FORBIDDING RAINBOW-COLORED STARS

Transcription

1 FORBIDDING RAINBOW-COLORED STARS CARLOS HOPPEN, HANNO LEFMANN, KNUT ODERMANN, AND JULIANA SANCHES Abstract. W cosidr a xtrmal problm motivatd by a papr of Balogh [J. Balogh, A rmark o th umbr of dg colorigs of graphs, Europa Joural of Combiatorics 7, 006, ], who cosidrd dg-colorigs of graphs avoidig fixd subgraphs with a prscribd colorig. Mor prcisly, giv r t, w look for -vrtx graphs that admit th maximum umbr of r-dg-colorigs such that at most t 1 colors appar i dgs icidt with ach vrtx. For larg, w show that, with th xcptio of th cas t =, th complt graph K is always th uiqu xtrmal graph. W also cosidr gralizatios of this problm. 1. Itroductio W cosidr dg-colorigs of graphs that satisfy a crtai proprty. Giv a umbr r of colors ad a graph F, a r-pattr P of F is a partitio of its dg st ito r (possibly mpty) classs. A dg-colorig (ot cssarily propr) of a host graph H is said to b (F, P )-fr if H dos ot cotai a copy of F i which th partitio of th dg st iducd by th colorig is isomorphic to P. If at most r colors ar usd, w call it a (F, P )-fr r-colorig of H. For xampl, if th pattr of F cosists of a sigl class, o moochromatic copy of F should aris i H. W ask for th -vrtx host graphs H (amog all -vrtx graphs) which allow th largst umbr of (F, P )-fr r-colorigs. Qustios of this typ hav b first cosidrd by Erdős ad Rothschild [6], who askd whthr cosidrig dg-colorigs avoidig a moochromatic copy of F would lad to xtrmal cofiguratios that ar substatially diffrt from thos of th Turá problm. Idd, F -fr graphs o vrtics ar atural cadidats for admittig a larg umbr of colorigs, sic ay r-colorig of thir dg st obviously dos ot produc a moochromatic copy of F (or a copy of F with ay giv pattr, for that mattr), so that (Turá) F -xtrmal graphs admit r x(,f ) such colorigs, whr, as usual, x(, F ) is th maximum umbr of dgs i a -vrtx F -fr graph. Erdős ad Rothschild [6] cojcturd that, for vry l 3 ad > 0 (l), ay -vrtx graphs with th largst umbr of K l -fr -colorigs is isomorphic to th (l 1)-partit Turá graph, which was prov for l = 3 by Yustr [13] ad for l 4 by Alo, Balogh, Kvash, ad Sudakov [1], who also showd that th sam coclusio holds i th cas r = 3. Howvr, for r 4 colors, th Turá graph for K l is o logr optimal, ad th situatio bcoms much mor complicatd; i fact, xtrmal cofiguratios ar ot kow ulss r = 4 ad F {K 3, K 4 }, s Pikhurko ad Yilma [11]. A similar phomo, i which (Turá) xtrmal graphs admit th largst umbr of r-colorigs if r {, 3}, but do ot for r 4, has b obsrvd for a fw othr classs of graphs ad hyprgraphs, such as th 3-uiform Fao pla [10]. (S [8] for a mor dtaild accout of istacs whr this phomo holds.) Balogh [] was th first to cosidr colorigs avoidig fixd pattrs that ar ot moochromatic. Mor prcisly, h showd that th (l 1)-partit Turá graph is still optimal for r = colors wh forbiddig ay -pattr of K l. O th othr had, h obsrvd that this dos This work was partially supportd by CAPES ad DAAD via Probral (CAPES Proc. 408/13 ad DAAD 56677). Th first author also ackowldgs th support by CNPq (Proc /01-0 ad /01-). Th fourth author was fudd y CNPq. 1

2 C. HOPPEN, H. LEFMANN, K. ODERMANN, AND J. SANCHES ot hold i gral for r = 3 colors ad arbitrary 3-pattrs of K l. Idd, cosidr F = K 3 ad lt P b a partitio of K 3 ito thr classs cotaiig o dg ach, so that w ar lookig for 3-colorigs with o raibow triagl. If w color th complt graph K with ay two of th thr colors availabl, thr is o raibow copy of K 3, which givs at last 3 ( ) 3 distict (K3, P )-fr colorigs, ad is mor tha 3 x(,k 3) = 3 /4+O(1). This suggsts that th study of colorigs that avoid gral pattrs, ad i particular raibow pattrs, dsrvs mor atttio. I coctio with this, w should mtio that it was rctly prov that th complt graph is idd optimal for raibow triagls [4]. Thr has also b a xtsiv dscriptio of xtrmal graphs wh o forbids matchigs with various forbidd pattrs [9], which icluds all raibow cass. Stars hav playd a importat rôl i ths dvlopmts. Moochromatic stars F = S t with t 3 dgs wr th first istacs for which it was show [8] that F -xtrmal graphs (i this cas, (t 1)-rgular graphs for v) do ot admit th largst umbr of r-colorigs with o moochromatic copy of F for ay fixd r. I particular, this implis that this trasitio btw th cass r {, 3} ad r 4 dscribd abov dos ot hold for arbitrary graphs F. O th othr had, xtrmal -vrtx graphs for forbidd moochromatic S t ar ot yt kow for ay r ad t 3. I this papr, our iitial motivatio was to study r-colorigs that avoid raibow-colord stars S t, that is, w lt F = S t ad w cosidr th pattr whr ach dg is i a diffrt class (i particular, r t). For t = ad ay giv umbr of colors r, it is asy to s that a matchig of siz / yilds th largst umbr of r-colorigs with o raibow S, as this rstrictio implis that ay colorig must hav moochromatic compots (for odd, both a additioal isolatd vrtx ad a coctd compot with thr vrtics grat a xtrmal cofiguratio). Th sam xtrmal cofiguratio had b obsrvd for moochromatic S wh r =, but ot for largr valus of r. Not that th st of r-colorigs avoidig a moochromatic S is prcisly th st of propr r-dg-colorigs of a graph, ad hc this problm cosists of fidig -vrtx graphs with th largst umbr of propr colorigs. Howvr, i cotrast to th moocromatic cas, w maagd to fid th optimal cofiguratio for larg ad vry fixd r, t 3, which, i all cass, turs out b th complt graph K. Sic th tchiqus usd to prov this rsult may b adaptd to othr pattrs, w stat our rsults i gratr grality. I particular, to driv this gralizatio, w show that i ay graph with may dgs, thr is a almost spaig subgraph with a larg umbr of subgraphs of ay boudd dgr squc satisfyig a dsity costrait, which sms to b of idpdt itrst (s Lmma 3.1 for a prcis formulatio). A dg-colord star S tl with tl dgs such that t distict colors ar ach assigd to xactly l dgs is calld a raibow-s t,l. Giv itgrs r, t ad l 1, ad a graph G, a raibow-s t,l -fr r-colorig of G is a dg-colorig of G with colors i [r] = {1,..., r} for which thr is o raibow-s t,l. Clarly, if l = 1, w forbid raibow stars S t, ad w call such colorigs raibow-s t -fr. For ay graph G, lt C r,t,l (G) b th st of all raibow-s t,l -fr r-colorigs of G. W writ c r,t,l () = max { C r,t,l (G) : V (G) = }, ad w say that a -vrtx graph G is C r,t,l -xtrmal if C r,t,l (G) = c r,t,l (). W prov th followig rsult. Thorm 1.1. For all r, t 3 ad l 1, thr xists 0 such that, for all 0, w hav c r,t,l () = C r,t,l (K ). Morovr, th complt graph K is th sigl C r,t,l -xtrmal graph o vrtics.

3 FORBIDDING RAINBOW-COLORED STARS 3 Th rmaidr of this work is orgaizd as follows. I Sctio, w dal with som asy cass ad w prov our rsult for raibow stars, which givs a ovrviw of th gral cas. Th proof for gral l is th subjct of Sctio 3.. Colorigs avoidig a raibow star Th mai objctiv of this sctio is to prov Thorm 1.1 i th cas l = 1. This cas was th mai motivatio for our work ad, as it turs out, its proof givs a accurat ovrviw of th gral cas. Bfor doig this, w first dal with som straightforward cass. Rcall that, for t = ad l = 1, th r-colorigs of a graph G avoidig a raibow-s ar such that adjact dgs hav th sam color. I such a graph dgs i th sam compot hav to b colord th sam, but dgs i diffrt compots might b colord diffrtly. Thus, if th graph G has j compots cotaiig at last o dg, w hav C r,,1 (G) = r j. I ordr to maximiz this, j has to b as larg as possibl. Hc, th umbr of such colorigs is at most C r,,1 (M), whr M is a maximum matchig i G. So th oly C r,,1 -xtrmal graphs o vrtics ar for v a matchig of siz /, ad, for odd, a matchig of siz ( 1)/ ad a isolatd vrtx, or a matchig of siz ( 3)/ ad vrtx-disjoit coctd compot o thr vrtics. Clarly i this cas w hav c r,,1 () = r. For r < t, o r-colorig ca produc a raibow-s t,l, so that c r,t,l () = r ( ), ad K is th oly C r,t,l -xtrmal graph. Usig this argumt mor carfully, w may xtd this coclusio for som additioal valus of r ad t. Lmma.1. Lt t 3 r t 3, l 1 ad b positiv itgrs. Th c r,t,l () = C r,t,l (K ) ad th complt graph K is uiqu with this proprty amog all -vrtx graphs. Proof. Lt G b a -vrtx graph whr som dg = {v, w} is missig. Cosidr a fixd raibow-s t,l -fr r-colorig of G. W show that w ca xtd to a colorig of G = G +. Lt S v ad S w b th sts of colors occurig o at last l dgs icidt with v ad w, rspctivly, hc S v, S w t 1. If S v S w, th w ca xtd to G by colorig with ay color i S v S w. Now lt S v S w =, i particular S v + S w r t 3. If t 1 = S v > S w, w ca color with ay color i S v. If S v, S w < t 1, th w ca color with ay color. I coclusio, ca b xtdd to a colorig of G. To fiish th proof, w show that at last o of th colorigs of G may b xtdd to G i mor tha o way. As t 3, ay moochromatic colorig of G ca b xtdd to a colorig of G + by colorig with ay color, so that C r,t,l (G) < C r,t,l (G + ). Rmark: Th proof of Lmma.1 also yilds th followig for ay r t 3 ad l. If r(l 1)/ + l(t 1) + 1, th it is possibl to xtd ay colorig to a missig dg {v, w}, v if S v S w = ad S v = S w = t 1. Idd, if a colorig caot b xtdd udr such coditios, th colors i S v must hav b assigd to xactly l 1 dgs icidt with w, ad vic-vrsa. Morovr, ay color i S v S w must appar at l 1 dgs icidt with v or l 1 dgs icidt with w, othrwis it could b usd to xtd th colorig. Howvr, th dgrs of v ad w (which ar at most ) ar too small for all of ths coditios to hold bcaus of our boud o. W ow focus o th proof of Thorm 1.1 i th cas l = 1. Th gral ida of th proof is as follows. Cosidr a fixd raibow-s t -fr r-dg colorig of a graph G. By dfiitio, for vry vrtx v of G, th umbr of colors apparig o dgs icidt with v is at most (t 1). For sts S 1,..., S [r], lt C r,t,(s1,...,s )(G) dot th st of all dg-colorigs of G whr o dgs icidt with vrtx v i ar assigd colors from th st [r] \ S i, for all i = 1...,.

4 4 C. HOPPEN, H. LEFMANN, K. ODERMANN, AND J. SANCHES Th th st C r,t (G) of all raibow-s t,l -fr r-colorigs of G satisfis C r,t (G) = (S 1,...,S ) S i =t 1,,..., C r,t,(s1,...,s )(G). (1) Obsrv that th uio is ot disjoit, as fwr tha t 1 colors could appar i dgs icidt with som vrtx. Bfor procdig, ot that this dcompositio ca b asily gralizd to l. Th diffrc, for a fixd raibow-s t,l -fr r-dg colorig of a graph G, is that th sts S i cotai th colors that appar at last l tims i dgs icidt with v i, which w call ordiary colors with rspct to v i, whil th rmaiig colors ar said to b rar for v i. I aalogy to th abov cas, C r,t,l,(s1,...,s )(G) dots th st of all dg-colorigs of G whr fwr tha l dgs icidt with vrtx v i ar assigd ach color from th st [r] \ S i, for all i = 1...,. As i (1), th st C r,t,l (G) of all raibow-s t,l -fr r-colorigs of G satisfis C r,t,l (G) = (S 1,...,S ) S i =t 1,,..., C r,t,l,(s1,...,s )(G). () Our proof cosists of four stps. W first show that ay xtrmal graph must hav a lot of dgs, as othrwis it caot bat th umbr of colorigs achivd by th complt graph. Nxt w prov that most colorigs i (1) aris from th cass wh almost all sts S i ar th sam. Usig ths facts, w ca prov that xtrmal graphs hav larg miimum dgr, which, i th last stp, allows us to prov that ay xtrmal graph coicids with K. Th followig lmma is th first stp i th abov dscriptio, which may b asily provd for gral l. Lmma.. For r t 3 ad l 1, thr is a costat D > 0 such that if G = (V, E) is a C r,t,l -xtrmal graph o tl + 1 vrtics, th E(G) ( ) D log t 1. Proof. Fix r t 3 ad lt G = (V, E) b a -vrtx C r,t,l -xtrmal graph with V = {v 1,..., v }. Not that G has at last (t 1) ( ) (3) S t,l -fr r-dg colorigs, as th complt graph K has at last ths may colorigs: choos a fixd (t 1)-subst S of [r] ad assig colors i S to all dgs of K. W cosidr th dcompositio i (), ad fix sts S 1,..., S. Colorigs i C r,t,l,(s1,...,s )(G) may b producd as follows: for ach vrtx v i w choos at most (r t + 1)(l 1) icidt dgs to b assigd colors that ar ot i S i ad color thm with ths colors. Th rmaiig dgs {v i, v j } E ar assigd colors i S i S j. For sufficitly larg, this implis that C r,t,l(s1,...,s )(G) (r t+1)(l 1) j=0 ( ) 1 r j j {v i,v j } E S i S j (r t+1)(l 1) r (r t+1)(l 1) (t 1) E. (4)

5 As (S 1,..., S ) ca b chos i ( r FORBIDDING RAINBOW-COLORED STARS 5 C r,t,l (G) t 1) ways, (S 1,...,S ) S i =t 1,,..., ( r t 1 C r,t,l(s1,...,s )(G) ) (r t+1)(l 1) (t 1) E (t 1) D log t 1 (t 1) E, (5) ( whr D = log r t 1 t 1) + (r t + 1)(l 1) is a costat. Combiig (3) ad (5), w hav ( ) (t 1) D log t 1 (t 1) E (t 1) ( ) = E D log t 1, as rquird. To prform th scod stp of th proof, for a costat A > 0, lt S A dot th st of all collctios (S 1,..., S ) of (t 1)-substs of [r] whr o st S i appars mor tha ( A log t 1 ) tims, i {1,..., }. W prov that w may fid A for which th umbr of colorigs i S A is gligibl. As i th prvious rsult, w prov this for gral l, as thr is littl additioal work. Lmma.3. Lt r t 3 ad l 1 b itgrs. For all D > 0 thr xists a positiv costat A with th followig proprty. Giv ε > 0 thr is a costat 0 such that, for all 0, ay -vrtx graph G = (V, E) with at last ( ) D logt 1 dgs satisfis C r,t,l,(s1,...,s )(G) (S 1,...,S ) S A ε(t 1) E(G). Proof. With forsight, fix { } 3(r t + 1)(l 1) A > max 1 log t 1 (t ), D, ad lt B b a itgr satisfyig / ( r t 1) B A logt 1, whr will b chos sufficitly larg latr i th proof. Giv a -vrtx graph G = (V, E) with E ( ) D log t 1, w provid a uppr boud o th umbr of raibow-s t,l -fr r-dg colorigs i a st C r,t,l,(s1,...,s )(G) such that max S {v V : S v = S, S = t 1} = B. To grat ths colorigs, w choos a st U V such that U = B ad a (t 1)-subst S of [r] which is assigd to all vrtics i U. W th assig othr (t 1)-substs to th rmaiig ( B) vrtics of G. Lt E(U, V \ U) dot th st of dgs with o vrtx i U ad th othr i V \ U. As i th proof of Lmma. (s (4)), for ach vrtx v i w choos at most (r t + 1)(l 1) dgs i at most (r t+1)(l 1) ) i=0 (r t+1)(l 1) ways for sufficitly larg. Ths dgs ar assigd colors that ar ot i S i i at most r (r t+1)(l 1) ways. Th rmaiig dgs {v i, v j } E ar assigd colors i S i S j. Ay such dg i E(U, V \U) may b assigd at most (t ) colors, sic th sts assigd to thir dvrtics ar distict. Hc th umbr of raibow-s t,l -fr r-colorigs of G is boudd abov by = ( B ( B ) ) ( 1 i ( ) r B+1 (r) (r t+1)(l 1) (t ) E(U,V \U) E(G) E(U,V \U) (t 1) t 1 ) B+1 (r) (r t+1)(l 1) (t 1) E(G) ( r t 1 ( ) t E(U,V \U). (6) t 1

6 6 C. HOPPEN, H. LEFMANN, K. ODERMANN, AND J. SANCHES Not that E(U, V \ U) ( ) D log t 1 = D log t 1 + B B { D log t 1 + mi ( ) B = (A D) log t 1 A log t 1 ( ) B A log t 1 A log t 1, ( r t 1 } ) ) for larg. As a cosquc, sic A > D ad is sufficitly larg (i particular dpds o A, r, l ad t), w obtai E(U, V \ U) (A log t 1 )/. If w sum (6) ovr all possibl valus of B, w obtai at most ( ) r ( ) r (r) (r t+1)(l 1) (t 1) E(G) t 1 t 1 + log ( r t 1) (t 1) E(G) +(r t+1)(l 1) log t 1 (r) ε(t 1) E(G) ( t t 1 ( t t 1 ) A log t 1 ( r t 1 ) A log t 1 raibow-s t,l -fr r-colorigs of G, whr ε > 0 is arbitrary as log as w choos sufficitly larg, sic w hav A > 3(r t+1)(l 1) 1 log t 1 (t ). Th xt stp i our proof of Thorm 1.1 for l = 1 is provig that ay xtrmal graph has larg miimum dgr. Ulik th prvious stps, w shall ow dal xclusivly with th cas l = 1, as tratig rar colors will rquir cosidrably mor work. Lmma.4. For all itgrs r t 3 thr is a 0 such that th miimum dgr of G satisfis δ(g) 3/4 1 for all C r,t,1 -xtrmal graphs G with 0 vrtics. Proof. Assum that a -vrtx C r,t,1 -xtrmal graph G has a vrtx v with dgr d(v) < (3/4 1). Lt w 1,..., w /4 b vrtics i G that ar ot adjact to v. Dfi th graph G by addig th dgs {v, w 1 },..., {v, w /4 } to G. Th basic ida of th proof is to show that G admits mor raibow S t -fr r-colorigs tha G, ad w do this by showig that, if w compar th umbr of colorigs cratd ad lost with th additio of th w dgs, thr ar mor of th formr. To b mor prcis, giv a collctio (S 1,..., S ) of (t 1)-substs of [r], it is clar that w may xtd all colorigs i C (S1,...,S )(G) to C (S1,...,S )(G ) whvr S v S wi for all i {1,..., /4 }, as w may assig ay color i th corrspodig itrsctio to {v, w i } without producig a raibow star S t. Morovr, this xtsio may b do i svral ways, dpdig o th sizs of th itrsctios, which lads to w colorigs of G, as opposd to colorigs that ar i o-to-o corrspodc with colorigs of G. O th othr had, colorigs of G for which S v S wi = for som i may ot b xtdd i this way, ad w say that ths colorigs ar lost wh th w dgs ar addd. To fid a lowr boud o th umbr of colorigs cratd, cosidr oly thos dg colorigs of G whr vry dg is assigd a color from a fixd (t 1)-st S i [r]. Each such colorig ca b xtdd to at last (t 1) /4 (t 1) E(G) raibow-s t -fr colorigs of G by assigig a arbitrary color of S to ach w dg. This crats at last ( (t 1) /4 1 ) (t 1) E(G) w colorigs. O th othr had, th raibow-s t -fr r-colorigs of G that caot b xtdd to colorigs of G ar thos whr th sts of colors availabl at v ad at w i do ot itrsct, for som

7 FORBIDDING RAINBOW-COLORED STARS 7 i {1,..., /4 }. By Lmma.3 with ε = 1, th umbr of colorigs of G whr vry (t 1)-st of colors is assigd to at most A log t 1 vrtics of G is at most (t 1) E(G). Hc w coctrat o colorigs whr som (t 1)-st S appars at last A log t 1 tims. Th umbr of such colorigs is at most ( ) ( ) ( ) ( ) ( ) r r (t 1) r r A logt 1 (t 1) E(G), (7) t 1 t 1 t 1 A log t 1 t 1 sic thr ar ( r t 1) ways to choos Sv, a o-ighbor w i ca b chos i at most ways ad it is assigd a st S wi of colors with S wi S v =, which ca b do i ( ) r (t 1) t 1 ways. Th st S ca b chos i ( r t 1) ways, th vrtics which ar assigd th st S ca b chos i at most ( ) ( A log t 1 = A log t 1 ) ways ad vry rmaiig vrtx is associatd with som arbitrary (t 1)-st of colors. (Not that this uppr boud taks car of all th colorigs whr th st S is assigd to m vrtics, whr A log t 1 m.) Clarly, w hav ( A log t 1 ) A log t 1, ad, for larg ( ) ( ) ( ) ( ) r r (t 1) r r A logt 1 < A log t 1. t 1 t 1 t 1 t 1 W coclud from (7) that, for sufficitly larg, th umbr of raibow-s t -fr r-colorigs of G that caot b xtdd to such colorigs of G is at most A log t 1 (t 1) E(G) + (t 1) E(G) ((t 1) /4 1) (t 1) E(G). I othr words, by addig th dgs {v, w 1 },..., {v, w /4 } to G, w icras th total umbr of colorigs, which cotradicts th choic of G. W rmark that th prvious proof may b asily adaptd so that α = 3/4 is rplacd by ay fixd 0 < α < 1. W ar ow rady to prform th last stp i th proof of Thorm 1.1 for l = 1, which shows that, i a xtrmal graph G o dg may b missig. Thorm.5. For r t 3, thr xists 0 such that c r,t,1 () = C r,t,1 (K ) holds for 0. Morovr, K is th uiqu -vrtx C r,t,1 -xtrmal graph. Proof. Assum that thr is a C r,t,1 -xtrmal graph G = (V, E) o vrtics with at last two o-adjact vrtics x ad y. As i th proof of Lmma.4, w prov that G = G + {x, y} has mor raibow-s t -fr r-colorigs tha G if is sufficitly larg. By Lmma. w kow that may b chos so that E(G) ( ) D logt 1, whr D is a costat. Evry colorig of G for which oly (t 1) colors ar usd ca b xtdd, assigig ay of ths (t 1) colors to {x, y}, to a colorig of G, which icrass th total umbr of colorigs by (t ) (t 1) E(G). (8) W show that th umbr of all raibow-s t,1 -fr r-colorigs of G that caot b xtdd to a colorig of G is smallr tha (8). By Lmma.3 with A = A(r, t, l, D) ad ε = 1/, w kow that w may choos 0 such that th umbr of colorigs associatd with assigmts i S A is at most 1 (t 1) E(G). Thrfor, i th followig w oly d to cosidr colorigs from th st A = C (S1,...,S )(G). (9) (S 1,...,S ) S A Th oly colorigs of G that caot b xtdd to colorigs of G ar thos whr th color sts S x ad S y assigd to x ad y, rspctivly, ar disjoit, so that w ar uabl to assig

8 8 C. HOPPEN, H. LEFMANN, K. ODERMANN, AND J. SANCHES a color to {x, y}. Fix (S 1,..., S ) such that S is assigd to at last ( A log t 1 ) vrtics of G. Rcall that both vrtics x, y hav dgr at last 3/4 1 by Lmma.4. Th coditio o th dgrs implis that th commo ighbourhood N({x, y}) of x ad y has siz at last /. For ay vrtx w i N({x, y}) w hav S w (S x S y ) t 1. Mor prcisly, w hav S w S x = a w ad S w S y t 1 a w, so that thr ar at most a w (t 1 a w ) ((t 1)/) ways to assig colors to th dgs {x, w} ad {y, w}. Hc all dgs btw {x, y} ad thir commo ighbourhood N({x, y}) may b colord i at most ((t 1)/) N({x,y}) ways. This lads to th followig uppr boud o th umbr of lmts i (9) that caot b xtdd to a colorig of G. Th st S may b chos i ( r t 1) ways ad, for larg, thr ar ( A log t 1 ) < /4 ways of choosig A log t 1 vrtics which ar assigd S. For sufficitly larg, th rmaiig vrtics may b assigd color sts i at most ( r A logt 1 t 1) < /4 ways, ad w ifr that ( ) ( ) ( ) r r A logt 1 A A log t 1 t 1 t 1 ( ) ( r A log t 1 t 1 ( ) r / (t 1) E(G) t 1 1 ( ) r (t 1) E(G). t 1 ) A logt 1 +1 (t 1) E(G) (t 1) E(G) 4 N({x,y}) Altogthr, th umbr of all colorigs of G that caot b xtdd by addig dg {x, y} to th graph G is o mor tha 1 (t 1) E(G) + 1 ( ) r (t 1) E(G), t 1 which is smallr tha (8) for sufficitly larg. 3. Colorigs avoidig a raibow S t,l I this sctio, w cosidr th proof Thorm 1.1 for gral l. Although w oly prst it for l >, th cas l = may b tratd similarly, but with asir calculatios, as w just d to aalys a grdy algorithm. As w rmarkd bfor, th stratgy for achivig this rsult is xactly th sam as for th cas l = 1, but th prsc of rar colors will mak th argumts mor tchical. Rcall that first ad scod mai stps of th proof, amly showig that xtrmal graphs hav a larg umbr of dgs, ad that most colorigs hav th proprty that almost all vrtics hav th sam st of ordiary colors, hav alrady b provd for gral l (Lmmas. ad.3). To prform th rmaiig stps, w us th stratgy mployd i Lmma.4, ad show that th umbr of colorigs cratd xcds th umbr of colorigs lost wh dgs ar addd. To rach this coclusio, w d a lowr boud o th umbr of colorigs cratd, ad a uppr boud o th umbr of colorigs lost, with th proprty that th lowr boud is largr tha th uppr boud. Howvr, valuatig ths bouds will b hardr i this cas bcaus of th rar colors. To dscrib th mai igrdit dd to trat rar colors, first cosidr colorigs for which S 1 = = S, so that th rar colors ar th sam for all vrtics. I ay S t,l -fr r-colorig i C (S1,...,S )(G), th graph iducd by ach rar color has maximum dgr lss tha l, so w d to cout th umbrs of subgraphs of G of this typ. A classical rsult

9 FORBIDDING RAINBOW-COLORED STARS 9 of Bdr ad Cafild [3] implis that th umbr N (d) of subgraphs of K with dgr squc d = (d 1,..., d ), whr th compots d i ar boudd abov by som absolut costat d, satisfis N (d) (m)! xp( λ ( ) λ ) m m m! m d i! d i! xp( λ λ ), (10) whr m = d i ad λ = 1 ( di ) m, ad whr th asymptotics ar i (th scod approximatio uss Stirlig s formula). Hr A() B() mas that lim A()/B() = 1. For simplicity, ad giv that thr is ough room i our approximatios, w oft writ that ( ) m m N (d) = d i! xp( λ λ ) for larg. As it turs out, this rsult for K is sufficit for th uppr boud, as w may assum that ay dg dd i our costructio lis i th graph. Howvr, this may ot b do for th lowr boud, whr w d a approximat vrsio of Bdr ad Cafild s rsult. I th followig, giv a graph H o vrtics ad a itgr squc d = (d 1,..., d ), lt N H (d) b th umbr of subgraphs with dgr squc d i H. Mor grally, giv a array d = (d 1,..., d k ), lt N H ( d) b th umbr of ways of slctig a k-tupl (H 1,..., H k ) of dg-disjoit subgraphs of H such that ach subgraph H i has dgr squc d i, for all i {1,..., k}. I th followig, w say that a itgr squc d = (d 1,..., d ) is ds if d i. Lmma 3.1. Giv positiv itgrs d ad k, ad a costat D > 0, thr xist positiv costats 0, M ad α satisfyig th followig proprty for all 0. For vry graph H with V (H) = ad E(H) ( ) D l, thr xists W V (H) with W M l such that, for all ds dgr squcs d 1,..., d k {0,..., d} W, whr k k, w hav N H[W ] (d 1,..., d k ) α N W (d i ). Ituitivly, this lmma stats that, i ay graph with may dgs, thr is a almost spaig subgraph with a larg umbr of subgraphs of ay boudd dgr squc that is sufficitly ds. Not that this would trivially fail if w rquird W = V, as a my would b abl to isolat vrtics wh rmovig D log t 1 dgs of K to produc G, so that N G (d) = 0 for ay positiv squc d = (d 1,..., d ). Th proof of Lmma 3.1, which adapts idas of Gao [7], lis i Sctio 4. Lmma 3.. For all itgrs r t 3 ad l 1, thr is 0 such that ay C r,t,l -xtrmal graph G o 0 vrtics satisfis δ(g) 3/4 1. Proof. Assum that a C r,t,l -xtrmal -vrtx graph G = (V, E) has a vrtx v with dgr d(v) < 3/4 1. W suppos that is sufficitly larg for all stps i th proof to hold. Lt w 1,..., w /4 b /4 vrtics i G that ar ot adjact to v, ad lt G b th graph obtaid by addig all th dgs {v, w i } to G, i = 1,..., /4. Lt D > 0 such that G has at last ( ) D logt 1 dgs (Lmma.) ad fix A > 0 with th proprty of Lmma.3. W show that th umbr N of w colorigs of G obtaid by xtdig colorigs of G is largr tha th umbr N of colorigs of G that caot b xtdd to colorigs of G. By Lmma.3 with ε = 1/3, th umbr of colorigs i (S 1,...,S ) S A C (S1,...,S )(G) is at most (t 1) E(G) /3 for sufficitly larg. Lt N A b th umbr of w colorigs of G associatd with -tupls (S 1,..., S ) i S A, ad lt N A b th umbr of colorigs associatd with such collctios that caot b xtdd. I th rmaidr of th proof, w fid a lowr boud o N A ad a uppr boud k

10 10 C. HOPPEN, H. LEFMANN, K. ODERMANN, AND J. SANCHES o N A to show that N A N A. Morovr, it turs out that N A (t ) (t 1) E(G) (s (16)), so that ( ) ( N N N A (t 1) E(G) (t ) N A + 1 ) (t 1) E(G) > 0, 3 3 as rquird. Bfor procdig, lt 0, M ad α giv by Lmma 3.1 applid for D, d = l 1 ad k = r l + 1 (adjustig th costats so that th logarithms i th statmt of th lmma hav bas t 1), ad fix a st W V with W M log t 1 such that G[W ] satisfis th coclusio of th lmma. Uppr boud: W giv a uppr boud o th umbr N A of raibow-s t,l -fr r-colorigs of G that ar associatd with collctios (S 1,..., S ) S A ad caot b xtdd to a colorig of G. If S v S wi for vry i, th colorigs of C r,t,l,(s1,...,s )(G) ca b asily xtdd to colorigs of C r,t,l,(s1,...,s )(G ) usig ordiary colors for ach w dg, so that w may assum that th sts of colors availabl at v ad availabl at w i do ot itrsct for som i {1,..., /4 }. Howvr, ot that S v S wi = dos ot imply that thr is o color availabl for th dg {v, w i }, sic it could possibly b colord with o of th rar colors. For th sak of simplicity, ad sic w ar lookig for a uppr boud, w shall igor this fact ad assum that S v S wi = always maks it impossibl to color th dg {v, w i }. To costruct colorigs of this typ w do th followig. First, w fix a collctio (S 1,..., S ) S A with th rquird proprtis: (i) choos a (t 1)-subst S v [r]; (ii) choos a vrtx w that is ot adjact to v; (iii) choos a (t 1)-subst st S w [r], which is disjoit from S v, to b assigd to w; (iv) choos th (t 1)-subst S [r] that is assigd to at last A log t 1 vrtics of G; (v) choos A log t 1 vrtics that ar assigd this st S; (vi) assig ay (t 1)-substs i [r] to th A log t 1 rmaiig vrtics. Not that stps (i), (ii) ad (vi) allow us to choos S, so it might wll b that mor tha A log t 1 vrtics ar assigd S. Th umbr of choics for th sts S v, S w, S abov is at most ( r 3, t 1) whil is a uppr boud o th umbr of choics of w. Stps (v) ad (vi) may b prformd i ( ) ( A log t 1 r A logt 1 t 1) ways, so that a uppr boud o th umbr of ways of fixig a collctio (S 1,..., S ) with th rquird proprtis is ( ) r 3 ( ) ( ) r A logt 1 ( ) r A logt 1 +1 A log t 1. (11) t 1 A log t 1 t 1 t 1 Now assum that such a collctio (S 1,..., S ) is fixd. Lt Y = {u V : S u = S} ad lt H = G[W Y ]. W ow costruct colorigs i C (S1,...,S )(G). To this d, w procd as follows: (i) color dgs icidt with vrtics u V \ Y with rar colors with rspct to u; (ii) color dgs icidt with th rmaiig vrtics of (V \ W ) Y with rar colors; (iii) color dgs icidt with V \ Y with ordiary colors (with rspct to som dpoit i V \ Y ); (iv) color dgs i H with rar colors (with rspct to S); (v) color dgs icidt with vrtics with both ds i Y with ordiary colors (with rspct to S). Obsrv that th dgs = {u, v} such that u is assigd S, but v is ot, may b colord i (i) if is assigd a rar color with rspct to v or i (iii), if is assigd a ordiary color with rspct to v. For simplicity, w shall assum that dgs colord i (i) may b rcolord i (iii).

11 FORBIDDING RAINBOW-COLORED STARS 11 For stp (i), thr ar at most ( (l 1)(r t+1) ) A log t 1 ways of choosig (l 1) dgs icidt with ach such vrtx w for ach of th (r t + 1) rar colors. Not that w do ot d to cosidr th possibility that fwr dgs ar assigd such colors bcaus, with our stimats, th dgs colord at this poit could b rcolord i latr stps. Stp (ii) may b prformd i at most (l 1)(r t+1)m log t 1, whil stp (iii) may b prformd i at most (t 1) γ ways, whr γ is th umbr of dgs icidt with vrtics i V \ Y. To assig rar colors (with rspct to S) to th dgs of H, w us th followig procdur. Procdur 3.3. Suppos w hav a q-vrtx iput graph H = (V, E) ad a st T ( [r] t 1). Assum that H q,0 is a st of isolatd vrtics labld by V. For i {1,..., r t + 1}, choos a graph H q,i i E \ i 1 j=1 E(H q,j) with th dgr squc d i = (d 1 i,..., dq i ), whr dj i l 1 ad assig th ith color i [r] \ T to th dgs of H q,i. As w ar lookig for a uppr boud, w shall assum that W Y = W, possibly big abl to rcolor som dgs that hav b colord i prvious stps). For simplicity, assum that W = p. For i {1,..., r t + 1}, fix dgr squcs d i = (d 1 i,..., dp i ), whr th ith dgr squc is associatd with th ith rar color: w fid r t + 1 dg-disjoit subgraphs H 1,..., H r t+1 of H such that H i has dgr squc d i, i = 1,..., r t + 1. Not that N H (d 1,..., d r t+1 ) is th umbr of ways i which this ca b do. Th umbr of ways of prformig stps (iv) ad (v) is boudd abov by d N H ( d) (t 1) E(G) u( d) γ, whr th sum rags ovr th arrays d = (d 1,..., d r t+1 ), whr ach d i = (d 1 i,..., dp i ) has compots boudd by l 1. Morovr, w dot u( d) = 1 r t+1 p j=1 dj i. As a cosquc, th umbr of colorigs costructd abov is boudd abov by (l 1)(r t+1)a log t 1 (l 1)(r t+1)m log t 1 (t 1) γ d N H ( d) (t 1) E(G) u( d) γ. (1) It is asy to s that, choosig sufficitly larg, w may choos a arbitrarily small costat δ > 0 such that th product of quatios (11) ad (1) is at most (1 + δ) N H ( d) (t 1) E(G) u( d) = (1 + δ) N H ( d) (t 1) E(G) u( d), (13) d s d+ whr s: [r t + 1] {0, 1} is a fuctio that, for ach i, idicats whthr th dgr squc d i is ds (that is, wh s(i) = 1, th compots of d i sum to p or mor) or spars (that is, wh s(i) = 0, th compots of d i sum to lss tha p), whil d + ad d ar th arrays of ds ad spars dgr squcs, rspctivly, that crat a array d with th distributio dtrmid by s. W split quatio (13) accordig to whthr all rar colors grat ds graphs, or whthr this dos ot hold. For simplicity, w writ d + to say =j that w sum ovr arrays of lgth j whos dgr squcs ar all ds. W obtai (1 + δ) N H ( d + ) (t 1) E(G) u( d+) + N H ( d) (t 1) E(G) u( d).(14) d + =r t+1 s 1 d d+ d

12 1 C. HOPPEN, H. LEFMANN, K. ODERMANN, AND J. SANCHES Obsrv that th scod summatio may b stimatd as N H ( d) (t 1) E(G) u( d) d+ s 1 d r t ( p ) p(r t+1 j)/ r t+1 l p(r t+1) (r t+1)/ r t l p(r t+1) p p(r t+1 j)/ j=1 j=1 d + =j (t 1) E(G) u( d +) d + =j (t 1) E(G) u( d +) N H ( d + ) j N Kp (d i +). (15) To s why th first iquality is tru, ot that s rags i a st with at most r t+1 lmts, th umbr of arrays d = (d 1,..., d r t+1 j ) is boudd abov by l p(r t+1 j) ad N H ( d ) r t+1 j N Kp (d i ) ( ( p ) ) p/ r t+1 j (r t+1)/ p(r t+1 j)/ pp(r t+1 j)/, whr N Kp (d i ) is boudd usig (10) ad th fact that j dj i p. Lowr boud: Nxt w giv a lowr boud o th umbr N A of raibow-s t,l-fr r-colorigs w gai by addig th dgs {v, w 1 },..., {v, w /4 } to our xtrmal graph G. Th ida hr is that, if w cosidr colorigs i C r,t,l,(s,...,s) (G) for som fixd (t 1)-subst S of [r], thos colorigs ca b xtdd to colorigs of G by assigig ay of ths (t 1) colors to ach of th dgs {v, w i }, i = 1,... /4. Hc th umbr of colorigs of this typ for G is at last (t 1) /4 C r,t,l,(s,...,s) (G). Rmovig colorigs that ar xtsios of th corrspodig colorigs of G, th t gai of colorigs is ( ) (t 1) /4 1 C r,t,l,(s,...,s) (G). W ow fid a lowr boud o C r,t,l,(s,...,s) (G). To assig rar colors to dgs of G, w apply Procdur 3.3 to H = G[W ] ad T = S. Sic w ow d a lowr boud, w may ot suppos that G[W ] = K p, but Lmma 3.1 guarats that, for all k-tupls (d 1,..., d k ) such that k r t + 1 ad p j=1 dj i p, for all i, w hav N G[W ](d 1,..., d k ) α k N K p (d i ). With this, a lowr boud o th umbr of colorigs gaid by addig th dgs {v, w 1 },..., {v, w /4 } to G is ((t 1) /4 1) d (t 1) E(G) u( d) N G[W ] ( d), (16) whr w agai sum ovr arrays d = (d 1,..., d r t+1 ), whr ach d i = (d 1 i,..., dp i ) has compots boudd by l 1. To coclud our argumt, w compar th uppr boud (14) ad th lowr boud (16). W hav (14) (16) (1 + δ) d + =r t+1 N H( d + ) (t 1) E(G) u( d + ) ((t 1) /4 1) d (t 1) E(G) u( d) N G[W ] ( (17) d) + (1 + δ) l p(r t+1) j pp(r t+1 j)/ d+ (t 1) E(G) u( d + ) j [l] j N K p (d i +) ((t 1) /4 1) d (t 1) E(G) u( d) N G[W ] (.(18) d)

13 FORBIDDING RAINBOW-COLORED STARS 13 For th first trm i th sum, w us that, for positiv umbrs a 1, b 1,..., a s, b s with a i /b i Z for i = 1,..., s, th iquality a a s b b s Z. holds. By our choic of W, th trm (17) bcoms (1 + δ) d + =r t+1 N H( d + ) (t 1) E(G) u( d + ) ((t 1) /4 1) d (t 1) E(G) u( d) N G[W ] ( d) (1 + δ) d + =r t+1 N H( d + ) (t 1) E(G) u( d + ) ((t 1) /4 1) d + =r t+1 (t 1) E(G) u( d + ) N G[W ] ( d + ) (1 + δ) r t+1 N Kp (d i ) ((t 1) /4 1) α r t+1 N Kp (d i ) = α (1 + δ) ((t 1) /4 1) 1 4, (19) for sufficitly larg. For th scod trm (18), w writ (1 + δ) l p(r t+1) j pp(r t+1 j)/ d+ (t 1) E(G) u( d + ) j N K p (d i +) = (1 + δ) l p(r t+1) (t 1) /4 1 (1 + δ) l p(r t+1) (t 1) /4 1 ((t 1) /4 1) d (t 1) E(G) u( d) N G[W ] ( d) r t j=0 d + =j r t j=0 d + =j p p(r t+1 j)/ (t 1) E(G) u( d + ) j N K p (d i +) d (t 1) E(G) u( d) N G[W ] ( d) p p(r t+1 j)/ (t 1) E(G) u( d + ) j N K p (d i +) (t 1) E(G) u( d + ) (r t+1 j)p(l 1)/ N G[W ] ( d +, d ), (0) whr d is a array of r t+1 j dgr squcs qual to d = (l 1,..., l 1) (o of th trms l 1 may b rplacd by l i ths squcs to dal with parity costraits). Not that, to rach (0), w rplacd th domiator withi th sums by a sigl trm, which dpds o ach particular trm big addd. Our choic of W implis that, wh d + = j, j r t+1 j N G[W ] ( d +, d ) α N Kp (d i +) N Kp (d ), α ( (l 1)! p α p p(l 1)(r t+1 j)/ ((l 1)!) (l 1)p(r t+1) ( ) p(l 1) p(l 1)/ ( ) ) r t+1 j 1 (l 1) j xp N Kp (d i 4 +) j N Kp (d i +). (1) To fid a uppr boud o (0), w combi (1) with th fact that th umbr of trms i th sums is at most (r t + 1)l p(r t+1) ad that, for l, th trm p (r t+1 j)p( l)/ is maximizd for j = r t, which lads to th followig uppr boud o (0): α (1 + δ) l p(r t+1) (r t + 1) ( (t 1) (l 1)!) (r t+1)(l 1)p = (t 1) O() p ( l)p/ < 1 4 ((t 1) /4 1)p (l )p/

14 14 C. HOPPEN, H. LEFMANN, K. ODERMANN, AND J. SANCHES for larg, as p M log t 1. Thrfor, w obtai (14) (16) < 1, () for t 3 ad sufficitly larg, which implis that N A N A, as rquird. Thorm 3.4. For itgrs r t 3 ad l, thr xists 0 such that c r,t,l () = C r,t,l (K ) holds for 0. Morovr, K is th uiqu -vrtx C r,t,l -xtrmal graph. Proof. Fix r t 3 ad l. Assum that thr is a C r,t,l -xtrmal graph G o -vrtics with at last two o-adjact vrtics x ad y. W prov that G = G + {x, y} has mor colorigs tha G if is sufficitly larg. Our proof uss th stratgy mployd for Lmma 3.. By Lmma. w kow that 0 may b chos so that E(G) ( ) D logt 1, whr D = D(r, t) is a costat. Fix A > 0 with th proprty of Lmma.3. Lt 0, M ad α giv by Lmma 3.1 applid for D, d = l 1 ad k = r l + 1 (adjustig th costats so that logarithms i th statmt of th lmma hav bas t 1), ad fix a st W V with W M log t 1 = p such that G[W ] satisfis th coclusio of th lmma. W show that th umbr N of w colorigs of G obtaid by xtdig colorigs of G is largr tha th umbr N of colorigs of G that caot b xtdd to colorigs of G. By Lmma.3 with ε = 1/3, th umbr of colorigs i (S 1,...,S ) S A C (S1,...,S )(G) is at most (t 1) E(G) /3 for sufficitly larg. Lt N A b th umbr of w colorigs of G associatd with -tupls (S 1,..., S ) i S A, ad lt N A b th umbr of colorigs associatd with such collctios that caot b xtdd. I th rmaidr of th proof, w fid a lowr boud o N A ad a uppr boud o N A to show that N A N A. Oc agai w hav N A (t ) (t 1) E(G) (s (3)), which lads to th dsird rsult: ( N N N A N A + ) ( (t 1) E(G) (t ) 1 ) (t 1) E(G) > With th argumts usd for (16), w may show that vry colorig of G for which vry vrtx is assigd th sam st of (t 1) colors ca b xtdd to colorigs of G, which icrass th total umbr of raibow-s t,l -fr r-colorigs by at last (t ) d N G[W ] ( d) (t 1) E(G) u( d), (3) whr w agai sum ovr arrays d = (d 1,..., d r t+1 ), whr ach d i = (d 1 i,..., dp i ) has compots boudd by l 1. Rcall that u( d) = 1 r t+1 p j=1 dj i. W show xt that th umbr of all colorigs of G that caot b xtdd to a colorig of G is smallr tha (3). W giv a uppr boud o th umbr N A of raibow-s t,l -fr r-colorigs of G that ar associatd with collctios (S 1,..., S ) S A ad caot b xtdd to a colorig of G. All such colorigs hav th proprty that th color sts S x ad S y assigd to x ad y, rspctivly, ar disjoit. Fix (S 1,..., S ) for which this holds, ad whr S is assigd ( to at last ( A log t 1 ) vrtics of G. Th (t 1)-subst S [r] may chos i r ) ( t 1 ways ad thr ar A log t 1 ) ways of choosig ( A logt 1 ) vrtics which ar assigd S. Th rmaiig vrtics may b assigd (t 1)-lmt color sts i at most ( r A logt 1 t 1) ways. As a cosquc, ( ) ( ) ( ) r r A logt 1 ( ) r A logt 1 +1 A log t 1 (4) t 1 A log t 1 t 1 t 1

15 FORBIDDING RAINBOW-COLORED STARS 15 is a uppr boud o th umbr of ways of fixig a collctio (S 1,..., S ) with th rquird proprtis. W driv a uppr boud o th umbr of colorigs i C (S1,...,S )(G) that caot b xtdd to G. Sic x, y hav dgr at last 3/4 by Lmma 3., thir commo ighbourhood N({x, y}) has siz at last /. For ay vrtx w i N({x, y}) w hav S w (S x S y ) t 1. Mor prcisly, w hav S w S x = a w ad S w S y t 1 a w, so that thr ar at most a w (t 1 a w ) ((t 1)/) ways to assig ordiary colors to th dgs {x, w} ad {y, w}. Hc all dgs btw {x, y} ad thir commo ighbourhood N({x, y}) may b colord i at most ((t 1)/) N({x,y}) ways with ordiary colors (with rspct to both ds). As i th proof of Lmma 3., w procd as follows: lt Y = {u V : S u = S} ad lt H = G[W Y ], (i) color dgs icidt with vrtics u V \ Y with rar colors with rspct to u; (ii) color dgs icidt with vrtics of (V \ W ) Y with rar colors; (iii) color dgs icidt with V \ Y with ordiary colors (with rspct to som dpoit i V \ Y ); (iv) color dgs i H with rar colors (with rspct to S); (v) color dgs icidt with ach vrtx i Y with ordiary colors (with rspct to S). W obtai th followig uppr boud o th umbr of colorigs of G that caot b xtdd to G : ( ) r A logt 1 +1 A log t 1 (l 1)(r t+1)a log t 1 (l 1)(r t+1)m log t 1 t 1 ( ) N H ( t 1 N(x,y) d) (t 1) E(G) u( d) N(x,y). (5) d It is asy to s that, choosig sufficitly larg, quatio (5) is at most / ( ) N H ( t 1 N(x,y) d) (t 1) E(G) u( d) N(x,y), (6) d which may b rwritt as / s d d+ N H ( d) (t 1) E(G) u( d) N(x,y), (7) whr, as i (13), s: [r t + 1] {0, 1} is a fuctio that, for ach i, idicats whthr th dgr squc d i is ds or spars, whil d + ad d ar th arrays of ds ad spars dgr squcs, rspctivly, that crat a array d with th distributio dtrmid by s. W split quatio (7) accordig to whthr all rar colors grat ds graphs, or whthr this dos ot hold, which lads to / d + =r t+1 d + =r t+1 N H ( d + ) (t 1) E(G) u( d + ) N(x,y) + s 1 N H ( d + ) (t 1) E(G) u( d + ) / + s 1 d d d+ N H ( d) d+ N H ( d) (t 1) E(G) u( d) N(x,y) (t 1) E(G) u( d) /. (8)

16 16 C. HOPPEN, H. LEFMANN, K. ODERMANN, AND J. SANCHES Usig th stimat (15) w hav s 1 d d+ N H ( d) r t l p(r t+1) p p(r t+1 j)/ j=1 (t 1) E(G) u( d) / d + =j (t 1) E(G) u( d + ) / j N Kp (d i +). (9) To coclud our argumt, w compar th uppr boud (8) ad th lowr boud (3). W hav (8) d + =r t+1 N H( d + ) (t 1) E(G) u( d + ) (3) / (t ) d (t 1) E(G) u( d) N G[W ] ( (30) d) + lp(r t+1) j pp(r t+1 j)/ d + =j (t 1) E(G) u( d +) j N K p (d i +) / (t ) d (t 1) E(G) u( d) N G[W ] (. (31) d) Usig th stimats of (19), th trm (30) may b boudd by d + =r t+1 N H( d + ) (t 1) E(G) u( d + ) / (t ) d + =r t+1 (t 1) E(G) u( d +) N G[W ] ( d + ) α / (t ) 1 4 For th scod trm (31), w rpat th argumts usd i th proof of Lmma 3.. Thrfor, w obtai for t 3 ad sufficitly larg, as rquird. (8) (3) < 1, (3) 4. Auxiliary rsults To coclud our papr, w prov Lmma 3.1. To do this, w d som prlimiary dfiitios ad rsults, which ar basd o th idas of Gao [7]. Lt (H ) 1 b a squc of graphs o vrtics ad lt (H ) dot th umbr of dgs i H. Color all dgs i H blu ad all dgs i its complmt H rd, so that K is th complt graph o vrtics with a -colorig K = H H. For ay rd dg x = uv, dfi th rd dgr of x d r (x) = d r (u) + d r (v), whr d r (u) dots th umbr of rd dgs icidt with u. Lt r (H ) = max x E(K) d r (x), whr w maximiz ovr rd dgs. Fix a dgr squc d = (d 1,..., d ), whr max d i d for som absolut costat d N, ad lt F b a graph with dgr squc d = (d 1,..., d ) chos uiformly at radom from all subgraphs of K with this dgr squc. Lt X = X (F ) b th radom variabl that accouts for th umbr of rd dgs i F. Rcall that Lmma 3.1 couts th umbr of subgraphs (with a giv dgr squc) of a complt graph that rmai aftr a my dlts a crtai umbr of dgs to produc a subgraph G. I th currt sttig, dgs that rmai ar rprstd by blu, whil dgs dltd ar rprstd by rd. As a cosquc, tratig subgraphs of th complt graph that ar ot affctd by th dltio of dgs is th sam as tratig colord subgraphs of a -colorig of K whos dgs ar all blu.

17 FORBIDDING RAINBOW-COLORED STARS 17 Th followig rsult is th mai igrdit i th proof of Lmma 3.1, as it givs a lowr boud o th probability of choosig a subgraph with o rd dgs for ay -colorig such that th rd dgr is small. Lmma 4.1. Fix a ds squc d = (d 1,..., d ) such that max d i d. Lt H b a graph o vrtics ad dot t = ( ) (H ). Cosidr th radom variabl X dfid abov. Assum a = lim sup r (H )/ < 1/d. Th, providd that t / 0 for, ( 9d ) t P(X = 0) xp. (1 da) Bfor provig this rsult, w show that it implis Lmma 3.1. First, giv a positiv itgr k, lt Y = Y (F ) dot th umbrs of rd dgs cotaid i a k-tupl F = (F, 1..., F k ) of dg-disjoit subgraphs of H such that F i has dgr squc d i boudd by d, i = 1,..., k. W assum that ths graphs ar chos squtially, uiformly at radom from th rmaiig graph. Corollary 4.. Fix a positiv itgr d ad ds squcs d 1 = (d 1 1,..., d1 ),..., d k = (d k 1,..., dk ) such that max d j i d. Lt H b a graph o vrtics ad lt t = ( ) (H ). Cosidr th radom variabl Y dfid abov. Assum a = lim sup r (H )/ < 1/d. Th, providd that t / 0 for, ( 9d ) t k P(Y = 0) xp. (1 da) Proof. W itrat Lmma 4.1. Lt H 1 = H b a graph satisfyig th hypothss, ad choos F 1 uiformly at radom from all subgraphs of K = H 1 H 1 with dgr squc d 1, so that, by Lmma 4.1, ( ) P(X 1 9d t (1) = 0) xp, (1 da) whr X 1 = X (F) 1 ad t (1) = t. Assum that such a graph F 1 has b chos so that X 1 = 0. Cosidr th graph H = H 1 \ E(F) 1 ad lt t () = ( ) (H ). I othr words, w cosidr a w colorig of K i which th dgs i F 1 ar rcolord rd. Not that t () t +d/ ad r (H) r (H)+d. 1 Thrfor t () / is sufficitly small for larg ad a = lim sup r (H)/ = a bcaus d/ 0 wh. As a cosquc, if w choos F uiformly at radom from all subgraphs of K = H H with dgr squc d, w may apply Lmma 4.1 W apply it to th w colorig iducd by H, which w hav show to satisfy th hypothss of Lmma 4.1. to obtai ( ) P(X = 0 X 1 9d t () = 0) xp, (1 da) whr X = X (F). Obsrv that th probability obtaid by th lmma is a coditioal probability, bcaus w choos F aftr of F 1 has alrady b chos. Itrativly, for 3 i k, w coditio upo havig chos F, 1..., F i 1 with th corrspodig dgr squcs, all of which cotaiig o rd dgs of th origial graph K. W lt H i = H i 1 \ E(F i 1 ), t (i) = ( ) (H i ) ad a i = lim sup r (H)/ i = a, ad w choos F i uiformly at radom from all subgraphs of K = H i H i with dgr squc d i. If w lt X i = X (F), i w obtai P ( X i = 0 X 1 =... = X i 1 = 0 ) xp ( 9d t (i) (1 da) ).

18 18 C. HOPPEN, H. LEFMANN, K. ODERMANN, AND J. SANCHES As a cosquc, P(Y = 0) = P as rquird. ( ) X 1 =... = X k = 0 = P(X 1 = 0) P(X = 0 X 1 = 0)... P(X k = 0 X 1 =... = X k 1 = 0) ( 9d t 1 ) ( 9d t k ) xp xp (1 da) (1 da) ( 9d ) t k xp, (33) (1 da) Usig th ida that th rd dgs ar th dgs missig from th graph G to tur it ito a complt graph, w prov Lmma 3.1, which w ow rstat. Lmma 3.1. Giv positiv itgrs d ad k, ad a costat D > 0, thr xist positiv costats 0, M ad α satisfyig th followig proprty for all 0. For vry graph H with V (H) = ad E(H) ( ) D l, thr xists W V (H) with W M l such that, for all ds dgr squcs d 1,..., d k {0,..., d} W, whr k k, w hav N H[W ] (d 1,..., d k ) α N W (d i ). Proof. Fix d, k, ad D as i th statmt of th lmma, ad lt H b a graph o vrtics with E(H) ( ) D l. As bfor, w shall assum that sufficitly larg. Cosidr th st { A = v V (G): d(v) 3d 1 } 3d, so that A 3d 1 + ( A ) ( 1) ( 1) D l 3d = A ( ) 3d + 1 D l k 6Dd l = A M l 3d for M = 7dD ad sufficitly larg. Lt W = V (H)\A ad fix boudd dgr squcs d 1,..., d k {0,..., d} W whr k k. W wish to apply Corollary 4. to th graph H W = H[W ] o W vrtics, whr th dgs i H W ar prcisly th blu dgs of K W. To this d, lt a = lim sup W r (H W )/ W ad t W = ( ) W (H W ). W d to prov that a < 1/d ad that t W / W 0 as W. Obsrv that, for ay vrtx v i W, th umbr of rd dgs icidt with v i K W is prcisly th umbr of dgs icidt with v i th complmt of H[W ] (with rspct to K W ), which is at most th umbr of dgs icidt with v i th complmt of H (with rspct to K ). By our choic of W, this is at most /(3d), ad w hav r ( W ) /(3d). Thus for larg, r (H W ) /(3d) W M l 4 5d < 1 d. Morovr, (H W ) ( ) D logt 1 M log t 1 = ( ) (D + M) logt 1, for larg. W hav ( ) W t W = (H W ) (D + M) log t 1,

19 FORBIDDING RAINBOW-COLORED STARS 19 so that t W / W 0 ad Corollary 4. applis to H W ad th dgr squcs d 1,..., d k. Sic k k, ( ) 9kd t W P(Y = 0) xp. (1 da) W Bcaus, W M log t 1 ad t W (D + M) log t 1, aftr som straighforward calculatios, w obtai P(Y = 0) β, for larg, whr β = 51(D+M)d k l(t 1), w obtai 10(D+M)d k (1 da) l(t 1) 50(D+M)d k l(t 1), as a 4/(5d). Hc, with α = N H[W ] (d 1,..., d k ) α N W (d i ). To coclud this sctio, w prov Lmma 4.1. Th followig switchig opratios will b particularly usful. W us th otatio itroducd at th bgiig of this sctio, whr w hav a complt graph K whos dg st is -colord with rspct of a giv subgraph H. (i) r-switchig: Giv a graph F cotaiig at last o rd dg, choos a rd dg x F, labl its d vrtics u ad v, ad choos a blu dg x F that is ot icidt with x, labl its d vrtics u ad v. Rplac ths two dgs by uu ad vv. Th r-switchig is applicabl if ad oly if uu ad vv ar blu dgs ad ar ot i F. (ii) ivrs r-switchig: Giv a graph F cotaiig at last two blu dgs, choos a blu dg i F ad labl its d vrtics u ad u, th choos aothr blu dg i F that is ot icidt with uu ad labl its d vrtics v ad v. Rplac ths two dgs by uv ad u v. Th ivrs r-switchig is applicabl if ad oly if uv ad u v ar ot i F ad uv is rd ad u v is blu. Lt d = (d 1,..., d ) b a array of ds squcs such that max d i d. Lt R(l) b th st of all subgraphs F of K with dgr squc d ad l rd dgs. Not that, for vry l 1, a r-switchig opratio covrts a graph F R(l) ito a graph F R(l 1). O th othr had, a ivrs r-switchig covrts a F R(l 1) ito a F R(l). Lt N(F ) b th umbr of r-switchigs applicabl o F ad N (F ) th umbr of ivrs r-switchig applicabl o F. Lt t = ( ) (H ) ad lt c = t /. Proof of Lmma 4.1. For th calculatios prstd hr, w furthr suppos that m = i d i. Th cas m < / must b tratd sparatly. Lt b = (1 da)/8. By th dfiitio of lim sup, thr xists 0 > 0 such that, for all 0, r () < a + 1 da d k = 1 4b, ad b 4d. d Claim 4.3. For all l such that m/d l/d 4d r > 0, ad giv F R(l) ad F R(l 1), w hav l(m l 4d d r ) N(F ) 4lm, 0 N (F ) d c. Proof. Giv F R(l), th umbr of ways to choos th rd dg x ad labl its d vrtics is l. Th umbr of ways to choos th blu dg y ad labl its vrtics is at most m. So N(F ) 4lm. For th lowr boud, oc x has b chos ad its vrtics hav

20 0 C. HOPPEN, H. LEFMANN, K. ODERMANN, AND J. SANCHES b labld, th umbr of ways to choos y ad labl its d vrtics such that y is a blu dg that is ot icidt with x, ad whr uu ad vv ar both blu dgs that do ot li i F, is at last m l 4d d r. Idd, havig chos th rd dg uv, th vrtics u, v caot b choos from ay of th at most r dpoits of th rd dgs lavig u or v. Morovr, w caot choos th dpoits of th at most d dgs i F lavig u or v. Thus, w caot choos at most d( r + d) dgs to sur th right coctio btw u, v ad th pair {u, v }. Th boud follows wh w tak ito accout that w d to labl u ad v ad that u v caot b o of th l rd dgs. Giv F R(l 1), th umbr of ways to choos th rd dg x ad labl its dvrtics is at most c. Morovr, thr ar at most d ways to choos u ad d ways to choos v, so that N (F ) d c. I our cas, th assumptio that m / implis with l b that ( ) 1 4b m l 4d d r m b b d b. d Giv l b, cosidr th auxiliary bipartit graph with bipartitio R(l) R(l 1) whr a lmt i R(l) is adjact to a lmt i R(l 1) if thy ca b obtaid from ach othr by switchig opratios. By coutig th umbr of dgs i this auxiliary graph, usig Claim 4.3, w obtai lb R(l) d c R(l 1), so that R(l) R(l 1) d c bl, ad hc ( R(l) d ) l R(0) c 1 b l!. W shall also mak us of th followig fact about th probability of F havig may rd dgs. Claim 4.4. P(X b) = o(1). Bcaus m s=0 P(X = s) = 1 ad P(X b) = o(1) by th abov claim, w hav s=0 P(X = s) = 1 o(1). Thus, b 1 P(X = 0) = (1 + o(1)) (1 + o(1)) (1 + o(1)) b s=0 b s=0 b s=0 (1 + o(1)) xp P(X = s) P(X = 0) R(s) R(0) ( d ) l c 1 b s! ( d c b Not that this implis th validity of Lmma 4.1, as P(X = 0) 1 ( ) xp d c = 1 ( 8d ) b xp t (1 da) for larg. ). (34) ( 9d ) t > xp (1 da)

21 FORBIDDING RAINBOW-COLORED STARS 1 Proof of Claim 4.4. Thr ar at most c rd dgs i K, so thr ar at most ( c ) b ways to choos b rd dgs to iclud i F. W d to choos th m b rmaiig dgs to form F. To fid a uppr boud o th umbr of ways i which this ca b do, w cosidr th followig way of producig subgraphs of K with dgr squc (d 1,..., d ), whr d i = m: cosidr bis labld 1,..., ad put (labld) balls ito ths bis i such a way that th ith bi cotais d i balls. A pairig of th st of balls is giv by P = {a 1,..., a m } such that ach a i is a uordrd pair of balls, ad ach ball is i prcisly o pair a i. Ay such pairig givs ris to a (multi)graph with dgr squc (d 1,..., d ) by thikig of th ith bi as a vrtx v i, ad statig that v i ad v j ar adjact whvr balls i ths bis ar paird to ach othr. Th raso why w d to mtio multigraphs is that, i a pairig, two balls i th sam bi could b paird (which would produc a loop), or two or mor balls i o bi could b paird to balls i som othr bi (which would produc multipl dgs). It is ot difficult to s that ach (simpl) graph with dgr squc (d 1,..., d ) corrspods to th sam umbr of pairigs (amly d 1! d!), which is th basic ida i th cofiguratio modl for radom rgular graphs (s [5, 1] for a dtaild xplaatio.) I our cotxt, sic w hav alrady chos b rd dgs to iclud i F, w may assum that, wh producig F usig pairigs, w hav paird b balls, so that w d to fid a pairig of th m b rmaiig balls. As th umbr of ways of doig this is th umbr of prfct matchigs i K m b, w coclud that it may b do i at most ( ) (m b)! m b m b m b (m b)! = (1 + o(1)) ways. Th approximatio usd Stirlig s formula. Cosqutly, th umbr of subgraphs of K with dgr squc d that cotai at last b rd dgs is for larg at most ( ) ( ) c m b m b. b is W kow that th umbr of such subgraphs with o rstrictio o th umbr of rd dgs N (d) = d i! ( ) m m xp( λ λ ). With d i d, i = 1,..., ad λ = (1/(m)) ) i < d w ifr that ( ) m m N (d) > (d!) xp( d d 4 ). Thrfor, usig ( k) (/k) k, w hav P(X b) ( c b ( di ) ( m b N (d) ) m b ) m b ( ) c b ( b m b ) m xp( d d 4 ) (d!) ( m ( c ) b b (d!) xp(d + d 4 ) ( m ) b = ( ) b c (d!) xp(d + d 4 ), bm