suprrgularity of G implis that thr is an dg E(G) btwn a prdcssor p of som vrtx y 1 Y 1 and a succssor s of som vrtx x 2 X 2. W thus obtain a cycl C whos vrtx st is V (P ) by rmoving th dgs py 1 and x 2 s from P and adding th thr dgs, xx 2 and yy 1. Th cas (ii) is idntical xcpt that w now considr th prdcssors of th vrtics in X 1 and th succssors of th vrtics in Y 2. Altogthr, w hav now constructd a 2-factor whr th numbr of cycls has dcrasd. Continuing in this way, w vntually arriv at a Hamilton cycl C. It is asy to chck that th symmtric diffrnc of C and R contains only at most 5(n/ log n) 1/5 ηνn/2 dgs. Togthr with (16) this shows that C is as rquird in th thorm. It rmains to dduc Thorm 1.3 from Thorms 1.1 and 1.2. Proof of Thorm 1.3. First suppos that n is vn. St n := n/2. Considr a random partition of th vrtx st of G into two sts A and B of qual siz. Lt G b th bipartit subgraph of G btwn A and B. Lmma 2.1 implis that w may assum that th graph G is (d, 2ε)-suprrgular (in th bipartit sns) if n is sufficintly larg compard to ε. Also, Lmma 2.1 implis that w may assum that th dnsity of th bipartit subgraph of H i btwn A and B is still clos to ν i d for all i k. Thus w can apply Thorms 1.1 and 1.2 in this cas. Now suppos that n is odd and st n := n/2. Dlt any vrtx x from th vrtx st of G. Again, Lmma 2.1 implis that w may assum that th bipartit graph G = (A, B) constructd as abov on th rmaining 2n vrtics 14
is (d, 3ε)-suprrgular if n is sufficintly larg compard to ε. Morovr, w may assum that for all i k th dnsity of th bipartit subgraph of H i btwn A and B is still vry clos to that of H i i.. clos to ν i d. Thus w may apply Thorm 1.2 to obtain a Hamilton cycl C which satisfis (1 η/2)2ν i n C E(H i x) (1 + η/2)2ν i n. Lt P b a Hamilton path obtaind from C by adding an dg btwn x and som vrtx y C and dlting on of th two dgs on C incidnt to y. As in th proof of Thorm 1.2, on can asily show that on can transform P into a Hamilton cycl C by dlting two and adding thr dgs. Thn C is as rquird in Thorm 1.3(i). Rfrncs [1] M.J. Albrt, A. Friz and B. Rd, Multicolourd Hamilton cycls, Elctronic J. Combinatorics 2 (1995), #R10. [2] N. Alon, V. Rödl and A. Ruciński, Prfct matchings in ε-rgular graphs, Elctronic J. Combinatorics 5 (1998), #R13. [3] N. Alon and J. Spncr, Th Probabilistic Mthod (2nd dition), Wily- Intrscinc 2000. [4] B. Bollobás, Random Graphs (2nd dition), Cambridg studis in Advancd Mathmatics 73, Cambridg Univrsity Prss 2001. [5] L.M. Brégman, Som proprtis of nonngativ matrics and thir prmannts, Sovit Mathmatics Doklady 14 (1973), 945 949. [6] C. Coopr and A. Friz, Multicolourd Hamilton cycls in random graphs; an anti-ramsy thrshold, Elctronic J. Combinatorics 2 (1995), #R19. [7] F.R.K. Chung, Spctral Graph Thory, CBMS monograph no. 92, Amrican Mathmatical Socity, 1997. [8] G.P. Egorichv, Th solution of th van dr Wardn problm for prmannts, Dokl. Akad. Nauk SSSR 258 (1981), 1041 1044. [9] D.I. Falikman, A proof of th van dr Wardn s conjctur on th prmannt of a doubly stochastic matrix, Mat. Zamtki 28 (1981), 931 938. [10] A. Friz and M. Krivlvich, On packing Hamilton cycls in ε-rgular graphs, prprint 2003. [11] S. Janson, T. Luczak and A. Ruciński, Random graphs, Wily-Intrscinc 2000. [12] D. Kühn and D. Osthus, Loos Hamilton cycls in 3-uniform hyprgraphs of larg minimum dgr, submittd. 15
[13] V. Rödl and A. Ruciński, Prfct matchings in ε-rgular graphs and th Blow-up lmma, Combinatorica 19 (1999) 437 452. [14] V. Rödl, A. Ruciński and E. Szmrédi, A Dirac-typ thorm for 3- uniform hyprgraphs, prprint. [15] A. Schrijvr, A short proof of Minc s conjctur, J. Combin. Thory A 25 (1978), 80 83. Danila Kühn & Dryk Osthus School of Mathmatics Birmingham Univrsity Edgbaston Birmingham B15 2TT UK E-mail addrsss: {kuhn,osthus}@maths.bham.ac.uk 16