EDGE-COLORINGS AVOIDING RAINBOW STARS

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1 EDGE-COLORINGS AVOIDING RAINBOW STARS CARLOS HOPPEN, HANNO LEFMANN, KNUT ODERMANN, AND JULIANA SANCHES Abstract. We cosider a extremal problem motivated by a paper of Balogh [J. Balogh, A remark o the umber of edge colorigs of graphs, Europea Joural of Combiatorics 7, 006, ], who cosidered edge-colorigs of graphs avoidig fixed subgraphs with a prescribed colorig. More precisely, give r t, we look for -vertex graphs that admit the maximum umber of r-edge-colorigs such that at most t 1 colors appear i edges icidet with each vertex, that is, r-edge-colorigs avoidig raibow-colored stars with t edges. For large, we show that, with the exceptio of the case t =, the complete graph K is always the uique extremal graph. We also cosider geeralizatios of this problem. 1. Itroductio We cosider edge-colorigs of graphs that satisfy a certai property. Give a umber r of colors ad a graph F, a r-patter P of F is a partitio of its edge set ito r (possibly empty classes. A edge-colorig (ot ecessarily proper of a host graph G is said to be (F, P -free if G does ot cotai a copy of F i which the partitio of the edge set iduced by the colorig is isomorphic to P. If at most r colors are used, we call it a (F, P -free r-colorig of G. For example, if the patter of F cosists of a sigle class, o moochromatic copy of F should arise i G. We ask for the -vertex host graphs G (amog all -vertex graphs which allow the largest umber of (F, P -free r-colorigs. Questios of this type have bee first cosidered by Erdős ad Rothschild [8], who asked whether cosiderig edge-colorigs avoidig a moochromatic copy of F would lead to extremal cofiguratios that are substatially differet from those of the Turá problem. Ideed, F -free graphs o vertices are atural cadidates for admittig a large umber of colorigs, sice ay r-colorig of their edge set obviously does ot produce a moochromatic copy of F (or a copy of F with ay give patter, for that matter, so that (Turá F -extremal graphs admit r ex(,f such colorigs, where, as usual, ex(, F is the maximum umber of edges i a -vertex F -free graph. Erdős ad Rothschild [8] cojectured that, for every l 3 ad > 0 (l, ay -vertex graph with the largest umber of K l -free -colorigs is isomorphic to the (l 1-partite Turá graph, which was prove for l = 3 by Yuster [17] ad for l 4 by Alo, Balogh, Keevash, ad Sudakov [1], who also showed that the same coclusio holds i the case r = 3. However, for r 4 colors, the Turá graph for K l is o loger optimal, ad the situatio becomes much more complicated; i fact, extremal cofiguratios are ot kow uless r = 4 ad F {K 3, K 4 }, see Pikhurko ad Yilma [15]. A similar pheomeo, i which (Turá extremal graphs admit the largest umber of r-colorigs if r {, 3}, but do ot for r 4, has bee observed for a few other classes of graphs ad hypergraphs, such as the 3-uiform Fao plae [14]. (See [11] for a more detailed accout of istaces where this pheomeo is kow to hold. This work was partially supported by CAPES ad DAAD via Probral (CAPES Proc. 408/13 ad DAAD ad The first author also ackowledges the support by CNPq (Proc /014- ad /01- ad FAPERGS (Proc /14-9. The fourth author was fuded by CNPq. Some results of this paper appeared i the Proceedigs of the VII Lati-America Algorithms, Graphs ad Optimizatio Symposium (015. 1

2 C. HOPPEN, H. LEFMANN, K. ODERMANN, AND J. SANCHES Balogh [4] was the first to cosider colorigs avoidig fixed patters that are ot moochromatic. More precisely, he showed that the (l 1-partite Turá graph is still optimal for r = colors whe forbiddig ay -patter of K l. O the other had, he observed that this does ot hold i geeral for r = 3 colors ad arbitrary 3-patters of K l. Ideed, cosider F = K 3 ad let P be a partitio of K 3 ito three classes cotaiig oe edge each, so that we are lookig for 3-colorigs with o raibow triagle. If we color the complete graph K with ay two of the three colors available, there is o raibow copy of K 3, which gives 3 ( 3 distict (K 3, P -free colorigs of K, ad is more tha 3 ex(,k3 3 /4. This suggests that the study of colorigs that avoid geeral patters, ad i particular raibow patters, deserves more attetio. I coectio with this, we should metio that it was recetly prove that, for large, the complete graph is ideed optimal for raibow triagles [6] i the case r = 3. O the other had, the balaced, complete bipartite Turá graph was show to be optimal [13] for all r 10. There has also bee a extesive descriptio of extremal graphs whe oe cosiders matchigs with various forbidde patters [1], which icludes all raibow cases. Stars have played a importat rôle i these developmets. Moochromatic stars F = S t with t 3 edges were the first istaces for which it was show [11] that F -extremal graphs (i this case, (t 1-regular graphs for eve do ot admit the largest umber of r-colorigs with o moochromatic copy of F for all values of r. I particular, this implies that the trasitio betwee the cases r {, 3} ad r 4 described above does ot hold for all graphs F. O the other had, extremal -vertex graphs for forbidde moochromatic stars S t are ot yet kow for ay r ad t 3. I this paper, our iitial motivatio was to study r-colorigs that avoid raibow-colored stars S t, that is, we let F = S t ad we cosider the patter where each edge is i a differet class (i particular, r t. Colorigs of this type have bee itroduced i the cotext of Ramsey Theory by Gyárfás, Lehel, Schelp ad Tuza [10], who called them local (t 1- colorigs. I other words, the iitial questio was to fid the -vertex graphs that admits the largest umber of local (t 1-colorigs usig at most r-colorigs. For t = ad ay give umber of colors r, it is easy to see that a matchig of size / yields the largest umber of r-colorigs with o raibow S, as this restrictio implies that ay colorig must have moochromatic compoets (for odd, both a additioal isolated vertex ad a coected compoet with three vertices geerate extremal cofiguratios. The same extremal cofiguratio had bee observed for moochromatic S whe r =, but ot for larger values of r. Note that the set of r-colorigs avoidig a moochromatic S is precisely the set of proper r-edge-colorigs of a graph, ad hece the problem would cosist of fidig -vertex graphs with the largest umber of proper colorigs. However, i cotrast to the moocromatic case, we were able to fid the optimal cofiguratio i the raibow case for large ad every fixed r, t 3, which is always the complete graph K. Sice the techiques used to prove this result may be adapted to other patters, we state our results i greater geerality. I particular, to derive this geeralizatio, we show that i ay graph with may edges, there is a almost spaig subgraph with a large umber of subgraphs of ay bouded degree sequece satisfyig a desity costrait, which seems to be of idepedet iterest (see Lemma 3.1 for a precise formulatio. Give positive itegers t ad l 1 l t, a raibow-s l1,...,l t is a edge-colored star with t i=1 l i edges such that t pairwise distict colors c 1,..., c t have the property that c i is assiged to exactly l i edges. For a host graph G, a raibow-s l1,...,l t -free r-colorig of a graph G is a edge-colorig of G with colors i [r] = {1,..., r} for which there is o raibow S l1,...,l t. A importat special case occurs whe l 1 = = l t = l for some costat l 1, whe we refer to a raibow-s t,l ad to raibow-s t,l -free r-colorigs of G. Clearly, if l = 1, we forbid raibow stars S t, ad we call such colorigs raibow-s t -free. This coditio resembles the cocept of a m-good colorig, which was itroduced by Alo, Jiag, Miller ad Pritiki []

3 EDGE-COLORINGS AVOIDING RAINBOW STARS 3 ad cosists of a edge-colorig where each color appears at most m times at ay vertex. Hece, we cosider colorigs such that each vertex is (l 1-good for all but (t 1 colors. For ay host graph G, let C r,l1,...,l t (G be the set of all raibow-s l1,...,l t -free r-colorigs of G. We write c r,l1,...,l t ( = max { C r,l1,...,l t (G : V (G = }, ad we say that a -vertex graph G is C r,l1,...,l t -extremal if C r,l1,...,l t (G = c r,l1,...,l t (. We prove the followig result. Theorem 1.1. For all r, t 3 ad l 1 l t 1, there exists 0 such that, for all 0, we have c r,l1,...,l t ( = C r,l1,...,l t (K. Moreover, the complete graph K is the sigle C r,l1,...,l t -extremal graph o vertices. As it turs out, this result will be a direct cosequece of the followig special case, where we cosider raibow-s t,l -free r-colorigs of G. For simplicity, let C r,t,l (G deote the set of raibow-s t,l -free r-colorigs of G, ad let c r,t,l ( be the correspodig extremal umber. Theorem 1.. For all r, t 3 ad l 1, there exists 0 such that, for all 0, we have c r,t,l ( = C r,t,l (K. Moreover, the complete graph K is the sigle C r,t,l -extremal graph o vertices. Note that the case t = 1 ad l 3 deals with colorigs avoidig moochromatic stars S l, which have bee cosidered i [11] ad for which the extremal cofiguratios are ot kow, eve whe r =. Moreover, our proof of Theorem 1. caot be exteded to the case t =, but we cojecture that the complete graph is also extremal i this case. The remaider of this work is orgaized as follows. I Sectio, we deal with some easy cases ad we prove our result for raibow stars, which gives a overview of the geeral case. The proof of Theorems 1. ad 1. is the subject of Sectio 3. Sectio 4 deals with the proof of our mai auxiliary result. We coclude the paper with ope questios.. Colorigs avoidig a raibow star The mai objective of this sectio is to prove Theorem 1. i the case l = 1. This case was the mai motivatio for our work ad, as it turs out, its proof gives a accurate overview of the geeral case. Before proceedig with the proof, we first deal with some straightforward cases. Recall that, for t = ad l = 1, the r-colorigs of a graph G avoidig a raibow-s are such that adjacet edges have the same color. I such a graph edges i the same compoet have to be colored the same, but edges i differet compoets might be colored differetly. Thus, if the graph G has j compoets cotaiig at least oe edge, we have C r,,1 (G = r j. I order to maximize this, j has to be as large as possible. Hece, the umber of such colorigs is at most C r,,1 (M, where M is a maximum matchig i G. So the oly C r,,1 -extremal graphs o vertices are for eve a matchig of size /, ad, for odd, a matchig of size ( 1/ ad a isolated vertex, or a matchig of size ( 3/ ad a vertex-disjoit coected compoet o three vertices. Clearly i this case we have c r,,1 ( = r. For r < t, o r-colorig ca produce a raibow-s t,l, so that c r,t,l ( = r (, ad K is the oly C r,t,l -extremal graph. Usig this argumet more carefully, we may exted this coclusio for some additioal values of r ad t. Lemma.1. Let t 3 r t 3, l 1 ad be positive itegers. The c r,t,l ( = C r,t,l (K ad the complete graph K is uique with this property amog all -vertex graphs. Proof. Let G be a -vertex graph where some edge e = {v, w} is missig. Cosider a fixed raibow-s t,l -free r-colorig of G. We show that we ca exted to a colorig of

4 4 C. HOPPEN, H. LEFMANN, K. ODERMANN, AND J. SANCHES G = G + e. Let S v ad S w be the sets of colors occurig o at least l edges icidet with v ad w, respectively, hece S v, S w t 1. If S v S w, the we ca exted to G by colorig e with ay color i S v S w. Now let S v S w =, i particular S v + S w r t 3. If t 1 = S v > S w, we ca color e with ay color i S v. If S v, S w < t 1, the we ca color e with ay color. I coclusio, ca be exteded to a colorig of G. To fiish the proof, we show that at least oe of the colorigs of G may be exteded to G i more tha oe way. As t 3, ay moochromatic colorig of G ca be exteded to a colorig of G + e by colorig e with ay color, so that C r,t,l (G < C r,t,l (G + e. Remark: The proof of Lemma.1 also yields the followig for ay r t 3 ad l. If r(l 1/ + l(t 1 + 1, the it is possible to exted ay colorig to a missig edge {v, w}, eve if S v S w = ad S v = S w = t 1. Ideed, if a colorig caot be exteded uder such coditios, all colors i S v must have appeared for at least l edges icidet with v, ad the same holds for w. Moreover, the colors i S v must have bee assiged to exactly l 1 edges icidet with w, ad vice-versa. Moreover, ay color i S v S w must appear at l 1 edges icidet with v or l 1 edges icidet with w, otherwise it could be used to exted the colorig. However, the degrees of v ad w (which are at most are too small for all of these coditios to hold because of our boud o. We remark that the case of r > t 3 is cosiderably more ivolved. Before dealig with the geeral case, we first focus o the proof of Theorem 1. i the case l = 1. The geeral idea of the proof is as follows. Cosider a fixed raibow-s t -free r-edge colorig of a graph G. By defiitio, for every vertex v of G, the umber of colors appearig o edges icidet with v is at most (t 1. For sets S 1,..., S [r], let C r,t,(s1,...,s (G deote the set of all edge-colorigs of G where o edges icidet with vertex v i are assiged colors from the set [r]\s i, for all i = 1...,. The the set C r,t (G of all raibow-s t,l -free r-colorigs of G satisfies C r,t (G = C r,t,(s1,...,s (G. (1 (S 1,...,S S i =t 1,i=1,..., Observe that the uio is ot disjoit, as fewer tha t 1 colors could appear i edges icidet with some vertex. Before proceedig, ote that this decompositio ca be easily geeralized to ay l 1. The differece, for a fixed raibow-s t,l -free r-edge colorig of a graph G, is that the sets S i cotai the colors that appear at least l times i edges icidet with v i, which we call ordiary colors with respect to v i, while the remaiig colors are said to be rare for v i. I aalogy to the above case, C r,t,l,(s1,...,s (G deotes the set of all edge-colorigs of G where fewer tha l edges icidet with vertex v i are assiged each color from the set [r] \ S i, for all i = 1...,. As i (1, the set C r,t,l (G of all raibow-s t,l -free r-colorigs of G satisfies C r,t,l (G = C r,t,l,(s1,...,s (G. ( (S 1,...,S S i =t 1,i=1,..., Our proof cosists of four steps. We first show that ay extremal graph must have a lot of edges, as otherwise it caot beat the umber of colorigs achieved by the complete graph. Next we prove that most colorigs i (1 arise from the cases whe almost all sets S i are the same. Usig these facts, we ca prove that extremal graphs have large miimum degree, which, i the last step, allows us to prove that ay extremal graph coicides with K. The followig lemma is the first step i the above descriptio, which may be easily proved for geeral l.

5 EDGE-COLORINGS AVOIDING RAINBOW STARS 5 Lemma.. For r t 3 ad l 1, there are costats 0 ad D > 0 such that if G = (V, E is a C r,t,l -extremal graph o 0 vertices, the ( E(G D log t 1. Proof. Fix r t 3, l 1 ad let G = (V, E be a -vertex C r,t,l -extremal graph with V = {v 1,..., v }. Note that G has at least (t 1 ( (3 raibow-s t,l -free r-edge colorigs, as the complete graph K has at least these may colorigs: choose a fixed (t 1-subset S of [r] ad assig colors i S to all edges of K. We cosider the decompositio i (, ad fix sets S 1,..., S. Colorigs i C r,t,l,(s1,...,s (G may be produced as follows: for each vertex v i we choose at most (r t + 1(l 1 icidet edges to be assiged colors that are ot i S i ad color them with these colors. The remaiig edges {v i, v j } E are assiged colors i S i S j. For sufficietly large, this implies that (r t+1(l 1 ( C r,t,l(s1,...,s (G 1 r j S i S j j As (S 1,..., S ca be chose i ( r C r,t,l (G j=0 {v i,v j } E (r t+1(l 1 r (r t+1(l 1 (t 1 E. (4 t 1 ways, (S 1,...,S S i =t 1,i=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 ( where D = log r t 1 t 1 + (r t + 1(l 1 is a costat. Combiig (3 ad (5, we have ( (t 1 D log t 1 (t 1 E (t 1 ( = E D log t 1, as required. To perform the secod step of the proof, for a costat A > 0, let S A deote the set of all collectios (S 1,..., S of (t 1-subsets of [r] where o set S i appears more tha A log t 1 times, i {1,..., }. We prove that we may fid A for which the umber of colorigs i S A is egligible. As i the previous result, we prove this for geeral l, as there is little additioal work. Lemma.3. Let r t 3 ad l 1 be itegers. For all D > 0 there exists a positive costat A with the followig property. Give ε > 0 there is a costat 0 such that, for all 0, ay -vertex graph G = (V, E with at least ( D logt 1 edges satisfies C r,t,l,(s1,...,s (G (S 1,...,S S A ε(t 1 E(G. Proof. With foresight, fix { } 3(r t + 1(l 1 A > max 1 log t 1 (t, D,

6 6 C. HOPPEN, H. LEFMANN, K. ODERMANN, AND J. SANCHES ad let B be a iteger satisfyig / ( r t 1 B A logt 1, where will be chose sufficietly large later i the proof. Give a -vertex graph G = (V, E with E ( D log t 1, we provide a upper boud o the umber of raibow-s t,l -free r-edge colorigs i a set C r,t,l,(s1,...,s (G such that max S {v V : S v = S ad S = t 1} = B. To geerate these colorigs, we choose a set U V such that U = B ad a (t 1-subset S of [r] which is assiged to all vertices i U. We the assig other (t 1-subsets to the remaiig B vertices of G. Let E(U, V \ U deote the set of edges with oe vertex i U ad the other i V \ U. As i the proof of Lemma. (see (4, for each vertex v i we choose at most (r t + 1(l 1 edges i at most (r t+1(l 1 i=0 (r t+1(l 1 ways for sufficietly large. These edges are assiged colors that are ot i S i i at most r (r t+1(l 1 ways. The remaiig edges {v i, v j } E are assiged colors i S i S j. Ay such edge i E(U, V \ U may be assiged at most t colors, sice the sets assiged to their edvertices are distict. Hece the umber of raibow-s t,l -free r-colorigs of G is bouded above 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 Note that for large 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, ( t E(U,V \U. (6 t 1 ( r t 1 } As a cosequece, sice A > D ad is sufficietly large (i particular depeds o A, r, l ad t, we obtai E(U, V \ U (A log t 1 /. If we sum (6 over all possible values of B, we obtai at most ( r (r (r t+1(l 1 (t 1 E(G t 1 ( t t 1 + log ( r t 1 (t 1 E(G +(r t+1(l 1 log t 1 (r ε(t 1 E(G A log t 1 ( t t 1 ( r t 1 A log t 1 raibow-s t,l -free r-colorigs of G, where ε > 0 is arbitrary as log as we choose sufficietly large, sice we have A > 3(r t+1(l 1 1 log t 1 (t. The ext step i our proof of Theorem 1. for l = 1 is provig that ay extremal graph has large miimum degree. Ulike the previous steps, we shall ow deal exclusively with the case l = 1, as treatig rare colors will require cosiderably more work. Lemma.4. For all itegers r t 3 there is a 0 such that the miimum degree of G satisfies δ(g 3/4 1 for all C r,t,1 -extremal graphs G with 0 vertices. Proof. Assume that a -vertex C r,t,1 -extremal graph G has a vertex v with degree d(v < (3/4 1. Let w 1,..., w /4 be vertices i G that are ot adjacet to v. Defie the graph G by addig the edges {v, w 1 },..., {v, w /4 } to G.

7 EDGE-COLORINGS AVOIDING RAINBOW STARS 7 The basic idea of the proof is to show that G admits more raibow-s t -free r-colorigs tha G, ad we do this by showig that, if we compare the umber of colorigs created ad lost with the additio of the ew edges, there are more of the former. To be more precise, give a collectio (S 1,..., S of (t 1-subsets of [r], it is clear that we may exted all colorigs i C (S1,...,S (G to C (S1,...,S (G wheever S v S wi for all i {1,..., /4 }, as we may assig ay color i the correspodig itersectio to {v, w i } without producig a raibow star S t. Moreover, this extesio may be doe i several ways, depedig o the sizes of the itersectios, which leads to ew colorigs of G, as opposed to colorigs that are i oe-to-oe correspodece with colorigs of G. O the other had, colorigs of G for which S v S wi = for some i may ot be exteded i this way, ad we say that these colorigs are lost whe the ew edges are added. To fid a lower boud o the umber of colorigs created, cosider oly those edge colorigs of G where every edge is assiged a color from a fixed (t 1-set S i [r]. Each such colorig ca be exteded to at least (t 1 /4 (t 1 E(G raibow-s t -free colorigs of G by assigig a arbitrary color of S to each ew edge. This creates at least ( (t 1 /4 1 (t 1 E(G ew colorigs. O the other had, the raibow-s t -free r-colorigs of G that caot be exteded to colorigs of G are those where the sets of colors available at v ad at w i do ot itersect, for some i {1,..., /4 }. By Lemma.3 with ε = 1, the umber of colorigs of G where every (t 1-set of colors is assiged to at most A log t 1 vertices of G is at most (t 1 E(G. Hece we cocetrate o colorigs where some (t 1-set S appears at least A log t 1 times. The umber 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 sice there are ( r t 1 ways to choose Sv, a o-eighbor w i ca be chose i at most ways ad it is assiged a set S wi of colors with S wi S v =, which ca be doe i ( r (t 1 t 1 ways. The set S ca be chose i ( r t 1 ways, the vertices which are assiged the set S ca be chose i at most ( ( A log t 1 = A log t 1 ways ad every remaiig vertex is associated with some arbitrary (t 1-set of colors. (Note that this upper boud takes care of all the colorigs where the set S is assiged to m vertices, where A log t 1 m. Clearly, we have ( A log t 1 A log t 1, ad, for large ( ( ( ( r r (t 1 r r A logt 1 < A log t 1. t 1 t 1 t 1 t 1 We coclude from (7 that, for sufficietly large, the umber of raibow-s t -free r-colorigs of G that caot be exteded 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 other words, by addig the edges {v, w 1 },..., {v, w /4 } to G, we icrease the total umber of colorigs, which cotradicts the choice of G. We remark that the previous proof may be easily adapted to the case where the coditio o the miimum degree is replaced by δ(g α for ay fixed 0 < α < 1. We are ow ready to perform the last step i the proof of Theorem 1. for l = 1, which shows that, i a extremal graph G o edge may be missig. Theorem.5. For r t 3, there exists 0 such that c r,t,1 ( = C r,t,1 (K holds for 0. Moreover, K is the uique -vertex C r,t,1 -extremal graph.

8 8 C. HOPPEN, H. LEFMANN, K. ODERMANN, AND J. SANCHES Proof. Assume that there is a C r,t,1 -extremal graph G = (V, E o vertices with at least two o-adjacet vertices x ad y. As i the proof of Lemma.4, we prove that G = G + {x, y} has more raibow-s t -free r-colorigs tha G if is sufficietly large. By Lemma. we kow that may be chose so that E(G ( D logt 1, where D is a costat. Every colorig of G for which oly (t 1 colors are used ca be exteded, assigig ay of these colors to {x, y}, to a colorig of G, which icreases the total umber of colorigs by (t (t 1 E(G. (8 We show that the umber of all raibow-s t,1 -free r-colorigs of G that caot be exteded to a colorig of G is smaller tha (8. By Lemma.3 with A = A(r, t, l, D ad ε = 1/, we kow that we may choose 0 such that the umber of colorigs associated with assigmets i S A is at most 1 (t 1 E(G. Therefore, i the followig we oly eed to cosider colorigs from the set A = C (S1,...,S (G. (9 (S 1,...,S S A The oly colorigs of G that caot be exteded to colorigs of G are those where the color sets S x ad S y assiged to x ad y, respectively, are disjoit, so that we are uable to assig a color to {x, y}. Fix (S 1,..., S such that S is assiged to at least A log t 1 vertices of G. Recall that both vertices x, y have degree at least 3/4 1 by Lemma.4. The coditio o the degrees implies that the commo eighbourhood N({x, y} of x ad y has size at least /. For ay vertex w i N({x, y} we have S w (S x S y t 1. More precisely, we have S w S x = a w ad S w S y t 1 a w, so that there are at most a w (t 1 a w ((t 1/ ways to assig colors to the edges {x, w} ad {y, w}. Hece all edges betwee {x, y} ad their commo eighbourhood N({x, y} may be colored i at most ((t 1/ N({x,y} ways. This leads to the followig upper boud o the umber of elemets i (9 that caot be exteded to a colorig of G. The set S may be chose i ( r t 1 ways ad, for large, there are ( A log t 1 < /4 ways of choosig A log t 1 vertices which are assiged S. For sufficietly large, the remaiig vertices may be assiged color sets i at most ( r A logt 1 t 1 < /4 ways, ad we ifer that A ( ( ( r r 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 A logt 1 (t 1 E(G (t 1 E(G 4 N({x,y} Altogether, the umber of all colorigs of G that caot be exteded by addig edge {x, y} to the graph G is o more tha 1 (t 1 E(G + 1 ( r (t 1 E(G, t 1 which is smaller tha (8 for sufficietly large.

9 EDGE-COLORINGS AVOIDING RAINBOW STARS 9 3. Colorigs avoidig a raibow S t,l I this sectio, we cosider the proof Theorem 1. for geeral l. Recall, that a edge-colored star S tl with tl edges such that t distict colors are each assiged to exactly l edges is called a raibow-s t,l. As we remarked before, the strategy for achievig this result is exactly the same as for the case l = 1, but the presece of rare colors will make the argumets more techical. Recall that first ad secod mai steps of the proof, amely showig that extremal graphs have a large umber of edges, ad that most colorigs have the property that almost all vertices have the same set of ordiary colors, have already bee proved for geeral l (Lemmas. ad.3. To perform the remaiig steps, we use the strategy employed i Lemma.4, ad show that the umber of colorigs created exceeds the umber of colorigs lost whe edges are added. To reach this coclusio, we eed a lower boud o the umber of colorigs created, ad a upper boud o the umber of colorigs lost, with the property that the lower boud is larger tha the upper boud. However, evaluatig these bouds will be harder i this case because of the rare colors. To describe the mai igrediet eeded to treat rare colors, first cosider colorigs for which S 1 = = S, so that sets of ordiary ad rare colors are the same for all vertices. I ay S t,l -free r-colorig i C (S1,...,S (G, the graph iduced by each rare color has maximum degree less tha l, so we eed to cout the umbers of subgraphs of G of this type. A classical result of Beder ad Cafield [5] implies that the umber N K (d of labeled subgraphs of K with degree sequece d = (d 1,..., d, where all compoets d i are bouded above by some absolute costat d, satisfies N K (d (m! exp( λ ( λ m m m! m i=1 d i! i=1 d i! exp( λ λ, (10 e where m = d i ad λ = 1 ( di m, ad where the asymptotics are i (the secod approximatio uses Stirlig s formula. Here A( B( meas that lim A(/B( = 1. As it turs out, the Beder ad Cafield formula may be applied directly whe fidig a upper boud o the umber of colorigs lost, as we may assume that ay edge eeded whe fixig the subgraph whose edges have some rare color lies i the graph. However, this may ot be doe for the lower boud o the umber of colorigs created, where we eed a approximate versio of Beder ad Cafield s result. I the followig, give a graph H o vertices ad a iteger sequece d = (d 1,..., d, let N H (d be the umber of subgraphs with degree sequece d i H. More geerally, give a array d = (d 1,..., d k, let N H ( d be the umber of ways of selectig a k-tuple (H 1,..., H k of edge-disjoit subgraphs of H such that each subgraph H i has degree sequece d i, for all i {1,..., k}. We clearly have N H ( d N H (d 1 N H (d k. I the followig, we say that a iteger sequece d = (d 1,..., d is γ-dese if i=1 d i γ. Lemma 3.1. Give positive itegers d ad k, costats D > 0 ad γ > 0, there exist positive costats 0, M ad α satisfyig the followig property for all 0. For every graph H with V (H = ad E(H ( D l, there exists W V (H with W M l such that, for all γ-dese degree sequeces d 1,..., d k {0,..., d} W, where k k, we have N H[W ] (d 1,..., d k α N K W (d i. Ituitively, this lemma states that, i ay graph with may edges, there is a almost spaig subgraph with a large umber of subgraphs of ay bouded degree sequece that k i=1

10 10 C. HOPPEN, H. LEFMANN, K. ODERMANN, AND J. SANCHES is sufficietly dese. Note that this would trivially fail if we required W = V, as a oppoet would be able to isolate vertices whe removig D log t 1 edges of K to produce G, so that N G (d = 0 for ay positive sequece d = (d 1,..., d. The proof of Lemma 3.1, which adapts ideas of Gao [9], lies i Sectio 4. Lemma 3.. For all itegers r t 3 ad l 1, there is 0 such that ay C r,t,l -extremal graph G o 0 vertices satisfies δ(g 3/4 1. Proof. Assume that a C r,t,l -extremal -vertex graph G = (V, E has a vertex v with degree d(v < 3/4 1. We suppose that is sufficietly large for all steps i the proof to hold. Let w 1,..., w /4 be /4 vertices i G that are ot adjacet to v, ad let G be the graph obtaied by addig all the edges {v, w i } to G, i = 1,..., /4. Let D > 0 such that G has at least ( D logt 1 edges (Lemma. ad fix A > 0 with the property of Lemma.3. We show that the umber N of ew colorigs of G obtaied by extedig colorigs of G is larger tha the umber N of colorigs of G that caot be exteded to colorigs of G. By Lemma.3 with ε = 1/3, the umber of colorigs i (S 1,...,S S A C (S1,...,S (G is at most (t 1 E(G /3 for sufficietly large. Let N A be the umber of ew colorigs of G associated with -tuples (S 1,..., S i S A, ad let N A be the umber of colorigs associated with such collectios that caot be exteded. I the remaider of the proof, we fid a lower boud o N A ad a upper boud o N A to show that N A (t 1/8 N A N A. Moreover, it turs out that N A (t (t 1 E(G (see (16, so that N N N A ( N A + (t 1 E(G 3 N A (t 1 E(G 3 ( (t 1 3 (t 1 E(G > 0, as required. Before proceedig, let (3.1 0, M ad α give by Lemma 3.1 applied for D, d = l 1, k = r t + 1 ad γ = 8/9 (adjustig the costats so that the logarithms i the statemet of the lemma have base t 1, ad fix a set W V with W M log t 1 such that G[W ] satisfies the coclusio of the lemma. Note that we use (3.1 0 to deote the value of 0 obtaied i this applicatio of Lemma 3.1. Upper boud: We give a upper boud o the umber N A of raibow-s t,l -free r-colorigs of G that are associated with collectios (S 1,..., S S A ad caot be exteded to a colorig of G. If S v S wi for every i, the colorigs of C r,t,l,(s1,...,s (G ca be easily exteded to colorigs of C r,t,l,(s1,...,s (G usig ordiary colors for each ew edge, so that we may assume that the sets of colors available at v ad available at w i do ot itersect for some i {1,..., /4 }. However, ote that S v S wi = does ot imply that there is o color available for the edge {v, w i }, sice it could possibly be colored with oe of the rare colors. For the sake of simplicity, ad sice we are lookig for a upper boud, we shall igore this fact ad assume that S v S wi = always makes it impossible to color the edge {v, w i }. To costruct colorigs of this type we proceed as follows. First, we fix a collectio (S 1,..., S S A with the required properties: (i choose a (t 1-subset S v [r]; (ii choose a vertex w that is ot adjacet to v; (iii choose a (t 1-subset set S w [r], which is disjoit from S v, to be assiged to w; (iv choose the (t 1-subset S [r] that is assiged to at least A log t 1 vertices of G; (v choose A log t 1 vertices that are assiged this set S; (vi assig ay (t 1-subsets i [r] to the A log t 1 remaiig vertices. Note that steps (i, (ii ad (vi allow us to choose S, so it might well be that more tha A log t 1 vertices are assiged S. The umber of choices for the sets S v, S w, S above is at most ( r 3, t 1 while is a upper boud o the umber of choices of w. Steps (v ad (vi may be performed

11 EDGE-COLORINGS AVOIDING RAINBOW STARS 11 i ( ( A log t 1 r A logt 1 t 1 ways, so that, for large, a upper boud o the umber of ways of fixig a collectio (S 1,..., S with the required properties 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 assume that such a collectio (S 1,..., S is fixed. Let Y = {u V : S u = S} ad let H = G[W Y ]. We ow costruct colorigs i C (S1,...,S (G. To this ed, we proceed as follows: (i color edges icidet with vertices u V \ Y with rare colors with respect to u; (ii color edges icidet with vertices u (V \ W Y with rare colors with respect to u; (iii color edges icidet with V \ Y with ordiary colors (with respect to some edpoit i V \ Y ; (iv color edges i H with rare colors (with respect to S; (v color edges with both eds i Y with ordiary colors (with respect to S. Observe that the edges e = {u, v} such that u is assiged S, but v is ot, may be colored i (i if e is assiged a rare color with respect to v or i (iii, if e is assiged a ordiary color with respect to v. For simplicity, we shall assume that edges colored i (i may be recolored i (iii. Also ote that edges e = {u, v} such that u is i W Y ad v (V \ W Y are colored i (ii or (v. For step (i, there are at most ( (l 1(r t+1 A log t 1 ways of choosig (l 1 edges icidet with each such vertex w for each of the (r t + 1 rare colors. Note that we do ot eed to cosider the possibility that fewer edges are assiged such colors because, with our estimates, the edges colored at this poit could be recolored i later steps. Step (ii may be performed i at most (l 1(r t+1m log t 1, while step (iii may be performed i at most (t 1 η ways, where η is the umber of edges icidet with vertices i V \ Y. To assig rare colors (with respect to S to the edges of H, we use the followig procedure. Procedure 3.3. Suppose we have a q-vertex iput graph H = (V, E ad a set T ( [r] t 1. Assume that H q,0 is a set of isolated vertices labeled by V. For i {1,..., r t + 1}, choose a graph H q,i i E \ i 1 j=1 E(H q,j with the degree sequece d i = (d i 1,..., di q, where d i j l 1 ad assig the ith color i [r] \ T to the edges of H q,i. As we are lookig for a upper boud, we shall assume that W Y = W, possibly beig able to recolor some edges that have bee colored i previous steps. For simplicity, assume that W = p. For i {1,..., r t + 1}, fix degree sequeces d i = (d i 1,..., di p, where the ith degree sequece is associated with the ith rare color: we fid r t + 1 edge-disjoit subgraphs H 1,..., H r t+1 of H such that H i has degree sequece d i, i = 1,..., r t + 1. Note that N H (d 1,..., d r t+1 is the umber of ways i which this ca be doe. We deote d = (d 1,..., d r t+1, where for each d i = (d i 1,..., di p all compoets are bouded above by l 1. Moreover, we deote u( d = 1 r t+1 p i=1 j=1 di j. The umber of ways of performig steps (iv ad (v is bouded above by N H ( d (t 1 E(G u( d η, d where the sum rages over the arrays d = (d 1,..., d r t+1. As a cosequece, the umber of colorigs costructed above is at most (l 1(r t+1a log t 1 (l 1(r t+1m log t 1 (t 1 η d N H ( d (t 1 E(G u( d η. (1

12 1 C. HOPPEN, H. LEFMANN, K. ODERMANN, AND J. SANCHES It is easy to see that, choosig sufficietly large, we may choose a arbitrarily small costat δ > 0 such that the product of equatios (11 ad (1 is at most (1 + δ d N H ( d (t 1 E(G u( d. Recall that a sequece d i is γ-dese if the compoets of d i sum to γp or more, otherwise it is γ-sparse (it will be coveiet for use γ = 8/9 i these calculatios. We ca write (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+ where s: [r t+1] {0, 1} is a fuctio that, for each i, idicates whether the degree sequece d i is γ-dese (whe s(i = 1 or sparse (whe s(i = 0, while d + ad d are the arrays of dese ad sparse degree sequeces, respectively, that create a array d with the distributio determied by s. We split equatio (13 accordig to whether all rare colors geerate dese graphs, or whether this does ot hold. For simplicity, we write d + to say that we sum =j over arrays of legth j whose degree sequeces are all dese. We obtai (1 + δ.(14 d + =r t+1 N H ( d + (t 1 E(G u( d+ + s 1 Observe that the secod summatio may be estimated as N H ( d (t 1 E(G u( d d+ s 1 d r t ( p γp(r t+1 j/ r t+1 l p(r t+1 (r t+1/ e r t l p(r t+1 p γp(r t+1 j/ j=0 i=1 j=0 d + =j d d (t 1 E(G u( d + d+ N H ( d (t 1 E(G u( d d + =j (t 1 E(G u( d + N H ( d + j N Kp (d i +. (15 To see why the first iequality is true, ote that s rages i a set with at most r t+1 elemets, the umber of arrays d = (d 1,..., d r t+1 j is bouded above by l p(r t+1 j ad r t+1 j N H ( ( ( p γp/ r t+1 j d N Kp (d i (r t+1/ e e γp(r t+1 j/ pγp(r t+1 j/, where N Kp (d i is bouded usig (10 ad the fact that j di j γp < p. Lower boud: Next we give a lower boud o the umber N A of raibow-s t,l-free r-colorigs we gai by addig the edges {v, w 1 },..., {v, w /4 } to our extremal graph G. The idea here is that, if we cosider colorigs i C r,t,l,(s,...,s (G for some fixed (t 1-subset S of [r], those colorigs ca be exteded to colorigs of G by assigig ay of these (t 1 colors to each of the edges {v, w i }, i = 1,... /4. Hece the umber of colorigs of this type for G is at least (t 1 /4 C r,t,l,(s,...,s (G. Removig colorigs that are extesios of the correspodig colorigs of G, the et gai of colorigs is ( (t 1 /4 1 C r,t,l,(s,...,s (G. We ow fid a lower boud o C r,t,l,(s,...,s (G. To assig rare colors to edges of G, we apply Procedure 3.3 to H = G[W ] ad T = S. Sice we ow eed a lower boud, i=1

13 EDGE-COLORINGS AVOIDING RAINBOW STARS 13 we may ot suppose that G[W ] = K p, but Lemma 3.1 guaratees that, for all k-tuples (d 1,..., d k such that k r t + 1 ad p j=1 di j γp, for all i, we have N G[W ](d 1,..., d k α k i=1 N K p (d i. With this, a lower boud o the umber of colorigs gaied by addig the edges {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 where we agai sum over arrays d = (d 1,..., d r t+1, where each d i = (d i 1,..., di p has compoets bouded by l 1. To coclude our argumet, we compare the upper boud (14 ad the lower boud (16. We have (14 (16 (1 + δ d + =r t+1 N K p ( 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 pγp(r t+1 j/ d + =j (t 1 E(G u( d + j i=1 N K p (d i + ((t 1 /4 1 d (t 1 E(G u( d N G[W ] (.(18 d By our choice of W, the term (17 becomes (1 + δ d + =r t+1 N K p ( 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 d + =r t+1 i=1 N Kp (d i (t 1 E(G u( d + ((t 1 /4 1 α d + =r t+1 (t 1 E(G u( d + r t+1 i=1 N Kp (d i α (1 + δ ((t 1 /4 1 1, (19 (t 1 /8 for sufficietly large. For the secod term (18, we write (1 + δ l p(r t+1 j pγp(r t+1 j/ d + =j (t 1 E(G u( d + j i=1 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 γp(r t+1 j/ (t 1 E(G u( d + j i=1 N K p (d i + d (t 1 E(G u( d N G[W ] ( d p γp(r t+1 j/ (t 1 E(G u( d + j i=1 N K p (d i + (t 1 E(G u( d + (r t+1 jp(l 1/ N G[W ] ( d +, d, (0 where d is a array of r t+1 j degree sequeces equal to d = (l 1,..., l 1 (oe of the terms l 1 may be replaced by l i these sequeces to deal with parity costraits. Note that, to reach (0, we replaced the deomiator withi the sums by a sigle term, which

14 14 C. HOPPEN, H. LEFMANN, K. ODERMANN, AND J. SANCHES depeds o each particular term beig added. Our choice of W implies that, whe d + = j, N G[W ] ( d +, d j r t+1 j α N Kp (d i + N Kp (d, i=1 α ( (l 1! p i=1 α p p(l 1(r t+1 j/ (e(l 1! (l 1p(r t+1 ( p(l 1 p(l 1/ ( r t+1 j 1 (l 1 j exp N Kp (d i e 4 + j N Kp (d i +. (1 i=1 To fid a upper boud o (0, we combie (1 with the fact that the umber of terms i the sums is at most (r t + 1l p(r t+1 ad γ = 8/9 ad that, for l, the term p (r t+1 j p(17/9 l/ is maximized for j = r t, which leads to the followig upper boud o (0: α (1 + δ l p(r t+1 (r t + 1 (e (t 1 (l 1! (r t+1(l 1p ((t 1 /4 1p (l (17/9p/ = (t 1 O( p (17/9 lp/ 1 < (t 1 /8 for large, as p M log t 1. Therefore, we obtai (14 (16 1 (t 1 /8 + 1 (t 1 /8 = 1, ( (t 1 /8 for t 3 ad sufficietly large, which implies that N A (t 1/8 N A, as required. Theorem 3.4. For itegers r t 3 ad l, there exists 0 such that c r,t,l ( = C r,t,l (K holds for 0. Moreover, K is the uique -vertex C r,t,l -extremal graph. Proof. Fix r t 3 ad l. Assume that there is a C r,t,l -extremal graph G o -vertices with at least two o-adjacet vertices x ad y. We prove that G = G + {x, y} has more colorigs tha G if is sufficietly large. Our proof uses the strategy employed for Lemma 3.. By Lemma. we kow that 0 may be chose so that E(G ( D logt 1, where D = D(r, t is a costat. Fix A > 0 with the property of Lemma.3. Let (3.1 0, M ad α give by Lemma 3.1 applied for D, d = l 1, k = r t + 1 ad γ = 8/9 (adjustig the costats so that logarithms i the statemet of the lemma have base t 1, ad fix a set W V with W M log t 1 = p such that G[W ] satisfies the coclusio of the lemma. We show that the umber N of ew colorigs of G obtaied by extedig colorigs of G is larger tha the umber N of colorigs of G that caot be exteded to colorigs of G. By Lemma.3 with ε = 1/3, the umber of colorigs i (S 1,...,S S A C (S1,...,S (G is at most (t 1 E(G /3 for sufficietly large. Let N A be the umber of ew colorigs of G associated with -tuples (S 1,..., S i S A, ad let N A be the umber of colorigs associated with such collectios that caot be exteded. I the remaider of the proof, we fid a lower boud o N A ad a upper boud o N A to show that N A /4 N A N A. Oce agai we have N A (t (t 1 E(G (see (3, which leads to the desired result: ( N N N A (t 1 E(G N A + 3 ( (t 1 3 i=1 (t 1 E(G > 0.

15 EDGE-COLORINGS AVOIDING RAINBOW STARS 15 With the argumets used for (16, we may show that every colorig of G for which every vertex is assiged the same set of (t 1 colors ca be exteded to colorigs of G, which icreases the total umber of raibow-s t,l -free r-colorigs by at least (t d N G[W ] ( d (t 1 E(G u( d, (3 where we agai sum over arrays d = (d 1,..., d r t+1, where each d i = (d i 1,..., di p has compoets bouded by l 1. Recall that u( d = 1 r t+1 p i=1 j=1 di j. We show ext that the umber of all colorigs of G that caot be exteded to a colorig of G is smaller tha (3. We give a upper boud o the umber N A of raibow-s t,l -free r-colorigs of G that are associated with collectios (S 1,..., S S A ad caot be exteded to a colorig of G. All such colorigs have the property that the color sets S x ad S y assiged to x ad y, respectively, are disjoit. Fix (S 1,..., S for which this holds, ad where S is assiged to at least A log t 1 vertices of G. The (t 1-subset S [r] may chose i ( r t 1 ways ad there are ( A log t 1 ways of choosig A logt 1 vertices which are assiged A logt 1 S. The remaiig vertices may be assiged (t 1-elemet color sets i at most ( r t 1 ways. As a cosequece, ( ( ( 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 is a upper boud o the umber of ways of fixig a collectio (S 1,..., S with the required properties. We derive a upper boud o the umber of colorigs i C (S1,...,S (G that caot be exteded to G. Sice x, y have degree at least 3/4 by Lemma 3., their commo eighbourhood N({x, y} has size at least /. For ay vertex w i N({x, y} we have S w (S x S y t 1. More precisely, we have S w S x = a w ad S w S y t 1 a w, so that there are at most a w (t 1 a w ((t 1/ ways to assig ordiary colors to the edges {x, w} ad {y, w}. Hece all edges betwee {x, y} ad their commo eighbourhood N({x, y} may be colored i at most ((t 1/ N({x,y} ways with ordiary colors (with respect to both eds. As i the proof of Lemma 3., we proceed as follows: let Y = {u V : S u = S} ad let H = G[W Y ], (i color edges icidet with vertices u V \ Y with rare colors with respect to u; (ii color edges icidet with vertices of (V \ W Y with rare colors; (iii color edges icidet with V \ Y with ordiary colors (with respect to some edpoit i V \ Y ; (iv color edges i H with rare colors (with respect to S; (v color edges with both eds i Y with ordiary colors (with respect to S. We obtai the followig upper boud o the umber of colorigs of G that caot be exteded to G : ( r A logt 1 +1 A log t 1 (l 1(r t+1a log t 1 (l 1(r t+1m 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 easy to see that, choosig sufficietly large, equatio (5 is at most / ( N H ( t 1 N(x,y d (t 1 E(G u( d N(x,y, (6 d

16 16 C. HOPPEN, H. LEFMANN, K. ODERMANN, AND J. SANCHES which may be rewritte as / s d d+ N H ( d (t 1 E(G u( d N(x,y, (7 where, as i (13, s: [r t + 1] {0, 1} is a fuctio that, for each i, idicates whether the degree sequece d i is γ-dese or ot, while d + ad d are the arrays of γ-dese ad sparse degree sequeces, respectively, that create a array d with the distributio determied by s. We split equatio (7 accordig to whether all rare colors geerate dese graphs, or whether this does ot hold, which leads 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 Usig the estimate (15 we have s 1 d d+ N H ( d (t 1 E(G u( d / r t l p(r t+1 p γp(r t+1 j/ j=1 d + =j d d d+ N H ( d d+ N H ( d (t 1 E(G u( d + / (t 1 E(G u( d N(x,y (t 1 E(G u( d /. (8 j N Kp (d i +. (9 To coclude our argumet, we compare the upper boud (8 ad the lower boud (3. We have (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 pγp(r t+1 j/ d + =j (t 1 E(G u( d + j i=1 N K p (d i + / (t d (t 1 E(G u( d N G[W ] (. (31 d Usig the estimates of (19, the term (30 may be bouded by d + =r t+1 N H( d + (t 1 E(G u( d + i=1 / (t d + =r t+1 (t 1 E(G u( d + N G[W ] ( d + α / (t 1 /4+1 For the secod term (31, we repeat the argumets used i the proof of Lemma 3.. Therefore, we obtai for t 3 ad sufficietly large, as required. (8 (3 1 / /4+1 = 1, (3 /4 To coclude this sectio, we show how the proof of Theorem 1.1 may be derived from our proof of Theorem 1..

17 EDGE-COLORINGS AVOIDING RAINBOW STARS 17 Proof of Theorem 1.1. The basic idea is to split the set C = C r,l1,...,l t (G of all raibow- S l1,...,l t -free r-colorigs of a graph G, l 1... l t, as a uio similar to (. However, give a colorig, we shall ow cosider that vertices may have two types: we will say that v has type 1 if there is o raibow S t,lt cetered at v, ad that it has type otherwise. The crucial observatio is that a vertex of type caot be the ceter of a raibow S t 1,l1, as otherwise there would be a raibow S l1,...,l t cetered i it. This motivates us to modify the defiitio of a ordiary color: for a vertex v of type 1, a color is ordiary if it appears at least l t times amog the with v adjacet edges; if v has type, a ordiary color must appear at least l 1 times. As before, the remaiig colors are said to be rare with respect to v. Oe importat feature of this defiitio is that vertices of type 1 may have up to t 1 ordiary colors, while vertices of type might have oly t ordiary colors. I particular, the aalogue of ( is give by C r,l1,...,l t (G = (S 1,...,S C r,l1,...,l t,(s 1,...,S (G, where (S 1,..., S is such that S i = t 1 for vertices of type 1 ad S i = t for vertices of type. With this i mid, it is easy to see that Lemma. would hold with essetially the same argumets for this more geeral forbidde cofiguratio. Moreover, we ca easily prove a versio of Lemma.3 where the class S A show to be egligible cotais all colorigs i C r,l1,...,l t,(s 1,...,S (G with the property that o set S for which S = t 1 appears at least A log t 1 times i the vector (S 1,..., S. I other words, colorigs i S A have at least A log t 1 vertices of type 1 which have the same set of ordiary colors. As was doe for S t,l -free colorigs, we show that -vertex C r,l1,...,l t -extremal graphs have miimum degree at least 3/4. To this ed, we imitate the proof of Lemma 3., i which we let G be a graph cotaiig some vertex v with degree at most 3/4 1, ad we defie G by addig edges betwee v ad all vertices ot adjacet to it, which form a set W such that W /4. Here, it is coveiet to split the set C = C r,l1,...,l t (G as the uio of C 1, cotaiig colorigs for which all vertices have type 1, ad C, which cotais the other colorigs. We treat these colorigs as i the proof of Lemma 3., where we assume that a colorig i G ca be exteded to a colorig i G provided that v shares a ordiary color with ay vertex w W. This is pessimistic, as some colorig of G might be extedible to a colorig of G usig rare colors. The discussio i Lemma 3. applies directly to colorigs i C 1 (G, which allows us to coclude that N A (t 1 /8 N (1 A, where N A is a lower boud o the umber of ew colorigs (i C 1(G created by the extesio ad N (1 A is a upper boud o the umber of colorigs i C 1(G that caot be exteded to G. We still eed to cosider the umber N ( A of colorigs i C (G that caot be exteded to G. Note that a colorig i C (G for which i vertices have type ca be tured ito a colorig where each rare color appears fewer tha l t times if we recolor at most l 1 l t edges icidet with each rare color. This may be doe by assigig them a fixed ordiary color (say the ordiary color of least idex amog the ordiary colors associated with that vertex. This gives rise to colorigs of G i classes D j (G, which are colorigs such that rare colors appear at most l t 1 times, each vertex is assiged at most t 1 ordiary colors, ad exactly j vertices have fewer tha t 1 ordiary colors (i.e. are icidet with l t or more edges of at

18 18 C. HOPPEN, H. LEFMANN, K. ODERMANN, AND J. SANCHES most t differet colors. Note that each colorig i D j may be tured ito at most ( j (l 1 l t(r t+i (l 1 l t(r t+i j i ( (l 1 l t(r t+i i colorigs i C (G for which i vertices have type, sice we first choose i vertices to have type, ad the, for each such vertex w ad each of the r t + rare colors, we choose up to l 1 l t edges icidet with w to be assiged the correspodig ordiary color (the factor (l 1 l t(r t+i takes care of the up to, as it allows us to decide, for each chose edge, whether to recolor it or ot. Summig over all 1 j ad all 1 i mi{j, A log t 1 }, we deduce that the umber of colorigs i C (G that caot be exteded to G is at most O(log t 1 D, where D is the umber of colorigs of D = A log t 1 j=1 D j (G that caot be exteded to G. But it is clear that D is bouded above by the umber N (1 A of colorigs i C 1(G that caot be exteded to G, as colorigs i D ca be viewed as colorigs of C 1 (G (ideed, they are colorigs of C 1 (G where some ordiary color fails to appear l t times for each vertex of type. As a cosequece, for sufficietly large, N (1 A + N ( A O(log t 1 N (1 A O(log t 1 N A (t 1 /8 < N A, which proves the extesio of Lemma 3. to the preset cotext. Clearly, Theorem 3.4 may be exteded aalogously, which leads to the desired result. 4. Auxiliary results To coclude our paper, we prove Lemma 3.1. To do this, we eed some prelimiary defiitios ad results, which are based o ideas of Gao [9]. Let H be a graph o vertices ad let e(h deote the umber of edges i H. Color all edges i H blue ad all edges i its complemet H red, so that K is the complete graph o vertices with a -colorig K = H H. For ay red edge e = {u, v}, defie the red degree of e d r (e = d r (u + d r (v, where d r (u deotes the umber of red edges icidet with u. Let r (H = max e E(H d r(e, where we maximize over red edges. Fix a degree sequece d = (d 1,..., d, where max d i d for some absolute costat d N, ad let F be a graph with degree sequece d = (d 1,..., d chose uiformly at radom from all subgraphs of K with this degree sequece. Let X = X (F be the radom variable that accouts for the umber of red edges i F. For k = 1, recall that Lemma 3.1 couts the umber of subgraphs (with a give degree sequece of a complete graph that remai after a oppoet deletes a certai umber of edges. I the curret settig, edges that remai are represeted by blue, while edges deleted are represeted by red. As a cosequece, treatig subgraphs of the complete graph that are ot affected by the deletio of edges is the same as treatig colored subgraphs of a -colorig of K whose edges are all blue. The followig result is the mai igrediet i the proof of Lemma 3.1, as it gives a lower boud o the probability of choosig a subgraph with o red edges for ay -colorig such that the red degree is small. Lemma 4.1. For γ > 0 ad a positive iteger d, there exists a positive iteger 0 such that the followig holds for 0. Fix a γ-dese sequece d = (d 1,..., d such that max d i d.