Equitable coloring of random graphs

Transcription

1 Equiable colorig of radom grahs Michael Krivelevich Balázs Paós July 2, 2008 Absrac A equiable colorig of a grah is a roer verex colorig such ha he sizes of ay wo color classes differ by a mos oe. The leas osiive ieger for which here exiss a equiable colorig of a grah G wih colors is said o be he equiable chromaic umber of G ad is deoed by χ = (G. The leas osiive ieger such ha for ay here exiss a equiable colorig of a grah G wih colors is said o be he equiable chromaic hreshold of G ad is deoed by χ =(G. I his aer we ivesigae he asymoic behavior of hese colorig arameers i he robabiliy sace G(, of radom grahs. We rove ha if /5+ǫ < < 0.99 for some 0 < ǫ, he almos surely χ(g(, χ = (G(, = ( + o(χ(g(, holds (where χ(g(, is he ordiary chromaic umber of G(,. We also show ha here exiss a cosa C such ha if C/ < < 0.99, he almos surely χ(g(, χ = (G(, (2 + o(χ(g(,. Cocerig he equiable chromaic hreshold, we rove ha if ( ǫ < < 0.99 for some 0 < ǫ, he almos surely χ(g(, χ =(G(, (2 + o(χ(g(, holds, ad if log+ǫ < < 0.99 for some 0 < ǫ, he almos surely we have χ(g(, χ =(G(, = O ǫ (χ(g(,. Keywords: grah colorig, equiable colorig, radom grahs AMS subjec classificaio: 05C5, 05C80 Iroducio ad mai resuls A equiable colorig of a grah is a roer verex colorig such ha he sizes of ay wo color classes differ by a mos oe. Oe of he firs resuls abou equiable colorigs School of Mahemaical Scieces, Raymod ad Beverly Sacler Faculy of Exac Scieces, Tel Aviv Uiversiy, Tel Aviv, 69978, Israel. rivelev@os.au.ac.il. Research suored i ar by a USA-Israeli BSF gra ad a gra from he Israeli Sciece Foudaio. Dearme of Mahemaics ad is Alicaios, Ceral Euroea Uiversiy, Budaes, 05 Nádor u. 9., Hugary. aos@reyi.hu. Research suored i ar by he Euroea Newor PHD, MCRN-5953.

2 is he celebraed Hajal-Szemerédi heorem [5] (recely reroved i a much simler way by Kiersead ad Kosoca [0] saig ha every grah wih maximum degree has a equiable colorig wih colors for ay +. Lih s aer [4] surveys some basic resuls o equiable colorigs ad how he boud of + ca be relaced by for cerai classes of grahs. Alicaios of he Hajal-Szemerédi heorem ad rece resuls o equiable colorigs of grahs ca be foud i (amog ohers [], [2], [9], [], [2], [9]. Equiable colorig ured ou o be useful i esablishig bouds o ails of sums of deede variables [6], [8], [8]. The roery of beig equiably colorable by colors is o moooe i, i.e. i is ossible ha a grah admis a equiable -colorig bu is o equiably ( + - colorable. Therefore here are wo arameers of a grah relaed o equiable colorigs. The leas osiive ieger for which here exiss a equiable colorig of a grah G wih colors is said o be he equiable chromaic umber of G ad is deoed by χ = (G, while he leas osiive ieger such ha for ay here exiss a equiable colorig of a grah G wih colors is said o be he equiable chromaic hreshold of G ad is deoed by χ =(G. (We follow he oaio of [4], hough equiable chromaic hreshold is someimes deoed by eq(g. I his aer, we rove resuls o he asymoic behavior of he above arameers i he radom grah G(,. By G(, we mea he robabiliy sace of all labeled grahs o verices, where every edge aears radomly ad ideedely wih robabiliy = (. We say ha G(, ossesses a roery P almos surely, or a.s. for breviy, if he robabiliy ha G(, saisfies P eds o as eds o ifiiy. Our mai aim is o address he followig cojecure: Cojecure: There exiss a cosa C such ha if C/ < < 0.99, he almos surely χ(g(, χ = (G(, χ =(G(, = ( + o(χ(g(, holds (he firs wo iequaliies are rue by defiiio. Before roceedig o our resuls le us summarize he asymoic behavior of χ(g(, i oe heorem which for some values of was discovered by Bollobás [4] ad ideedely by Maula ad Kučera [7], ad for oher values of by Lucza [5] (see also Chaer 7 of [7]. Here ad hroughou he aer log sads for he logarihm i he aural base e. Theorem. The followig saemes are rue for he chromaic umber χ(g(,. (a If log 8 < < 0.99, he almos surely where b =. 2 log b log b log b ( χ(g(, 2 log b 8 log b log b (, 2

3 (b There exiss a cosa C 0 > 0 such ha for every = ( saisfyig C 0 / log 8 almos surely 2 log( 2 log log( χ(g(, 2 log( 40 log log(. 2(log( log log(. Noe ha if eds o 0, he 2(log b log b log b ( is asymoically Also oe ha if is as i case (a, he log b log b log b (, while if is as i case (b, he log( log log(, so chagig he coefficie of log b log b ( (or log log( i he laer case i he deomiaor has o effec o he asymoics of he exressios i Theorem.. I our heorems, all lower bouds o χ = (G(, or χ =(G(, follow from he lower boud of Theorem.. By aalyzig a greedy algorihm we obai he followig heorem. Theorem.2 Almos surely he equiable chromaic hreshold χ =(G(, saisfies he followig iequaliies. (a If < 0.99 ad = o(, he where b =. 2 log b log b log b ( χ =(G(, log b 2. log b log b (, (b If here exiss a osiive ǫ such ha +ǫ log 8, he 2 log( log log( χ =(G(, log( 2. log log(. (c If 0 ad here is a osiive ǫ such ha log+ǫ, he for ay ǫ wih 0 < ǫ < ǫ we have +ǫ 2 log( log log( χ =(G(, ( ǫ +ǫ ǫ log(. Alyig case (c of he above heorem wih ǫ edig o ifiiy, we ge he followig resul. Corollary.3 If < 0.99 ad log log log(, he almos surely we have χ(g(, χ = (G(, χ =(G(, (2 + o(χ(g(,. 3

4 Alhough he above resul is o asymoically igh, he algorihm used i he roof gives us almos surely a equiable colorig i olyomial ime, while he oher resuls jus rove he exisece of a such colorig. The followig heorem is a urely deermiisic oe, which we will use i our robabilisic roofs; we sae i amog he mai resuls for i ca be of ideede ieres. Theorem.4 Le G be a grah o verices i which every iduced subgrah G[U] wih U m coais a ideede se of size s. Suose furher ha (G m ms2 m s holds, where (G deoes he maximum degree of he grah G. The G ca be roerly colored usig color classes oly of size s ad s. Our ex resul gives he asymoic value of χ = (G(, for dese radom grahs. Theorem.5 If /5+ǫ < < 0.99 for some ǫ > 0, he he followig holds almos surely: χ(g(, χ = (G(, ( + o(χ(g(,. Our las heorem gives he same uer boud as Theorem.2; is imorace is ha is roof wors also whe eds o 0 very quicly (i.e., whe G(, is very sarse. Theorem.6 There exiss a cosa C such ha if C χ = (G(, ( o( log(. log 8, he a.s. The res of his aer is orgaized as follows: i he ex secio we iroduce some oaio ad gaher some basic facs ha we will use i our roofs. I Secio 3, we rove Theorem.2 aalyzig a greedy algorihm. Theorem.5 will be roved i Secio 4 which cosiss of wo subsecios. I he firs subsecio, we give he roof of Theorem.4 ad deduce he very dese case of Theorem.5, i he secod subsecio we rove he dese, bu o very dese case of Theorem.5. Secio 4 coais he roof of Theorem.6. 2 Prelimiaries I his secio, we iroduce some (sadard oaio ad gaher some basic resuls cocerig biomial disribuios, radom grahs ad equiable colorig ha we will use i our roofs. Le α(g deoe he ideedece umber (he size of a larges ideede se of G. We will say ha a grah G is d-degeerae if every subgrah of G coais a verex of degree a mos d ad he degeeracy umber of G is he smalles ieger d such ha G is d-degeerae. We will also say ha he ses S,S 2,...,S s are almos equal if 4

5 S i S j for ay i,j s. The eighborhood of a se of verices U V (G is {v V (G \ U : u U such ha (u,v E(G} ad is deoed by N(U. The followig well-ow boud (see e.g. Theorem 2.. i [7] o he ails of biomial disribuios will be used several imes for rovig some roeries of radom grahs. Cheroff boud: If X is a biomially disribued radom variable wih arameers ad ad λ =, he for ay 0 we have ( 2 P(X EX + ex 2(λ + /3 ad P(X EX ex ( 2 2λ. Also, he followig resul of Kosocha, Narasi ad Pemmaraju will be quoed. Theorem 2. (Kosocha, Narasi, Pemmaraju [] For every d,, if a grah G is d-degeerae, has verices ad saisfies (G /5, he χ =(G 6d. Le us collec a coule of facs, easy iequaliies ha will be used durig he roofs. Fac 2.2 If > log2, he almos surely (G <.0. Corollary 2.3 If > log2, he almos surely for every se of verices U of size U =, we have N(U.0. For biomial coefficies we will have he followig uer boud. ( ( e = ex (O ( log(/. Le us fiish his secio wih iroducig he sadard oaios used for comarig he order of magiude of wo o-egaive sequeces. We will wrie g( = o(f( g( (g( = ω(f( o deoe he fac ha lim = 0 (lim f( g( =, while g( = f( O(f( (g( = Ω(f( will mea ha here exiss a osiive umber K such ha g( < K ( g( > K for all iegers. Fially, we will wrie g( f( for lim f( f( g( = f(. 3 Colorig greedily Proof of Theorem.2 We will use he followig greedy algorihm: le us fix a ieger, he fuure umber of color classes ad a ariio of he verex se V = V V 2... V such ha V = V 2 =... = V = ad if does o divide V = V \ i= V i. Our algorihm cosiss of rouds. I he ih roud we exose 5

6 all he edges havig oe edoi i V i ad he oher i i j= V j. I such a way, our grah will be ruly radom afer he h roud. Suose ha afer roud (i we have a roer colorig of he subgrah saed by i j= V j i colors so ha each color class has exacly oe verex i each V j ad has hus i verices. We say ha he ih roud succeeds if i is ossible o exed his colorig o a roer colorig of G[ i j= V j] by addig oe verex of V i o each of he color classes; if his is imossible he he roud fails. Observe ha he edges iside V i have o bearig o he oucome of he ih roud ad ca hus be igored. If all rouds are successful, he clearly we have roduced a equiable colorig of G i colors. Formally we ca defie a auxiliary radom biarie grah G i o 2 verices: verices sad for he verices i V i ad he oher verices rerese he color classes ha we have already buil. There is a edge bewee a verex rereseig a color class C ad a verex rereseig a verex v V i if ad oly if all he airs exosed i his roud bewee v ad C are o-edges. So our auxiliary grah is a radom biarie grah wih edge-robabiliy ρ i = ( i. By Remar 4.3 i Chaer 4 i [7], we ow ha he robabiliy ha here is o erfec machig i our auxiliary grah (ha is, we cao exed our colorig of he origial grah G(, roerly is O(e ρ i. There are rouds ad he robabiliy of a failure (i.e. here is o machig i he auxiliary grah is he bigges i he las roud. So by he uio boud, we ge ha he robabiliy ha our algorihm fails o give us a equiable colorig is O(e ( = O(e log (. I is clear ha he robabiliy of a failure decreases as icreases, so agai usig he uio boud, we obai ha he robabiliy ha our algorihm does o roduce a equiable -colorig for a leas oe is O(( e ( = O(e 2log (. So if for some value of we have ( = ω(log, he almos surely χ =(G(,. Case (a Le =, where b =. The we have log b 2. log b log b ( ( = log b 2. log b log b ( ( log b 2.log b log b ( = log b 2. log b log b ( log 2. b ( = log 2. b ( log b 2. log b log b (. 6

7 By he assumio ha > δ for every δ > 0, he above exressio is asymoically equal o log. b = Ω(log. log. Case (b Le = log( 2.log log(. The we have ( = log( 2. log log( ( (log( 2. log log( log( 2. log log( e2. log log( log( log. (. By our assumio o, his las exressio is asymoically bigger ha log. Case (c Fix a ǫ saisfyig he assumio of he heorem ad le = we have ( log( = ( (ǫ/(+ǫ ǫ (ǫ/( + ǫ ǫ log( (ǫ/( + ǫ ǫ log( e (ǫ/(+ǫ ǫ log( = log +ǫ /2, (ǫ/(+ǫ ǫ log(. The (ǫ/( + ǫ ǫ log( ( (ǫ/(+ǫ ǫ where for he las iequaliy we used he assumio log+ǫ <. 4 The equiable chromaic umber of dese radom grahs I his secio we rove Theorem.4. The roof will be divided io wo ars. I he followig subsecio we rove he heorem whe > log 8. For his urose, we firs rove Theorem.3 ad deduce his case of Theorem.4 as a corollary alog wih a alicaio o (, d, λ-grahs (see he defiiio here. I he secod subsecio we rove he dese, bu o very dese case. 4. The very dese case Proof of Theorem.4: Usig he roery of G assured by he assumio of he heorem, we ic airwise disjoi ideede ses I,I 2,...,I of size s as far as he umber of remaiig verices is less ha m, i.e. V (G \ i=i j < m. So we have m. Sice we are allowed o have color classes of size s, we may remove s oe verex from each I j. We will use his o creae ideede ses of size s for each v V (G \ j= I j. For he firs such verex v, le us ic verices u I j,u 2 7

8 I j2,...,u m I jm such ha i i 2 imlies j i j i2 ad v is o adjace o ay of he u i s. The by he roery of G, here exiss a ideede se of size s amog he u i s, which ogeher wih v is a ideede se of size s. We would lie o reea his rocedure for all verices i V (G \ j= I j. Sice we are allowed o have ideede ses oly of size s ad s we cao use he verices i he color classes from which we have already removed a verex. We have o mae sure ha we ca ic he u i s i he above meioed way eve for he las verex v l i V (G \ j= I j. Afer he firs hase (icig ideede ses greedily due o he roery of G, we had a leas m verices i he I j s. For each revious verex i V (G \ j= I j we have used s ideede ses, each coaiig s verices. We cao use hese verices, jus as we cao use a mos (G verices coeced o v l. Sice we have o ic verices from differe I j s, he umber of ossible I j s we are sill allowed o ic from is a leas (G m ms2. By he assumio of he heorem, s his is a leas m, so our rocedure ever fails. Now, we are ready o rove he very dese case of Theorem.5. I order o aly Theorem.4 o he radom grah G(,, we eed o fid he corresodig values of m ad s. This was he crucial se i Bollobás roof [4] for he asymoic value of χ(g(,. Here we cie a resul of Krivelevich, Sudaov, Vu ad Wormald [3] which we will use seig ǫ = 0.0. Le 0 = max{ : ( ( ( 2 4 }. I is well ow ha 2 log b 0 2 log b C log b log b ( for some cosa C. Theorem 4. [3] Le ( saisfy 2/5 log 6/5 ( ǫ for a absolue cosa 0 < ǫ <. The P[α(G(, < 0 ] = e Ω 2 0 «. 4 Now chagig o ad alyig he uio boud we ge ha he robabiliy log 3 b ha some subgrah of G(, o verices does o coai a ideede se of log 3 b size ( ( 2 log b log 3 C log b log b b log 3 2 log b C log b log b ( b is a mos ( «( ( e Ω ex O( Ω, log b which eds o zero, sice log b γ ad herefore 0 γ for all γ > 0 if > log 8. Therefore, usig Fac 2.2, we ca aly Theorem.4 wih s = 2 log b C log b log b ( ad m =. This fiishes he roof of he very dese case of Theorem.5. log 3 b 8

9 For aoher alicaio of Theorem.4 we eed he followig defiiio. Defiiio. A (, d, λ-grah is a d-regular grah o verices wih eigevalues d = λ λ 2... λ such ha λ max{ λ i : 2 i }. Theorem 4.2 Le G be a sequece of (,d,λ-grahs where d( 0.9 ad d3 Ω( α holds for some α > 0. The χ = (G = O( d. log d Proof. We will eed he followig resul of Alo, Krivelevich ad Sudaov: 2 λ = Theorem 4.3 (Alo, Krivelevich, Sudaov [3] Le G be a (, d, λ-grah such ha λ < d < 0.9. The he iduced subgrah G[U] of G o ay subse U, U = m, coais a ideede se of size a leas α(g[u] 2(d λ log ( m(d λ (λ + +. We would lie o aly Theorem.4 wih m = d2 ad s = log 3 clog for some c > 0, d so we have o verify ha he codiios of he Theorem are me. By he assumio d 3 = 2 λ Ω(α, we have λ = o(d, so = Θ( 2(d λ d. Agai by d 3 = 2 λ Ω(α, we have ( m(d λ log + = Ω(log, herefore i is ideed rue ha every subgrah of G (λ+ wih m verices coais a ideede se of size s. I remais o verify he iequaliy (G m ms2 m. (G = d 0.9, ms 2 = s c (G m ms2 = o(, so we have = Θ( = Θ( d d2 = m. Theorem.3 log ( s ( s log log 3 gives ha χ = (G = Θ d d = Θ, where he las equaliy follows from s log log d d 3 = 2 λ Ω(α (sice his rivially imlies d > 2/ The dese, bu o very dese case Bollobás argume [4] for deermiig he asymoic behavior of χ(g(, for dese radom grahs was o fid may ideede ses of size close o α(g(, ad he o color he remaiig small se of verices wih few addiioal colors. The colorig obaied his way is very much o equiable. To overcome his difficuly we will iroduce he oio of a ideede (, -comb which iformally cosiss of airwise disjoi ideede ses each of size ad a addiioal rasversal ideede se of size ha coais / verices from each of he airwise disjoi ideede ses (see he ex subsecio for he formal defiiio. Afer rovig ha every large eough subgrah of G(, coais a ideede (, -comb wih ad aroriaely chose, we will roceed as follows: we will ic ideede (,-combs C,C 2,...,C s uil he umber of remaiig verices will be small. The usig Theorem 2., we will ariio hese remaiig verices io exacly as may ideede ses I,I 2,...,I s as he umber of ideede (,-combs. Fially, we will mach he I i s o he ideede (,-combs i such a way ha if C i is mached o I j, he here are hardly 9

10 ay edges bewee I j ad he rasversal ideede se of C i. Thus we will be able o obai a equiable colorig of G(, by ariioig all mached airs io + ideede ses where he ( + -s se will be formed from he verices of I j ad mos verices of he rasversal ideede se of C i. I he ex subsecio we rove several roeries of G(, icludig he exisece of he ideede combs ad i Subsecio we rove he remaiig case of Theorem Proeries of radom grahs I his subsecio we collec all auxiliary lemmas ha we will use i he roof of Theorem.5. Lemma 4.4 Le 30/ ad le c > 0 a cosa. The i G(, almos surely 2 every s verices sa a mos s edges. Therefore almos surely every log c ( subgrah of G(, of size s is log c ( 4 log c ( -degeerae. Proof: Le r = 2. The robabiliy of he exisece of a subse V log c ( 0 V violaig he asserio of he lemma is a mos log c ( i=r where a i = ( i Noig ha (( i 2 ri log c ( i=r ri log c ( i=r [ e i ( r ] i ei r = 2r [ ( ei log c 3 log c ( ( 4 [ ( 3 log c ( eilog c 2 ] i ( log c (. 4 log c ( i=r 2 log c ( ] i = [ e 2 2r log c ( i=r ( ] r i ei 2r a i, a i+ a i ( e log c = 3 log c ( ( 4 2 log c ( ( (i + i+ i i 2 log c ( is moooe icreasig wih resec o i, we obai ha he erms of he las sum form a covex sequece, so eiher he firs oe or he las oe is he larges. Therefore he sum is a mos max { log c ( [ 2 ( e 2 ] 3 log c log ( c log c ( (, 2 [ ( 2 e log c 3 log c ( ( 4 ] log c log c ( ( }, which eds o zero as eds o ifiiy. 0

11 Lemma 4.5 For every air of osiive cosas 0 < γ < γ here exiss aoher cosa C = C(γ,γ such ha if C/ ad x = x(,y = y( saisfy xy =,x y, he he followig holds almos surely i G(,: log γ ( log γ ( for ay wo disjoi ses of verices U ad U 2 wih U = x ad U 2 = y, he umber of edges bewee U ad U 2 is a mos. log γ ( Proof: The execed umber of edges bewee wo fixed disjoi ses of he rescribed size is xy =. Alyig he Cheroff boud, we ge ha if C is chose such log γ ( ha 2, he he robabiliy ha for a fixed air of ses here are oo log γ ( log γ ( may edges bewee hem is a mos ( 3 ex. 6 log γ ( Therefore, usig he uio boud for all ossible airs of ses, we ge ha he robabiliy ha here is a air of ses coradicig he lemma is a mos ( ( ( 3 ex = x y log 6γ ( ( ( 3 3 ex x log(/x + x + y log(/y + y ex 4y log(/y. 6 log γ ( 6 log γ ( y log(/y for some log γ ( γ < γ < γ if C is large eough, herefore he 3 which is rue if C is 6 log γ ( 4 exressio above eds o zero rovided log γ ( chose large eough. Defiiio: A se E 0 ( V (G 2 \ E(G (E0 E(G of o-edges (edges forms a ideede (clique (,-comb i a grah G if here exis 2 airwise disjoi ses of verices I,...,I,J,...,J wih I i =, ad J i = for every i ( i such ha E 0 = ( Ii J i ( i= 2 Ji 2. A edge of a clique (,-comb is called horizoal if i lies iside some I i J i ad is called crossig oherwise (i.e. if i is a edge bewee some J i J i2. The umber of (all edges i a clique (,-comb will be deoed by E(, ( ad will be omied if heir value is clear from coex. Le us wrie m i he form m = +s +l for some 0 s ad 0 l <. A oimal subcomb of a clique (,-comb is a iduced subgrah of a clique (,- comb saed by all verices i j= J j s i= I i ad l addiioal verices from I s+. If m <, he a oimal subcomb is a iduced subgrah saed by m verices each of which are i I i J i for some i. The umber of edges saed by a oimal subcomb of m verices will be deoed by L(,,m.

12 I J I J J I J I J Figure : The verex se of a (,-comb ad a oimal subcomb. Lemma 4.6 The umber of edges saed by ay se of verices U i a clique (,- comb wih U = m is a mos L(,,m. Proof: If m, he a oimal subcomb is a clique, hus he lemma is rue rivially, herefore we ca assume ha m >. Le U be a se of m verices i a clique (,-comb saig he mos umber of edges amog all such ses. We may assume ha for ay i ( i, if U I i, he J i U. Ideed, if u U I i ad v J i,v U, ha U = U \ {u} {v} coais he same umber of horizoal edges ad coai a leas as may crossig edges as U. We ca assume ha here is a mos oe I i such ha U I i ad I i U. Ideed, if here were wo such I i s, he we could ic a verex v from he oe of which he iersecio wih U is o larger ha ha of he oher, ad remove v from U ad add a verex o U amog he verices of he oher I i which are o ye i U. (I is clear ha he umber of saed edges will icrease. If U saisfies he above assumios, ad if U coais all verices from j= J j, he U is a oimal subsrucure. If U does o coai all verices from j= J j, he we ca remove mi{ I i U, j= J j \ U } verices from he oly I i which is o comleely coaied i U ad add he same umber of verices o U from j= J j \U. Agai i is easy o see ha he umber of saed edges cao decrease. As a las se, if sill j= J j U, we ca remove j= J j \ U verices from ay of he I i s coaied i U ad add he verices of j= J j \ U. Lemma 4.7 If here exiss a 0 < ǫ such ha /5+ǫ log 8 holds, he almos surely every subgrah of G(, wih a leas verices coais a ideede log 7 ( (, -comb wih = log( 7 log log( ad = (2 log( 24 log log(. Proof: Le X deoe he umber of ideede (,-combs i G(,, where = log( ad = (2 log( 0 log log(. We esablish a uer boud for he robabiliy of he eve ha X = 0 by usig he geeralized Jaso iequaliy (see 2

13 e.g. Theorem 2.8 i [7]: ( (EX 2 P(X = 0 ex A B E(I, AI B where I A sads for he idicaor variable ha A is a se of o-edges formig a ideede (, -comb. (Formally we should aly he iequaliy for he idicaor variable, ha he se A of edges form a clique (,-comb i G(,, he comleme of G(,. Usig Lemma 4.6, we have ( ( A B E(I ( AI B m=2 m( m( (EX 2 ( ( ( = m=2 ( (... ( ( m( m ( E L(,,m = ( E ( (... ( ( E L(,,m ( E m=2 a m ( ( E, where a m = ( ( m m ( E L(,,m. We wa o rove ha amog he a i s a 2 is he larges. Usig Lemma 4.6 ad he defiiio of a oimal subcomb, we ge ha if 2 m <, he b m = a m+ ( m 2 = a m (m + ( 2 + m + ( m, ad if m = + s + l for some 0 s ad 0 l <, he b m = a m+ a m = ( m 2 (m + ( 2 + m + ( (l+. Elemeary calculaios show ha i each ierval (2 m < or + s m < +(s+ b m decreases (wih sarig value less ha ad he icreases (wih edig value larger ha. This imlies ha he maximum of he a m s is aaied a a 2 or a some a m, where m = + s for some 0 s. Before coiuig he roof, le us remar ha by he choice of ad we have Ideed, ( ( = 3 ( ( ( ( 2 log 4 ( 0 log 3 ( log log( (2log( 0log log(log( = 3

14 while ( [ ex 2 log 3 ( 8 log 2 ( log log( o(log 2 ( log log( ], ( ( (+ 2 2 ex [ 2 log 3 ( 20 log 2 ( log log( + o(log 2 ( log log( ]. Le c s = a +s. The for 0 s ( 2 d s = c s+ c s = [( s ] [ (2 + (s + ] ( ( +(s+ ( (s+ ( ( +s ( s ( ( 2+( / 2... [( (s + + ] 2+( / [ (2 + s + ] ( ( 2, which, by aig he 2 h ower of (, is less ha. This imlies ha he larges a m is eiher a 2 or a. To comare a 2 ad a, observe ha ( a ( ( = ( a ( 2 2( ( 2+ ( 2 ( 2...(( + ( (...( + 3 ( 2+ (, 2 which is (agai usig ( less ha. So we fially ge ha a 2 is he larges summad, herefore we have A B E(I AI B ( a 2 (EX = ( ( ( 2 2 ( 2 ( E ( ( ( 5 ( ( 2 33 log0 ( By chagig o, we ge ha he robabiliy ha here is a subgrah of G(, log 7 ( of size o coaiig a ideede log 7 ( (, -comb wih ( = log log 7 = log( 7 log log(, ( = ( 2 log( log 7 0 log log( ( log 7 ( (2 log( 24 log log( is a mos ( ( ex 5 2 / log 4 ( log 7 ( 33 log 0 ( 4 log 7 ( 0.

15 4.2.2 Proof of Theorem.5 I is eough o rove ha we ca color equiably ay grah G havig he roeries assured by Lemma 4.4, 4.5 ad 4.7 ad he Corollary 2.3 wih a mos ( (2 log( 24 log log( log( 7log log( colors, where i Lemma 4.5 we se γ = 5,γ = 4, x = α ad y = β log2 ( log 7 ( Give such a grah G, we ic sequeially usig Lemma 4.7 ideede (, -combs (which we will deoe by S i i s wih = log( 7 log log(, = (2 log( 24 log log( uil we are lef wih a mos verices. Noe ha he umber of log 7 ( combs s is s = = Θ(. Wih he hel of Theorem 2. (he resul of Kosocha log 2 ( e al. we ca ariio he verices lef io almos equal ideede ses A,...,A s, i.e. he umber of ses is equal o he umber of ideede combs. Ideed, he assumio log 8 assures ha he maximum degree amog he verices lef is a mos which is much smaller ha, he umber of remaiig verices, log 8 ( log 7 ( ad Lemma 4.4 assures( ha he degeeracy umber of he subgrah saed by he remaiig verices is O. log 7 ( ( Noe ha he size of he ideede ses A,...,A s is Θ. log 5 ( We are looig for a machig bewee he ideede ses ad he ideede combs, such ha if A i is mached wih S i2, he (wih he oaio of he defiiio ( of a (,-comb for ay j ( j a mos I 2 log 2 ( j J j = = Θ 2log 2 ( log( verices i I j J j have eighbors i A i. To esure he exisece of a such machig, we have o verify ha Hall s codiio holds. Firs oe ha ay A i ca be mached o a leas half of he ideede combs. ( Ideed, if o, he i a leas half of he ideede combs here are a leas Θ verices havig a leas oe eighbor i A log(. So alogeher, here are ( ( 2 Θ s = Θ verices wih his roery, which coradics he roery log( log 3 ( ( assured by Corollary 2.3 which i his case saes ha here ca be a mos O. log 5 ( such verices. This gives ha Hall s codiio holds for every family cosisig of a mos half of he ideede ses A i ( i s. We claim ha for ay family A coaiig a leas half of he ideede ses A,...,A s ad for ay (,-comb S, here is a A A such ha A ca be mached wih S (his of course would imly ha Hall s codiio holds for A, oo. Suose ( o. The for ay A A here are Ω edges bewee A ad S. Therefore log( ( ( here are s Ω = Ω edges bewee S ad log( log 3 ( A A A coradicig Lemma 4.6 accordig o which here should be a mos such edges. 5 log 4 (

16 Havig realized a machig wih he roery above, we would lie o roceed as follows: for every air of a ideede se A ad a ideede (,-comb S ha are mached i our machig, we would lie o ariio A S io + almos equal ideede ses. Sice all (,-combs have he same size ad he A i s are almos equal, for ay wo mached airs he size of A S may differ by a mos, so he resulig ideede ses will be almos equal. Le us suose ha i our machig a ideede se A is mached wih a ideede (,-comb S = j= I j J j. By defiiio S =, herefore he ideede ses we are looig for should be of size + A. We would lie o have ses J + i J i ( i wih J = = J such ha i= J i A is ideede ad is of size + A. Therefore we eed J + i o be + A + A = A +. By he defiiio of our machig here are a mos verices i I.8(+ i J i ha have a leas oe eighbor i A. Eve if all hese verices were i J i, here would be J i =.8(+ verices ou of which we could form J.8(+.8(+ + i. 2log 2 ( 5 The order of he equiable chromaic umber I his secio we will rove Theorem.6 usig very similar ideas o hose of he revious secio. The mai differece is ha we are o able o rove he exisece of large ideede combs ad herefore we have o use ideede ses ha we obai by Lucza s roof [5] of he chromaic umber for sarse radom grahs. I he ex subsecio we gaher some echical lemmas corresodig o he oes of he dese case i Subsecio 4.2. ad he i Secio 5.2 we rove Theorem Proeries of radom grahs The Corollary from he Iroducio is valid oly if log2, herefore we eed a similar saeme ha ca be used i he sarse case as well. Lemma 5. For every cosa c > 0 here exiss a cosa C > 0 so ha he radom grah G(, wih C = ( = o( has almos surely he followig roery: For every se J of size c log 0.5 ( he umber of verices ha are o adjace o ay verex i J is a leas 2 3, i.e. he eighborhood of J is of size a mos 3. Proof: For a fixed se J of size c log 0.5 ( he execed umber of such verices is c ( J ( log 0.5 c ( ( J e log 0.5 ( ( 00, 6

17 if is large eough. So by he Cheroff boud we ge ha he robabiliy ha here are less ha 2 such verices for his fixed J is a mos ex(. Therefore, aig 3 32 he uio boud, he robabiliy ha here is a se coradicig he saeme of he claim is a mos ( ex ( 32 ( (c log 0.5 c ( log 0.5 ( ex 32 = o(. c log 0.5 ( Lemma 5.2 There exiss a cosa C such ha if C log 8 he almos surely he verex se of G(, ca be covered by airwise disjoi ideede ses larger ha (2 log( 38 log log( ad all of he same size wih he exceio of a mos 4 log.5 ( verices. Proof: Lucza (usig Maula s exose-ad-merge mehod [6] showed i [5] (see also Lemma 7.7 ad Lemma 7.8 i Chaer 7 of [7] ha wih robabiliy a leas o(log (, oe ca ic airwise disjoi ses of verices I,I 2,...,I each of size (2 log( 37 log log( such ha V (G \ i= I i log 3 ( ad he oal umber of edges coaied i some I i ( i is a mos log 3 (. I follows ha. log( Le A be he se of hose I i s which coai more ha edges. The log 0.5 ( A log 2.5 ( ad so I A I 2 log.5 (. The umber of he remaiig I i s ha do o coai more ha edges ca be bouded by he oal umber log 0.5 ( of I i s which is log (. Therefore by deleig oe verex from each edge ad ossibly some addiioal verices, we ca ge ideede ses (all of he same size, which is larger ha (2 log( 38 log log( if is large eough wih removig a mos log.5 ( verices. So we covered he grah by ideede ses of he same size wih he exceio of a mos 2 log.5 ( + log.5 ( + log 3 ( 4 log.5 ( verices. 5.2 Proof of Theorem.6 The saemes of Lemmas 4.4, 4.5, 5. ad 5.2 hold almos surely, so i is eough o rove he heorem for grahs havig roeries assured by hese lemmas, ad his ime i Lemma 4.5 we se γ = 0.5,γ log( = 0.3, x = α,y = β. By Lemma log.5 ( 5.2 we ca cover all he verices of such a grah by ideede ses each of he same size, (2 log( 38 log log(, wih he exceio of a mos 4 log.5 ( verices. Le us deoe hese ideede ses by I. By Lemma 4.4, he degeeracy umber of he grah iduced by he verices o covered is O( log.5 ( (ad he maximum degree is a mos he maximum degree of he origial grah, which is a mos max{.0, log 2 } log.5 ( by he assumio o, so by Theorem 2. (Kosocha e al. [], we ca color hem equiably wih as may colors as he umber of ideede ses i I. I such a way we ge ideede ses J,J 2,...,J I, 7

18 such ha heir size may differ by a mos oe ad heir size is a mos some cosa c > 0. c for log 0.5 ( Our aim is o fid a machig bewee he ideede ses i I ad he J j s (le us deoe he family coaiig hem by J i such a way ha wheever a J j is mached wih some I i, he a leas half of he verices of I i ca be added o J j reservig he sabiliy of J j. To assure he exisece of such a machig, we have o verify ha Hall s codiio holds. As a cosequece of Lemma 5. we ge ha ay J J ca be mached wih a leas I I s from I. Ideed, if o he i a leas 4 I I s a leas half of he verices 5 5 cao be added o J, so here are a leas 2 verices wih his roery - coradicig 5 he lemma. Now we are ready o chec ha Hall s codiio holds for I ad J. If a subfamily J J has size less ha J he by he revious aragrah (ad cosiderig oly 5 oe se from J he umber of ses i I ha ca be mached wih some se J i J is a leas I J. Oherwise J J, ad we claim ha all ses i I are ca 5 5 be mached wih some se J i J. We oly have o chec his for J = J. I his 5 case log( J J J β ad for ay ideede se I I we have I = γ log.5 (.. log 0.3 ( Therefore by Lemma 4.5 he umber of edges bewee I ad J J J is a mos If o J J ca be mached wih I, he here are Ω edges bewee I ad ( log( ay J J, so here are Ω( edges bewee I ad J J J a coradicio. For ay wo mached airs (I,J ad (I 2,J 2, he size of I J ad I 2 J 2 differ by a mos, so if we ca sli all mached airs io wo almos equal ideede ses, he we ariio he verex se io almos equal ideede ses each of size a leas (log( 9 log log(. By he assumio o he machig we may add a leas half of he verices of I o J, while o sli I J io almos equal ars, we eed oly I J J verices. 2 Acowledgeme. We would lie o ha he aoymous referees for heir careful readig ad valuable commes. Refereces [] Alo, N.; Caalbo, M.; Kohayaawa, Y.; Rödl, V; Rucińsi, A.; Szemerédi, E., Uiversaliy ad Tolerace, Proc. 4 s IEEE FOCS, IEEE (2000, 4 2. [2] Alo, N.; Füredi, Z. Saig subgrahs of radom grahs, Grahs ad Combiaorics 8 (992,

19 [3] Alo, N.; Krivelevich, M.; Sudaov, B. Lis colorig of radom ad seudo-radom grahs, Combiaorica 9 (999, o. 4, [4] Bollobás, B. The chromaic umber of radom grahs, Combiaorica 8 (988, o., [5] Hajal, A.; Szemerédi, E. Proof of a cojecure of P. Erdős, Combiaorial heory ad is alicaios, II (Proc. Colloq., Balaofüred, 969, Norh-Hollad, Amserdam, 970. [6] Jaso, S. Large deviaios for sums of arly deede radom variables, Radom Srucures ad Algorihms 24 (2004, o. 3, [7] Jaso, S.; Lucza, T.; Rucińsi, A. Radom grahs. Wiley-Iersciece Series i Discree Mahemaics ad Oimizaio. Wiley-Iersciece, New Yor, [8] Jaso, S.; Rucińsi, A. The ifamous uer ail, Radom Srucures ad Algorihms 20 (2002, [9] Kosocha, A. V. Equiable colorigs of ouerlaar grahs, Discree Mah. 258 (2002, o. -3, [0] Kosocha, A.V.; Kiersead H. A Shor Proof of he Hajal-Szemerédi Theorem o Equiable Colorig, Combiaorics, Probabiliy ad Comuig 7 (2008, o.2, [] Kosocha, A. V.; Narasi, K.; Pemmaraju, S. V. O equiable colorig of d-degeerae grahs, SIAM J. Discree Mah. 9 (2005, o., [2] Kosocha, A. V.; Pelsmajer, M. J.; Wes, D. B. A lis aalogue of equiable colorig, J. Grah Theory 44 (2003, o. 3, [3] Krivelevich, M.; Sudaov, B.; Vu, V.H.; Wormald, N.C. O he robabiliy of ideede ses i radom grahs, Radom Srucures ad Algorihms 22 (2003, 4. [4] Lih, K.-W. The equiable colorig of grahs, Hadboo of combiaorial oimizaio, Vol. 3, , Kluwer Acad. Publ., Boso, MA, 998. [5] Lucza, T. The chromaic umber of radom grahs, Combiaorica (99, o., [6] Maula, D. W. Exose-ad-merge exloraio ad he chromaic umber of a radom grah, Combiaorica 7 (987, o. 3,

20 [7] Maula, D. W.; Kučera, L. A exose-ad-merge algorihm ad he chromaic umber of a radom grah, Radom grahs 87 (Pozań, 987, 75 87, Wiley, Chicheser, 990. [8] Pemmaraju, S. V. Equiable colorig exeds Cheroff-Hoeffdig bouds, Aroximaio, radomizaio, ad combiaorial oimizaio (Bereley, CA, 200, , Lecure Noes i Comu. Sci., 229, Sriger, Berli, 200. [9] Pemmaraju, S. V.; Narasi, K.; Kosocha, A. V. Equiable colorigs wih cosa umber of colors, Proceedigs of he Foureeh Aual ACM-SIAM Symosium o Discree Algorihms (Balimore, MD, 2003, , ACM, New Yor,