Self-similarity of graphs

Transcription

1 Self-similariy of graphs Choogbum Lee Po-She Loh Bey Sudakov Absrac A old problem raised idepedely by Jacobso ad Schöheim asks o deermie he maximum s for which every graph wih m edges coais a pair of edge-disjoi isomorphic subgraphs wih s edges. I his paper we deermie his maximum up o a cosa facor. We show ha every m-edge graph coais a pair of edge-disjoi isomorphic subgraphs wih a leas cm log m /3 edges for some absolue cosa c, ad fid graphs where his esimae is off oly by a muliplicaive cosa. Our resuls improve bouds of Erdős, Pach, ad Pyber from Iroducio The decomposiio of a give graph io smaller subgraphs is a old problem i graph heory ha has bee sudied from umerous perspecives. A celebraed resul of Wilso [16] assers ha give ay fixed graph H, he edge se of ay sufficiely large complee graph K ca be pariioed io edge-disjoi copies of H, as log as he obvious ecessary divisibiliy codiios eh ad g 1 where g is he greaes commo divisor of he degrees of H are saisfied. A facor of a graph is a spaig subgraph, ad a facorizaio is a pariio of is edges io facors. A series of papers by Graham, Harary, Robiso, Wallis, ad Wormald see, e.g., [7, 9, 10, 11, 15] iroduced he sysemaic sudy of isomorphic facorizaios, i which he resulig facors are required o be isomorphic o each oher as graphs. I his lieraure, a graph G is said o be divisible by a ieger, or -divisible, if G admis a isomorphic facorizaio io pars, alhough he aalogy wih he umber-heoreic oio of divisibiliy is oly syacical. The oio of -divisibiliy has also bee ermed bisecable, wih some auhors aggig o he exra codiio ha he resulig facors were also coeced graphs. The earlies work cocered he divisibiliy of he complee graph. Exedig a parial resul of Guidoi [8], Harary, Robiso, ad Wormald [10] proved ha he complee graph K is divisible by ay ieger which saisfies he obvious ecessary codiio. Mos oher exisig research o divisibiliy coceraes o rees ad foress, perhaps because heir simple srucure appears more racable. Algorihmically, Graham ad Robiso proved i [7] ha i is NP-hard o decide Deparme of Mahemaics, UCLA, Los Ageles, CA, choogbum.lee@gmail.com. Research suppored i par by a Samsug Scholarship. Deparme of Mahemaical Scieces, Caregie Mello Uiversiy, Pisburgh, PA 1513, ploh@cmu.edu. Research suppored i par by a NSA Youg Ivesigaors Gra ad a USA-Israel BSF gra. Deparme of Mahemaics, UCLA, Los Ageles, CA bsudakov@mah.ucla.edu. Research suppored i par by NSF gra DMS , NSF CAREER award DMS ad by USA-Israeli BSF gra. 1

2 wheher a ree is -divisible, while Harary ad Robiso [9] discovered a polyomial-ime algorihm o decide wheher a ree admis a isomorphic facorizaio io wo coeced graphs. The bes geeral resul o rees is due o Alo, Caro, ad Krasikov [1], who showed ha every m-edge ree ca be made -divisible by deleig oly Om/ log log m edges. Oce oe cosiders geeral graphs, however, i becomes esseially impossible o hope for - divisibiliy or eve closeess o -divisibiliy. I is herefore aural o ask wha is he larges -divisible subgraph which mus exis i a give graph. This problem saed below i geeraliy for hypergraphs was origially raised idepedely by Jacobso ad Schöheim. Problem 1.1. Le he self-similariy of a r-uiform hypergraph G, deoed ιg, be he larges ieger s for which G coais a pair of edge-disjoi isomorphic sub-hypergraphs wih s edges each. For each posiive ieger m, le ι r m be he miimum of ιg over all r-uiform hypergraphs wih m edges. Deermie ι r m. Remark. This paper focuses o graphs r =, so we will wrie ιm isead of ι m hroughou. The firs mai resul i his area was due o Erdős, Pach, ad Pyber []. Specifically, hey proved ha here were absolue cosas c r ad C r for which c r m /r 1 ι r m C r m /r+1 log m log log m. Their upper boud cosrucio is based o a appropriaely chose radom r-uiform hypergraph. For graphs r =, he powers of m coicide a m /3, so heir lower boud deviaed oly by a logarihmic facor from heir upper boud cosrucio, which was esseially he Erdős-Réyi radom graph. A aroud he same ime, similar resuls were obaied idepedely by Alo ad Krasikov upublished, ad by Gould ad Rödl. The laer group deermied i [6] ha for 3-uiform hypergraphs, ι 3 m 3 1 m, which mached he upper boud expoe, bu agai fell shor by a logarihmic facor. Very recely, Hor, Koubek, ad Rödl [13] aouced lower bouds for ι m, ι 5 m, ad ι 6 m which also came wihi poly-logarihmic facors of he correspodig upper bouds derived from radom hypergraphs. The mai resul of our paper compleely solves he graph case, deermiig he asympoic rae of growh of ιm = ι m. Theorem 1.. There are absolue cosas c ad C for which cm log m /3 < ιm < Cm log m /3. The key idea is o exploi rare large deviaios eves hrough a cosrucive algorihm, raher ha o aemp o erase hem wih uio bouds. Icideally, our upper boud cosrucio is sill based o a radom graph, bu wih a slighly modified edge probabiliy. Ispired by he asympoic opimaliy of radom graphs i he problem of Jacobso ad Schöheim, our ex resul explicily sudies he self-similariy of radom graphs. The Erdős-Réyi radom graph G,p is cosruced o he verex se [] = {1,..., } by akig each poeial edge idepedely wih probabiliy p. We say ha G,p possesses a graph propery P asympoically

3 almos surely, or a.a.s. for breviy, if he probabiliy ha G,p possesses P eds o 1 as grows o ifiiy. Sice is firs appearace i he 1960 s, his beauiful objec has bee a ceral opic of sudy i graph heory. Surprisigly, may problems abou radom graphs arose from research i various oher areas of mahemaics ad heoreical compuer sciece. Ye despie he grea amou of work devoed o his opic over he pas fify years, may ieresig uresolved quesios sill remai o be aswered. For more o radom graphs, we refer he reader o he books [3, 1]. Whe p < 0.99, i is well kow ha a.a.s. all coeced compoes of G,p are eiher rees or uicyclic are rees wih a sigle addiioal edge. Applyig he previously meioed resul of Alo, Caro, ad Krasikov, or eve Proposiio.3 below, i is he easy o see ha he selfsimilariy of G,p i ha regime is Θm a.a.s., where m is he umber of edges. Our secod resul asympoically deermies ιg,p for he remaiig rage of p. Theorem i If p 1 e 6 ii If p > 1 e 6, he ιg,p = Θ, he ιg,p = Θ p a.a.s. log γ a.a.s., where γ = 1 p. We will prove his heorem i he ex secio. Is proof illusraes he mai ideas of he argume for Theorem 1., which follows i Secio 3. Noaio. Le G be a graph wih verex se V. For a subse of verices X V, le G[X] be he subgraph of G iduced by X. For a verex v V, we use Nv o deoe he se of eighbors of v. Give a bijecio f : V V, le fg be he graph wih verex se V, where x, y V are adjace if ad oly if here exis wo adjace verices x, y V such ha fx = x ad fy = y. For wo graphs G 1 ad G defied o he same verex se, le G 1 G be he graph obaied by akig he uio of he edge ses of he wo graphs, ad le G 1 G be he graph obaied by akig he iersecio of he edge ses of he wo graphs. The followig sadard asympoic oaio will be uilized exesively. For wo fucios f ad g, we wrie f = og if lim f/g = 0, ad f = Og or g = Ωf if here exiss a cosa M such ha f M g for all sufficiely large. We also wrie f = Θg if boh f = Og ad f = Ωg are saisfied. All logarihms will be i base e.718. Radom graphs We will use he followig well-kow coceraio resul, which is a cosequece of Theorems A.1.11 ad A.1.13 i he book []. Le Bi, p deoe he biomial radom variable wih parameers ad p. Theorem.1. If X Bi, p ad λ p, he P [ X p λ] e λ 15p. 3

4 We begi by aalyzig he self-similariy of radom graphs. I addiio o beig a ieresig quesio i is ow righ, his ivesigaio also suggess good iuiio for geeral graphs. The upper bouds o ιg,p follow from relaively sraighforward uio bouds. Proof of upper boud i Theorem 1.3. Suppose ha we are seekig a pair of edge-disjoi isomorphic subgraphs wih edges. This ask is equivale o fidig subgraphs H wih edges ha ca be pariioed io he uio H πh, for some -edge subgraph H ad a permuaio π of he verex se. The expeced umber of such subgraphs H i G,p is a mos! p < e p e, 1 where he firs biomial coefficie cous he umber of ways o selec edges for H ou of all available, ad he! bouds he umber of permuaios π of he verex se. Togeher, hese choices deermie he edges which make up H, which appear wih probabiliy p. Thus, if we selec a value of for which he righ had side of 1 becomes o1, we will esablish ha he umber of such H is zero a.a.s., ad hece ιg,p < a.a.s. We separaely specify suiable choices for for he wo regimes of p ha we cosider i his heorem. For par i, where 1 p 1 e 6, we use = log γ, where γ = 1 p i his rage we have e 6 γ. The he righ had side of 1 becomes e γ log γ e = log γ Sice γ e 6, we have log γ e log γ e 3 e = o1. For par ii, where p 1 e 6 becomes e log γ > 3 γ log γ e = e log γ log γ e log γ. Noe ha e. log γ, ad hece he righ had side of 1 is a mos, we specify = e1 p. The righ had side of 1 he 1 e 1 p 1 e e = o1. e 11 e 11 The remaider of his secio is devoed o cosrucig large self-similar subgraphs i G,p. The srucure give i he followig defiiio urs ou o be exremely useful boh for his secio ad he ex secio. Defiiio.. Le d ad k be posiive iegers. i A d-sar is a graph cosisig of d + 1 verices ad d edges, where oe of he verices has degree d. We someimes simply refer o hese graphs as sars. ii A d, k-sar-fores is a collecio of k verex-disjoi d-sars. We deoe a d, k-sar-fores by he se of pairs {v, N v : v B}, where B is a se of k verices, ad for each v, he se N v Nv is a disjoi se of d eighbors of v.

5 The followig wo proposiios were he key ideas i []. We iclude heir proofs for compleeess, as well as o illumiae he pois a which we iroduce our ew argumes. The firs claim assers ha he self-similariy of a graph is large if here are may o-isolaed verices. Proposiio.3. Le G be a graph o verices wih o isolaed verices. The ιg. Proof. We firs prove ha G coais verex-disjoi sars ha cover all he verices of he graph. Give a graph G, ieraively remove edges ha coec wo verices of degree a leas wo i a arbirary order. Clearly, his process ever creaes isolaed verices, ad he fial graph cosiss oly of sars because all remaiig verices of degree wo or more are o-adjace. I remais o show ha ay -verex sar fores coais wo large edge-disjoi isomorphic subgraphs G 1 ad G. We cosider he sars i he fores by heir ype. Noe ha 1-sars are ohig more ha sigle edges, so for every wo 1-sars, we ca pu oe of hem i G 1 ad he oher i G. We accou for his as a coribuio of +1 oward ιg from he four verices i he wo 1-sars. O he oher had, for d, we ca spli he edges of every d-sar io wo ses of size d, possibly wih oe edge lef over. By addig oe par o G1 ad he oher o G, we see ha he d + 1 verices of each d-sar coribue + d o ιg. Accumulaig he coribuios from all verices, excep possibly for a mos wo verices from a sigle upaired 1-sar, we fid ha { { }} 1 d/ ιg mi, mi =. d d + 1 Alhough our problem cosiders he self-similariy wihi a sigle graph, our lower boud argume firs separaes he give graph io wo disjoi subgraphs, ad cosrucs a suiable mappig bewee hem which overlaps may edges. Defiiio.. Le G 1 ad G be wo edge-disjoi graphs, o possibly overlappig verex ses V 1 ad V of he same cardialiy. Le heir similariy ιg 1, G be he maximum ieger s such ha here exiss a bijecio f : V 1 V for which fg 1 G coais s edges. The ex proposiio uses a radom mappig as he ipu i Defiiio., i order o measure similariy of wo radom biparie graphs. Proposiio.5. For i = 1,, le G i be edge-disjoi biparie graphs wih pars A i ad B i, where A 1 = A = 1 ad B 1 = B =. Suppose ha A 1 A ad B 1 B are disjoi, bu A 1 may iersec A ad B 1 may iersec B. The ιg 1, G EG 1 EG 1. Proof. Idepedely sample uiformly radom bijecios from A 1 o A ad from B 1 o B, ad le f be heir combiaio. For each pair of edges e 1 EG 1 ad e EG, he probabiliy ha 1 e 1 ges mapped o e by f is exacly 1. Such a siuaio coribues +1 o he iersecio size fg 1 G. Therefore, by lieariy of expecaio, he expeced umber of edges i fg 1 G is a leas EG 1 EG 1, ad here exiss a suiable f which achieves ha boud. Corollary.6. Le G be a biparie graph wih pars A ad B such ha EG 10. ιg EG 5 A B. 5 The

6 Proof. Arbirarily pariio G io wo edge-disjoi subgraphs G 1 G wih 1 EG 1 EG 9 EG 0 edges, ad apply Proposiio.5. Corollary.7. Le G be a graph wih verices ad m edges, where m 0. The ιg m 5. Proof. Le A B be a bipariio of he verex se of G chose uiformly a radom. The probabiliy of a sigle edge iersecig boh pars is exacly 1, ad hus by averagig, here exiss a bipariio A B for which he biparie graph H bewee A ad B coais a leas m edges. Sice A B ad m/ 10, by Corollary.6, we have ιg m/ 5 / = m. 5 To prove Proposiio.5, we cosidered a radom bijecio bewee he wo verex ses, as here exiss a map such ha he resulig umber of overlappig edges is a leas is expecaio. This sraegy urs ou o be srog eough whe he graph is dese. O he oher had, for sparse graphs, Proposiio.3 produces a reasoable boud. These were he key seps used by Erdős, Pach, ad Pyber i []. I order o esablish Theorem 1.3, however, we eed somehig slighly more powerful for he iermediae edge desiy regime. The key ew igredie is o desig a verex permuaio ha performs beer ha a uiformly radom oe. To skech our argume, cosider he illusraive case p = 1/, which represes he mos delicae rage. We firs radomly spli he verices io four pars A 1, A, B 1, B of equal size, ad le G i be he biparie graph formed by he edges bewee A i ad B i. We discard all oher edges, ad boud oly he similariy bewee G 1 ad G. Raher ha searchig for a usrucured permuaio of he whole verex se, we build a favorable bijecio f : A 1 B 1 A B which seds A 1 o A ad B 1 o B wih may overlappig edges. Noe ha if we le f be a uiformly radom bijecio from A 1 B 1 o A B, he we esseially recover Proposiio.5, hus producig a lower boud of order oly Θ, which falls shor of Theorem 1.3 by a logarihmic facor. We sar wih a uiformly radom bijecio from B 1 o B, ad carefully exed i from A 1 o A as follows. Cosider a fixed verex v 1 i A 1 ad a fixed verex v i A. If we mapped v 1 o v, we would icrease he umber of overlappig edges by exacly fnv 1 Nv, where Nv i represes he se of eighbors of v i i B i. Recall ha we discarded all oher edges, so he v i oly have eighbors i heir correspodig B i. Sice we have p = 1/, if v is chose uiformly a radom, he expeced size of he se fnv 1 Nv is some cosa λ, ad his observaio led o he Θ lower boud whe cosiderig a uiformly radom bijecio. The crucial observaio is ha for each idividual pair of v i, he overlap fnv 1 Nv asympoically has he Poisso disribuio wih mea λ. Therefore, wih probabiliy a leas ε, i will be of size a leas ε log for some small cosas ε ad ε. Sice A has verices, he expeced umber of verices v A ha will give his high gai ogeher wih v 1 is Ω 1 ε. I paricular, i is very likely ha here exiss a suiable verex v for v 1 such ha fnv 1 Nv ε log, ad we will map v 1 o v i such a siuaio. By repeaig his for a cosa proporio of verices i A 1, we will obai ιg,p Ω ιg,p Ω log γ log. Sice γ =, his gives for our choice of p. Our ex wo lemmas formalize his iuiio. 6

7 Lemma.8. Le ad p saisfy 1 0 p 1 e 6, ad defie γ = 1 p. Le N 1,..., N s B be s 1/3 disjoi ses of size p 16, ad cosider he radom se B p, where we ake each eleme of B idepedely wih probabiliy p. The wih probabiliy a leas 1 e Ω1/1, here is a idex 0 log γ. i such ha B p N i Proof. Le = 0 log γ. I our rage of p, we always have 10, so i paricular 10 log γ. For a fixed idex i, he probabiliy ha B p N i 0 log γ is a leas N i p 1 p Ni. Usig he bouds k k k ad 1 p e p for small p, we have Ni p p 1 p Ni e p /15 = 16 16γ e p /15 10 log γ /10 log γ 16γ 1/15e1 = e 10 log γ log 16γ 10 log γ which by log 16γ 10 log γ log γ deduced from γ e 6, is a leas e 5 1/15e1 1/. 1/15e1, Hece he expeced umber of idices i such ha B p N i 0 log γ is a leas s 1/ 1/1. Sice he ses N i are disjoi, he above eves for differe choices of i are muually idepede. Therefore, by Cheroff s iequaliy, wih probabiliy a leas 1 e Ω1/1, we ca fid a idex i ideed, several for which B p N i 0 log γ. The previous esimae eables us o boud he similariy bewee radom biparie graphs. Lemma.9. Le ad p saisfy 1 0 p 1 e 6, ad le γ = 1 p. Le A 1, B 1, A, B be disjoi ses of size each, ad for each i = 1,, le G i be a radom biparie graph wih pars A i ad B i, where each edge appears idepedely wih probabiliy p. The ιg 1, G 160 log γ a.a.s. Proof. Sar wih a uiformly radom bijecio f from B 1 o B, ad also expose all edges i he radom biparie graph G. Sice p 1 0, Cheroff s iequaliy ad a uio boud esablish ha a.a.s., all degrees i G are bewee p 8 ad p. Codiio o his eve. We expose he edges i he biparie graph G 1 by ieraig over he verices i A 1, exposig each verex s icide edges i ur. Cosider he followig greedy algorihm for fidig a bijecio bewee A 1 ad A. Le A 1 be he se of verices i A 1 whose edges have bee exposed, ad suppose ha we have a ijecive map f : A 1 A such ha for all x A 1, fnx ad Nfx iersec i a leas 0 log γ verices. Le A = fa 1, ad le A i = A i \ A i for i = 1,. Suppose ha A 1 = A A 1 a some poi of he process. We firs prove ha he graph A B coais a p 16, 1/3 -sar-fores. Ideed, le k be he larges ieger such ha here exiss a p 16, k-sar-fores {x, N x : x X} for some se X A of 7

8 size X = k, ad suppose ha k < 1/3. Le NX be he uio of all eighborhoods of verices i X. We kow ha for every verex w A \ X, we have Nw NX p p 16 as oherwise we fid a p 16, k + 1-sar-fores, coradicig maximaliy. Therefore, here are a leas p 16 A X p 18 edges bewee he ses A \ X ad NX, ad i paricular, he se NX has a leas p 18 icide edges i G. Noe ha NX kp /3 p, sice we codiioed o all degrees i G beig a mos p, ad by he same reaso, he umber of edges icide o NX mus be a mos 7/3 p < p 18, coradicio. Therefore, we have k 1/3, as claimed. Now ake ay verex v 1 A 1, ad expose is edges o B 1. Is eighborhood Nv 1 is a radom subse of B 1, where each verex of B 1 appears idepedely wih probabiliy p. Sice he bijecio f : B 1 B was fixed from he ouse, he image of he eighborhood fnv 1 is also a radom subse of B wih he same produc disribuio. By Lemma.8, wih probabiliy a leas 1 e Ω1/1, we ca fid a verex v X A such ha fnv 1 N v 0 log γ, where X ad N v were from he sar fores cosruced above. Defie fv 1 = v ad repea he procedure. Sice he probabiliy of success a each roud is 1 o 1, we ca successfully ierae A 1 imes a.a.s., ad he fiish by exedig f by a arbirary bijecio bewee he o-mapped verices of A 1 ad A. I his way, we obai a bijecio f such ha he umber of edges i fg 1 G is a leas A 1 0 log γ = 160 log γ, as desired. We are ow ready o prove he lower bouds of Theorem 1.3. Proof of lower boud i Theorem 1.3. Par i has wo subcases. Firs, for 1 p 1/0, oe ha γ = 1 p 1/0, so he desired lower boud is of order log γ = Θ. I his rage, he umber of o-isolaed verices is Θ a.a.s., so Proposiio.3 complees his case. For he ex rage 1 0 p 1 e 6, we apply Lemma.9 afer spliig he verex se io four pars. Par ii follows direcly from Corollary.6. 3 Self-similariy of geeral graphs Alhough geeral graphs are o irisically radom, we apply probabilisic echiques o fid large edge-disjoi isomorphic subgraphs. The oulie of our proof for geeral graphs is similar o ha for radom graphs see he discussio followig Corollary.7 i he previous secio. The key idea is o exploi ail eves i he Poisso disribuio. However, esablishig his was somewha easier for radom graphs sice we had idepedece, ad could expose edges i a corolled maer. For geeral graphs, here are o radom edges o expose. Isead, we ur o sar foress, which were also a impora compoe i he proof of Lemma.9. Le G be a give graph o verices wih average degree d. As before, we begi by radomly spliig he verices io four pars A 1, A, B 1, B, ad cosider he biparie graphs G i formed by he edges bewee A i ad B i. We aemp o fid a oal of Ω 1 α may d 8, α -sar-foress S i,j = {v, N v : v X i,j } for i = 1,, 1 j Ω 1 α, where he ses X i,j A i are disjoi for differe idices. Noe ha X i,j he cover a cosa fracio of each A i, ad hece he edges i hese sar foress cosiue a cosa fracio of he edges i he eire graph G. If we 8

9 fail o fid such sar foress, he we will be able o pass o a subgraph where we ca fid eve larger isomorphic subgraphs. O he oher had, oce we fid such sar foress, we ake a radom bijecio f B from B 1 o B, ad exed i by idepede bijecios from X 1,j o X,j. To his ed, we declare f B o be good for he idex j if i ca be exeded o a bijecio bewee he ses B 1 X 1,j ad B X,j so ha he wo sar-foress overlap i Ω X 1,j edges uder he map. If some bijecio f B happes o be good for a cosa proporio of idices j, he we ca exed he bijecio f B o he ses X 1,j for hese idices, ad hereby cosruc a map f ha overlaps may edges of G 1 ad G. To begi his program, our firs lemma esablishes he ail probabiliy of he mai radom variable i our seig. I is he aalogue of Lemma.8. Lemma 3.1. Le α < 1 be a fixed posiive real umber, ad le d ad saisfy 1 α 16 d α. Le N1,..., N s [] be fixed disjoi ses of size for some s 1 5 α, ad le N be a uiformly radom subse of [] wih exacly d elemes. The wih probabiliy a leas 1 e Ωα/, α here exiss a idex i such ha N N i 8. Proof. Le N be a radom subse of [] obaied by idepedely akig each eleme wih probabiliy d. The disribuio of N codiioed o he eve N d ca be coupled wih he radom variable N, so ha N N give N, le N be a se of size d coaiig N chose uiformly a radom. By Cheroff s boud, he probabiliy of N > d is a mos e Ωd < e Ωα/, sice d 1 α 16 ad α < 1. Therefore, i order o prove our lemma, i suffices o show ha wih probabiliy a leas 1 e Ωα/, here exiss a idex i such ha N α N i Defie γ = ad = α. 8 log γ 8. Sice 1 α 16 d α, we have from which i follows ha < 1 α γ α 8, α 8 log γ α 8 log α 8 = α α + 8 log 1, for sufficiely large. Therefore, he roudig effec i he defiiio of a mos doubles he value, ad we have 1 α log γ. For each idex i, le E i be he eve ha N N i. As N N i is biomially disribued, jus as i he proof of Lemma.8, we may use he bouds k k k ad 1 p > e p for small p o fid Ni P [E i ] d 1 d Ni d/ d e d d d = e d. 9

10 Subsiue α log γ Sice α < 1 o ge d P [E i ] log γ α e d α, log γ > log, ad log γ, his is a leas log γ = e γ. αγ α 1 log γ e γ = α 1 γ α α = 3α. γ The E i are idepede because he N i are disjoi. Therefore he umber of E i ha occur sochasically domiaes a biomial radom variable wih mea s 3α/ 1 5 α/, ad we coclude by he Cheroff boud ha a leas oe E i ideed, several occurs wih probabiliy 1 e Ωα/, as desired. I he previous secio, i Lemma.9, we exploied he fac ha he give graph was radom ad he edges were idepede. This rick is oo resricive o be applied o geeral graphs. However, he ex lemma says ha for sar-foress, oe ca obai a lemma similar o Lemma.9. Lemma 3.. Le α < 1 be a fixed posiive real umber, ad suppose ha ad d saisfy 1 α 16 d α, ad are sufficiely large. For i = 1,, le G i be a d, α -sar-fores {v, N v : v X i } i he verex se X i B i, where X i = α ad B i =. The bijecio f B from B 1 o B chose uiformly a radom saisfies he followig propery wih probabiliy a leas 1 e Ωα/ : α f B ca be exeded o X 1 B 1 so ha he graph f B G 1 G has a leas X 1 edges. 36 Proof. Cosider a uiformly radom bijecio f B from B 1 o B. As i he proof of Lemma.9, we will pick verices of X 1 oe a a ime, mappig each oe o some verex i X i such a way α ha heir eighbors iersec i a leas verices uder he map f B. By repeaig his 9 for X 1 / seps, we he exed f B o form a oal of a leas X 1 α 9 overlappig edges, as required. To his ed, suppose ha we have already embedded some se X 1 X 1 of size less ha X 1 /, ad le X be he image of X 1. Furher suppose ha we have oly exposed he oucome of f B o he eighbors of X 1. Le B 1 = x X 1 N x ad B be is image which is already fully deermied by our parial exposure. The uexposed remaider of f B, codiioed o he previous oucome, is a radom uiform bijecio from B 1 \ B 1 o B \ B. Choose a arbirary verex x 1 X 1 \ X 1. Call a verex x X \ X available if N x \ B, or equivalely, N x B. Sice each uavailable verex accous for a leas verices of B, ad hose ses are disjoi for differe uavailable verices because G is a sar fores, we coclude ha he umber of uavailable verices is a mos B d/ = d X d/ = X X 1, ad hece he umber of available verices i X \ X is a leas X 1 /. 10

11 We ow expose he images of he d eighbors of x 1. This is a uiformly radom d-eleme subse of B \ B, where 1 o1 = d X 1 B \ B. For each available verex x, is deermiisically kow eighborhood i B \ B has size a leas d/, ad here are a leas X 1 / = α / such eighborhoods, all disjoi, comig from differe available verices. We are herefore i he seig of Lemma 3.1 wih 1 o1 isead of, ad so we coclude ha wih probabiliy 1 e Ωα/, here is a available verex x α such ha f B N x1 N x 9. Furhermore, we oly eed o expose he oucome of f B o N x1. We ca coiue he process for a leas X 1 imes, wih probabiliy a leas 1 X 1 e Ωα/ = 1 e Ωα/. This proves he lemma. Our ex proposiio bouds he self-similariy of a graph i erms of is media degree. To prove he proposiio, we will fid may sar-foress i our graph, ad apply Lemma 3. several imes. Proposiio 3.3. Le α 1 5 be a fixed posiive real umber. The for every sufficiely large ad d saisfyig 6 1 α 16 d α, every -verex graph G wih a leas verices of degree α a leas d has ιg > 59. Proof. Take a uiformly radom pariio A 1 A B 1 B of he verex se, where A 1 = A = B 1 = B =. For i = 1,, le G i be he biparie graph formed by he edges bewee A i ad B i. Sice d > 1/3, by he coceraio of he hypergeomeric disribuio see, e.g., Theorem.10 of [1] ad a uio boud, oe ca see ha a.a.s. each A i coais a leas 9 verices ha have a leas d 5 eighbors i B i i he graph G i. Codiio o his eve. Le d = d 10 ad = 1, ad oe ha sice α 5, α 3 < α < α. Le k 1 be he larges ieger for which we ca fid a collecio of d, α -sar-foress S 1,j = {v, N v : v X 1,j } i G 1, where he ses X 1,j are disjoi subses of A 1, for 1 j k 1. We claim ha k 1 1 α 18. Ideed, if o, he here exis over 9 k 1 α 18 verices i A 1 ha are o covered by he ses of he form X 1,j, ad have degree a leas d 5 i he se B 1. Le A 1 be he se of hese verices. By our maximaliy assumpio, we kow ha he graph G 1 [A 1 B 1] does o coai a d, α - sar-fores. Le S = {v, N v : v X} be a d, h-sar-fores i G 1 [A 1 B 1], where X A 1 ad h is as large as possible. By our assumpio, we kow ha h < α. The all he verices i A 1 \ X have degree a leas d 10 i he se N = v X N v. Noe ha N = d h < d 10 α ad A 1 \ X 18 h > 19. I his case, Corollary.6 applied o G[A 1 \ X N] already gives ιg ιg[a 1 \ X N] d/10 A 1 \ X 5 N A 1 \ X = d A 1 \ X > d1 α 500 N 950 > /3 950, which for large is already far more ha eough. Therefore, we may assume ha k 1 1 α 18. Similarly, here is a collecio of 1 α 18 may d, α -sar-foress S,j = {v, N v : v X,j } i G, where X,j are disjoi subses of A. 11

12 Le f B be a bijecio from B 1 o B chose uiformly a radom. Our iiial codiios o ad d imply ha ad d saisfy he requiremes of Lemma 3., so for each fixed j, wih probabiliy a leas 1 e Ωα/, f B ca be exeded o a bijecio bewee B 1 X 1,j ad B X,j such ha f B G 1 [B 1 X 1,j ] ad G [B X,j ] overlap i a leas α X 1,j > 36 log d α 3 36 log α log 5 α α 1 log edges, where we used 1 α 5. Sice he ses X 1,j are disjoi for disic j, ad X,j are also disjoi for disic j, a uio boud shows ha we ca idepedely exed he bijecio f B by each X 1,j X,j o cosruc a map f : A B which esablishes compleig he proof. ιg 1, G > 1 α 18 α α α =, 1 59 We are ow ready o prove Theorem 1., ad esablish he correc order of magiude of he fucio ιm. Proof of Theorem 1.. Cosider he radom graph G,p wih p =. For m = 1 3/, we a.a.s. have eg,p = 1 + o1m, ad by Theorem 1.3, ιg,p = Θ = Θm log m /3. Sice he fucio ι is moooe, his shows ha ιm Om log m /3, ad esablishes he upper boud. I he remaider of he proof, we focus o provig he lower boud. Le G be he give graph wih verices ad m edges. Wihou loss of geeraliy, we may assume ha G coais o isolaed verices. Le 0 =, m 0 = m, G 0 = G, ad le V 0 be he verex se of G 0. Le 0 = a m /3 0 0 for some real a log m 0 1/3 0. Le = 1 i he begiig ad cosider he followig ieraive process. A each sep, we will eiher fid wo large isomorphic edge-disjoi subgraphs, or will fid a iduced subgraph G o he verex se V such ha for = V, m = EG, ad a saisfyig = a m /3, we have he followig properies: log m 1/3 i G has o isolaed verex, ii m 0 m 1 1 i=0 a i m 0 > m 3, ad iii a a for 1. Noe ha he properies ideed hold for = 0. Suppose ha we are give parameers as above for some 0. If m log m /3, he by Proposiio.3, we have ιg m log m/3 = Ωm log m /3. O he oher had, if 8m/3, he by Corollary.7, we have log m 1/3 ιg m 5 Ωm log m /3. 1

13 Therefore, we may assume ha from which i follows ha 3 < a < log log m. Defie 8m /3 log m 1/3 < < m log m /3. 3 d = m a = a m log m 1/3, ad le V be he subse of verices which have degree a leas d i he graph G. Usig he upper boud of 3 ogeher wih a < log log m, oe ca see ha d > 3/ / log m > 1/ a log m > 6 1 for α = 1 5. The lower boud of 3 gives m < 8 3/ log m 1/, so usig a > 3, we fid ha d /8 3/ log m < 1 < α. a 17 Cosequely, if V V, he by Proposiio 3.3 we have α 16 ιg > α 59 log = 6800 log. Sice = 5a log m ad log m > = a log + 3 log m 1 3 log log m > 1 log m, we have ιg > 6800 log > log m log a = a log a m /3 log m /3. Sice a > 3, we have a a >, ad hus ιg > log m/3 log m /3 = Ωm log m /3. Oherwise, we have V < V. Le V +1 be he se of o-isolaed verices i he iduced subgraph G[V ]. Le +1 = V +1 ad le m +1 be he umber of edges i he iduced subgraph G +1 = m /3 +1 G[V +1 ]. Defie a +1 so ha +1 = a +1. Noe ha sice we oly removed verices log m +1 1/3 whose degree i G was less ha d, our ew umber of edges is m +1 > m d = 1 a m, ad i paricular is well above m / because a > 3. Propery i follows from he defiiio. For Propery ii, oe ha 1 m +1 > 1 a m 1 a 1 i=0 a i m 0 > 1 i=0 a i m 0, 13

14 ad moreover, sice a i > 3 ad a i+1 a i 1 3 for all i, we have 1 a i m > 1 m = i=0 Fially, sice m / < m +1 m we have i=0 3 i /3 m > m < = m /3 a 1 log m 1/3 < m +1 /3 a 1 log m +1 1/3 = a 1 m / log m +1 1/3, from which Propery iii follows. Noe ha by Propery iii, a some ime s we will reach a s 3, ad will be doe by Corollary.7, i he middle of he process a ime s. Cocludig remarks I his paper, we proved ha ιm = Θm log m /3. The upper boud followed by cosiderig he radom graph G,p wih p =. For his rage of p, we have m = Θ3/ 1/, or equivalely = Θ m /3 log m. By carefully sudyig he proof of Theorem 1., oe ca oice 1/3 ha every graph G wih ιg Om log m /3 has o be somewha similar o he above radom graph. Ideed, by choosig differe parameers i he proof, oe ca see ha for every ε > 0, such graphs G mus coai a subgraph o = Θ m /3 log m verices wih a leas 1 εm edges, where 1/3 he degree of a leas 1 ε verices is Ωd, for d beig he average degree of he subgraph hus d = Θm log m 1/3. Moreover, he edges of his subgraph are well-disribued, i he sese ha here does o exis a pair of disjoi verex subses X, Y saisfyig ex, Y d X Y sice i his case we ca direcly apply Corollary.7. For a posiive ieger s, le ι s G be he maximum for which G coais a s-divisible subgraph wih edges, ad le ι r,s m be he miimum of ι s G over all r-uiform hypergraphs wih m edges hus we have ι r m = ι r, m. By slighly adjusig our proof of he boud ιm = Θm log m /3, we ca also prove for fixed cosa s ha ι,s m = Θm s s s 1 log m s 1. 1/s. The upper boud follows by cosiderig he radom graph G,p wih p = For he lower boud, if ms/s 1, he we ca use a argume similar o ha of Corollary.7, ad if log m 1/s 1 m s s s 1 log m s 1, he we ca use a argume similar o ha of Proposiio.3. I he α remaiig rage of parameers, we ca proceed as i Secio 3. The value i Lemma 3.1 will be replaced by Ω Ackowledgme s 1 d s. 8 log We would like o hak he referee for careful readig of his paper, ad for useful feedback. 1

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