On subgraphs of C 2k -free graphs and a problem of Kühn and Osthus

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1 O subgraphs of C k -free graphs ad a proble of Küh ad Osthus Dáiel Grósz Abhishek Methuku Casey Topkis arxiv: v [ath.co] 7 Aug 07 Abstract Let c deote the largest costat such that every C 6 -free graph G cotais a bipartite ad C 4 -free subgraph havig c fractio of edges of G. Győri et al. showed that 3 8 c 5. We prove that c = 3 8. More geerally, we show that for ay ε > 0, ad ay iteger k, there is a C k -free graph G which does ot cotai a bipartite subgraph of girth greater tha k with ore tha k k +ε fractio of the edges of G. There also exists a C k -free graph G which does ot cotai a bipartite ad C 4 -free subgraph with ore tha k k +ε fractio of the edges of G. Oe of our proofs uses the followig stateet, which we prove usig probabilistic ideas, geeralizig a theore of Erdős: For ay ε > 0, ad ay itegers a, b, k, there exists a a-uifor hypergraph H of girth greater tha k which does ot cotai ay b-colorable subhypergraph with ore tha b +ε fractio of the hyperedges of H. We also prove further geeralizatios of this a theore. I additio, we give a ew ad very short proof of a result of Küh ad Osthus, which states that every bipartite C k -free graph G cotais a C 4 -free subgraph with at least /k fractio of the edges of G. We also aswer a questio of Küh ad Osthus about C k -free graphs obtaied by pastig together C l s with k > l 3. Itroductio Let eh deote the uber of hyperedges i a hypergraph H. For a faily of graphs F, let ex,f deote the axiu uber of edges i a -vertex graph which does ot cotai ay F F as a subgraph. I the case whe F = {F}, we write siply ex,f. The girth of a graph is defied as the legth of a shortest cycle if it exists, ad ifiity otherwise. I [6], Győri proved that every bipartite, C 6 -free graph cotais a C 4 -free subgraph with at least half as ay edges. Later Küh ad Osthus [9] geeralized this result by showig Departet of Matheatics, Uiversity of Pisa. e-ail: groszdaielpub@gail.co Departet of Matheatics, Cetral Europea Uiversity, Budapest, Hugary. e-ail: abhishekethuku@gail.co Alfréd Réyi Istitute of Matheatics, Hugaria Acadey of Scieces. e-ail: ctopkis496@gail.co

2 Theore. Küh ad Osthus [9]. Let k 3 be a iteger ad G a C k -free bipartite graph. The G cotais a C 4 -free subgraph H with eh eg k. I Sectio we give a ew short proof of their result. The coplete bipartite graphs K k, for large eough show that the factor caot be replaced by aythig k larger see Propositio 5 i [9]. Füredi, Naor ad Verstraëte [5] gave aother geeralizatio of Győri s theore by showig that every C 6 -free graph G has a subgraph of girth larger tha 4 with at least half as ay edges as G. Agai, K, shows that this factor caot be iproved. It follows that ex,c 6 ex,{c 4,C 6 }. Sice ay graph has a bipartite subgraph with at least half as ay edges, Theore. shows that ex,c k k ex,{c 4,C k }. These results cofir special cases of the copactess coecture of Erdős ad Sioovits [4] which states that for every fiite faily F of graphs, there exists a F F such that ex,f = Oex,F. Sice ay C 6 -free graph cotais a bipartite subgraph with at least half as ay edges, usig ay of the results above it is easy to show that ay C 6 -free graph G has a bipartite, C 4 -free subgraph with at least of the edges of G. Győri, Kesell ad Topkis [7] 4 iproved this factor by showig that Theore. Győri, Kesell ad Topkis [7]. If c is the largest costat such that every C 6 -free graph G cotais a C 4 -free ad bipartite subgraph B with eb c eg, the 3 8 c 5. The coplete graph K 5 as well as a graph cosistig of vertex disoit K 5 s gives that c. To show that 3 c they use a probabilistic deletio procedure where they first 5 8 radoly two-color the vertices, ad the delete soe additioal edges carefully i order to reove the reaiig C 4 s. I this paper we show that c = 3. I fact, we prove the 8 followig two geeral results; puttig k = 3 i either of the stateets below gives that c = 3. To prove these theores we will costruct graphs by replacig the hyperedges of 8 certai probabilistically costructed hypergraphs with fixed sall graphs. Theore.3. For ay ε > 0, ad ay iteger k, there is a C k -free graph G which does ot cotai a bipartite subgraph of girth greater tha k with ore tha eg+ε edges. k k Note that K k isc k -free, ad the oly subgraphs with girth greater thak are forests, givig a upper boud of eg+ε. The factor is the probability that a k k rado two-colorig of K k is ot oochroatic. Theore.4. For ay ε > 0, ad ay iteger k, there is a C k -free graph G which does ot cotai a bipartite ad C 4 -free subgraph with ore tha eg+ε k k edges. Theore.4 iproves the upper boud of eg+ε, which is give by the coplete k bipartite graphs K k,. Take a rado bipartitio of the vertices of K k, ad cosider the bipartite subgraph B betwee the colour classes of this bipartitio. The factor i the above theore is the liit of the expected value of the fractio of k k

3 edges of K k, i the biggest C 4 -free subgraph of B as. Note that because ay graph has a bipartite subgraph with at least half of its edges, Theore. iplies that every C k -free graph cotais a bipartite ad C 4 -free subgraph with at least k fractio of its edges. Iterestigly, our proofs use theores about hypergraphs that are geeralizatios of the followig theore of Erdős []. Every graph G has a bipartite subgraph with at least as ay edges as G, ad the coplete graph K shows that the factor caot be iproved. Iterestigly, Erdős showed that eve if oe requires girth to be large, the factor still caot be iproved. More precisely, Theore.5 Erdős []. For ay ε > 0, ad ay iteger k, there exists a graph G with girth greater tha k which does ot cotai a bipartite subgraph with ore tha eg+ε edges. I Sectio 3, we prove a series of leas about hypergraphs which are broad geeralizatios of Theore.5, ad which ay be of idepedet iterest. These leas have the thee that for ost hypergraphs, every fixed colorig behaves like a rado colorig with color classes of the sae sizes as i the fixed colorig. Our proof of Theore.4 uses these geeral leas directly. The proof of Theore.3 uses a ore direct aalogue of the above stateet for hypergraphs: Theore.7, which we preset below. We will prove Theore.7 fro the ore geeral leas. A Berge-cycle of legth l i a hypergraph H is a subhypergraph cosistig of l distict hyperedges e,...,e l ad cotaiig l distict vertices v,...,v l called its defiig vertices, such that v i e i e i+, i =,...,l, where additio i the idices is take odulo l. The girth of a hypergraph H is the legth of a shortest Berge-cycle if it exists, ad ifiity otherwise. Note that havig girth greater tha iplies that o two hyperedges share ore tha oe vertex. A hypergraph is b-colorable if there is a colorig of its vertices usig b colors so that oe of its hyperedges are oochroatic. Erdős ad Haal [3] showed the existece of hypergraphs of ay uifority, arbitrarily high girth ad arbitrarily high chroatic uber. Lovász [0] gave a costructive proof for this; several ewer proofs exist as well. The followig siple propositio is easy to see. We iclude its proof for copleteess. Propositio.6. For ay itegers a,b, every a-uifor hypergraph H cotais a b-colorable subhypergraph with at least b a eh hyperedges. Proof. Color each vertex of H radoly ad idepedetly, usig b colors with equal probability. For each hyperedge f of H, the probability that f is oochroatic is b b a = b a. Therefore, the expected uber of oochroatic hyperedges i H is eh b a. So there exists a colorig of the vertices ofh such that there are at ost eh b a oochroatic hyperedges i that colorig. Thus, the subhypergraph of H cosistig of all the ooochroatic hyperedges of H cotais at least b a eh hyperedges ad is b- colorable, as desired. Agai the coplete a-uifor hypergraph shows that the factor b a caot be iproved i the above propositio. We show that as i case of graphs, this factor caot be iproved eve if oe requires the girth to be large. 3

4 Theore.7. For ay ε > 0, ad ay itegers a,b,k, there exists a a-uifor hypergraph H of girth ore tha k which does ot cotai a b-colorable subhypergraph with ore tha b a eh+ε hyperedges. Clearly, lettig a = b = i the above theore, we get Theore.5. The hypergraph leas i Sectio 3 ca be used to prove stateets aalogous to Theore.7 with differet otios of colorability. As a exaple applicatio, we will prove the aalogous Propositio 3.8 about strog or raibow colorable subhypergraphs. More geerally, a graphgis called H-colorable where H is a fixed graph if there is a hooorphis G H. Our Lea 3.4 ca be said to geeralize the otio of H-colorig to hypergraphs, ad allow for provig stateets siilar to Theore.7 for H-colorability or aalogous hypergraph coditios. I Sectio 5, we aswer a questio of Küh ad Osthus i [9]. A graph is said to be pasted together fro C l s if it ca be obtaied fro a C l by successively addig ew C l s which have at least oe edge i coo with the previous oes. Questio.8 Küh, Osthus [9]. Give itegers k > l, does there always exist a uber d = dk such that every C k -free graph which is pasted together fro C l s has average degree at ost d? Küh ad Osthus show i [9] that a affirative aswer to the above questio, eve whe restricted to bipartite graphs, would iply that ay C k -free graph G cotais a C l -free subgraph cotaiig a costat fractio of the edges of G. They gave a positive aswer to the questio whe l = ad the graph is bipartite: they showed that if k 3 is a iteger ad G is a bipartite C k -free graph which is obtaied by pastig together C 4 s, the the average degree of G is at ost 6k. We aswer Questio.8 egatively by showig two differet pastigs of C 6 s to for a C 8 -free graph with high average degree. These two exaples show i two very differet ways that ay C 6 s ca be packed ito a graph while still keepig it C 8 -free. We will show that the first exaple ca be easily geeralized to ay pair k,l with k > l 3, showig that l = is the oly case whe ay C k -free graph obtaied by pastig together C l s has average degree bouded by a costat d = dk. Our paper is orgaized as follows: I Sectio, we give a short proof of Theore.. I Sectio 3, we prove a series of hypergraph leas ad Theore.7. Our proofs i Sectio 3 use coutig arguets ad probabilistic ideas very siilar to Erdős s proof. I Sectio 4 we prove Theore.3 ad Theore.4. I Sectio 5, we give two exaples of pastig together C 6 s to for a C 8 -free graph with high average degree, aswerig Questio.8. A siple proof of a theore of Küh ad Osthus Theore. Proof of Theore.. Let G be a C k -free bipartite graph with color classes A := {a, a,...,a } ad B := {b,b,...,b } for soe,. Order the vertices i A ad B 4

5 a a a 3 a k a k... b b b 3 b k b k Figure : The solid edges for a C k as a < a <... < a ad b < b <... < b respectively. A edge ab EG with a A ad b B is deoted by the ordered pair a,b. We defie a partial order P = EG, p o the edge set of G as follows. For ay two edges a,b,a,b EG, we say that a,b p a,b if ad oly if there exists a iteger r ad edges p i,q i EG, i =,,...,r such that a = p,b = q ad a = p r,b = q r, ad the followig coditios hold: p i < p i+ ad q i < q i+ ad the vertices p i,q i,p i+,q i+ iduce a C 4 for all i r. It is easy to see that if there is a chai of legth k i P the G cotais a cycle of legth k, a cotradictio see Figure. So the legth of a logest chai i P is at ost k which iplies that the size of a largest atichai i P is at least EG k by Mirsky s theore []. Sice G is bipartite, ay C 4 i G cotais two edges p,q, p,q EG such that p,q < p p,q, so the subgraph H of G cosistig of the edges i this largest atichai is C 4 -free, copletig the proof of the theore. 3 Hypergraph leas ad proof of Theore.7 Let Ha,, deote the faily of all a-uifor hypergraphs with vertices ad hyperedges for soe a. Ha,, = a. Give a colorig C : [] [b] of the vertex set [] with b colors with b, let C be the uber of vertices of color. The ultiset of the colors of the vertices of a hyperedge e with the ultiplicity with which they occur i e is called the color ultiset of e with respect to C. For a a-eleet ultiset of colors T, let p C T be the probability that the color ultiset of a rado hyperedge of the coplete a-uifor hypergraph o vertices, with the colorig C, is T. Note that i this paper, whe we etio a colorig, we ea a arbitrary colorig of the vertex set, ot ecessarily a proper colorig of a hypergraph, uless idicated. Propositio 3.. For, asyptotically p C T = b C = I T a b = C IT a where I T deotes the ultiplicity of i the ultiset T. a! b = I T! We will also use the followig tail boud o the bioial ad the hypergeoetric distributios. Hoeffdig proves this boud i a ore geeral settig, see Sectio i [8] for the bioial distributio ad Sectio 6 for the hypergeoetric distributio. If a rado 5

6 variable X has bioial distributio with trials ad success probability p, we write X Bioial,p. If X has hypergeoetric distributio with a populatio of size N cotaiig pn successes, ad with draws, we write X HypergeoetricpN, N,. Propositio 3.. Let,N N ad p,ε [0,], ad let X be a rado variable with X Bioial, p or X HypergeoetricpN, N,. The P X p > ε e ε. Lea 3.3. Let ad. For ay fixed ε > 0, for every hypergraph H i Ha,,, with the exceptio of o a hypergraphs, the followig holds: For ay colorig C of the vertex set [] with b colors, ad ay a-eleet ultiset of colors T, the uber of hyperedges of H whose color ultiset is T is p C T±ε. Note: I this paper, wheever we write X = Y ±ε, we ea X [Y ε,y +ε]. Proof. Let u be the uber of hypergraphs i Ha,, for which does ot hold. Correspodig to each such hypergraph H there is at least oe b-colorig C of its vertices, ad a ultiset of colors T, such that does ot hold for C ad T. Therefore u {H,C,T : H Ha,,, does ot hold for H, C ad T}. The uber of b-colorigs of vertices with b fixed colors is C = b. The uber of ultisets of a eleets of b colors is a+b a. Therefore { } a+b u b H Ha,, : a does ot hold for H, C ad T. ax colorig C ultiset of colors T Fix a b-colorig C ad a ultiset of colors T. A hypergraph H Ha,, cosists of hyperedges, out of a possibilities. Out of all possible hyperedges, p C T a have T as their color ultiset. So { e H : c C e = T } Hypergeoetric p C T a, a,. fails to hold for H, C ad T if { e H : c C e = T } p C T > ε. By the tail boud for the hypergeoetric distributio i Propositio 3., the uber of hypergraphs H Ha,, for which this holds is at ost a e ε, so a+b u b e ε a = o a a as. 6

7 The followig lea is a corollary of Lea 3.3. Lea 3.4. Let T be a faily of ultisets of a eleets which are i [b]. Let ad. For a b-colorig of vertices C, let pc T = T T pc T that is, the probability that the color ultiset of a rado hyperedge of the coplete a-uifor hypergraph is i T ; ad let C M be a b-colorig for which p C T takes its axiu. For a hypergraph H Ha,,, let qh be the uber of hyperedges i the biggest subhypergraph of H which is colorable i such a way that the color ultiset of every hyperedge of H is i T. For ay fixed ε > 0, for every hypergraph H i Ha,,, with the exceptio of o a hypergraphs, qh p C M T+ε. Proof. If T =, the qh = 0 for ay H. Fro ow we assue that T. We show that we ay also assue that p C M T > T whe is sufficietly large. Let T T, ad b a let C E be a b-colorig i which every color class has a size. The, by Propositio 3., b asyptotically p C E a! T = b a b = I T! b a. Sice p C M T p C E T = T T pc E T, for sufficietly large, p C M T T. b a A equivalet defiitio of the fuctio q is qh = ax { e H : c C e T }. b-colorig C We use Lea 3.3 with ε b a i place of ε. For alost every hypergraph H Ha,,, for every colorig C, { e H : c C e T } = = usig that p C M T T b a. T T p C T + ε T b a { e H : c C e = T } T T p C M T + ε T b a p C T+ ε b a p C M T+ε We defie a orieted hypergraph as a set of ordered sequeces without repetitio called hyperedges over a vertex set, such that two hyperedges are ot allowed to differ oly i their order. The order of the vertices o differet hyperedges is idepedet of each other. A orieted hypergraph is thus equivalet to a hypergraph alog with a total order o the vertices of each hyperedge. Let Oa,, deote the faily of all a-uifor orieted hypergraphs with vertices ad hyperedges. Note that other eaigs of the ter orieted hypergraph exist i the literature. Let C : [] [b] be a colorig of the vertex set [] with b colors b. We call the color sequece with respect to C of a a-tuple of vertices e = v,...,v a the sequece c C e = Cv,...,Cv a. If we choose a rado a-tuple of the vertex set V 7

8 without repetitio, the probability that its color sequece is a give sequece of colors s = s,...,s a is a b a! = C! a C {i [a] : s i = }! if. The followig lea is a variat of Lea 3.3 for orieted hypergraphs. Lea 3.5. Let ad. For ay fixed ε > 0, for every orieted hypergraph O i Oa,,, with the exceptio of o Oa,, hypergraphs, the followig holds: i= C c i For ay colorig C of the vertex set [] with b colors, ad ay a-tuple of colors a s, the uber of hyperedges of O whose color sequece is s is C s i i=. ±ε 3 Proof. We use Lea 3.3 with ε i the place of ε, i.e. that holds with ε for alost 4 4 every hypergraph H Ha,,. I every hypergraph i Ha,,, the hyperedges ca be ordered i the sae uber of ways: a!. So for alost every O Oa,,, holds for the correspodig hypergraph obtaied by forgettig the orders o the hyperedges. Let Õa,, Oa,, be the faily of orieted hypergraphs for which holds forgettig the orders with ε i the place of ε. Let u be the uber of orieted hypergraphs i Õa,, for which 3 does ot hold. Correspodig to each such orieted 4 hypergraph O Õa,,, there is at least oe b-colorig C of its vertices, ad a a-tuple of colors s, such that 3 does ot hold for C ad s. Therefore { }. u O,C,s : O Õa,,, 3 does ot hold for O, C ad s The uber of b-colorigs of vertices with b fixed colors is C = b. The uber of a-tuples of b colors is b a. Therefore { } u b +a O,C,s : O ax Õa,,, colorig C 3 does ot hold for O, C ad s. a-tuple of colors s Fix a b-colorig C ad a a-tuple of colors s. Let T be the ultiset cosistig of the eleets of s with the ultiplicity with which they occur i s that is, T is s forgettig the order. If holds for a H Ha,, with ε, the uber of hyperedges whose 4 color ultiset is T is M H := p C T± ε b = C IT a! = 4 a b = I T! ± ε a = i= C s i a! b = I T! ± ε usig the Propositio 3. for large eough. Chagig ε to ε accouts for the fact 4 that Propositio 3. is asyptotic. We ca obtai a orieted hypergraph fro H by 8

9 orderig its hyperedges i oe of the a! possible ways, idepedetly fro each other. If we take a hyperedge whose color ultiset is T, soe of these orders yield the color sequece s. The uber of such orders is b = I T!, so if we take a rado orderig of a hyperedge whose color ultiset is T, the probability that it has color sequece s is b = I T!. a! So if we obtai a orieted hypergraph O by radoly orderig every hyperedge of H, the { e O : c C e = s } b Bioial = M H, I T!, ad the expected value of the uber of hyperedges whose color sequece is s is b = E H := I a T! M H = a! a! i= C s i ± ε If the uber of hyperedges whose color sequece issis i the rage [ E H ε,e H + ε], the 3 holds for O, C ad s, sice a E H ± ε = C s i ±ε. i= We wat to boud the probability that i a radoly selected orieted hypergraph obtaied fro H, the uber of hyperedges whose color sequece is s is ot i the rage [ [ EH ε,e H + ε] = E H ε M H M H,E H + ε M H M H ]. By the tail boud for the bioial distributio i Propositio 3., this probability is at ost e ε/m H M H = e ε / /M H e ε /, so u b +a e ε / Õa,, = o Oa,, as. Lea 3.6. Let, k ad = o + k. Every hypergraph H i Ha,,, with the exceptio of o a hypergraphs, has at ost Berge-cycles with k or fewer hyperedges. Proof. A Berge-cycle of legth l has l defiig vertices, ad each of its l hyperedges cotais a additioal vertices. So the uber of Berge-cycles of legth l is less tha a l. The uber of hypergraphs i Ha,, which cotai a fixed Bergecycle of legth l is a l l, sice the l hyperedges of the Berge-cycle ca be arbitrarily exteded to a hypergraph of hyperedges. Therefore the uber of pairs H,B where H Ha,, ad B is ay Berge-cycle of legth l i H, is less tha a l a l l < a l l a = O a. l a 9.

10 Let f k H deotes the uber of Berge-cycles of legth k or less i H. Usig = o + k, we have H Ha,, f k H = k l k O a = O a l= = o a. The uber of hypergraphs H Ha,, with ore tha Berge-cycles of legth k or less is clearly o a = o a, provig Lea 3.6. Propositio 3.7. For ay ε > 0 ad k, there exists a a-uifor hypergraph H of girth ore tha k for which i Lea 3.3 ad i Lea 3.4 hold. There also exists a a-uifor orieted hypergraph O of girth ore tha k usig the usual eaig of girth, ot takig the orders o the hyperedges ito cosideratio for which 3 i Lea 3.5 holds. Proof. Take a sufficietly large, ad = o + k but such that as. The there is a hypergraph H Ha,, such that i Lea 3.3 holds with ε 4b a i place of ε, ad H cotais at ost Berge-cycles with k or fewer hyperedges ideed, all but o a hypergraphs have both properties. Now reove a hyperedge fro every Berge-cycle of legth k or saller i H. The resultig hypergraph H has hyperedges. Fix ay colorig C ad a a-eleet ultiset of colors T. I H, the uber of hyperedges whose color ultiset with respect toc ist is p C T± ε 4b. The uber a of such hyperedges i H is at least p C T ε 4b ad at ost p C T+ ε a 4b, a so it is i the rage p C T± ε b for big eough because. So i a Lea 3.3 holds for H, eve with ε i the place of ε. Fro the proof of Lea 3.4 it b a is clear that if holds with ε, the holds. b a I every hypergraph i Ha,,, the hyperedges ca be ordered i the sae uber of ways, so Lea 3.6 holds for orieted hypergraphs too. The proof i the previous paragraph works siilarly for orieted hypergraphs, provig the existece of O. Now we use Propositio 3.7 to prove Theore.7. Proof of Theore.7. By Propositio 3.7, there is a a-uifor hypergraph H of girth ore tha k for which i Lea 3.4 holds. We use with T cosistig of those ultisets which cotai at least two differet colors, ad with ε i the place of ε. With the otatio of Lea 3.4, qh < p C M T + ε = p C M T + ε T T a b {{}}{ = p C M,,...,} + ε b C a M +ε = = 0

11 usig the asyptotic Propositio 3. for large eough. b = C M power ea iequality we get that b b = C M a a b. So b C M a =, which iplies the stateet. b a =, ad usig the We show aother exaple applicatio of Lea 3.4 ad Propositio 3.7. A b-colorig of the vertices of a hypergraph is called a raibow or strog colorig if all the vertices have differet colors i every hyperedge. For a-uifor hypergraphs, this is oly possible if a b. Propositio 3.8. Let ad. For ay fixed ε > 0 ad itegers a b, every hypergraph H i Ha,,, with the exceptio of o a hypergraphs, cotais o subhypergraph that is raibow colorable with b colors with ore tha b a! eh+ε a b a hyperedges. Furtherore, for ay ε > 0 ad itegers k ad a b, there exists a a-uifor hypergraph H of girth ore tha k which does ot cotai a subhypergraph that is raibow colorable with b colors with ore tha b a! eh+ε hyperedges. a b a Proof. A hypergraph colorig is a raibow colorig if the color ultiset of every hyperedge is a covetioal set i.e., every color appears at ost oce i the ultiset. Let T = [b] a. We will prove that if i Lea 3.4 holds for a hypergraph H with this T ad with ε i the place of ε, the it does ot cotai a subhypergraph that is raibow colorable with b colors with ore tha b a! eh+ε hyperedges. The first stateet of the a b a propositio the follows directly fro Lea 3.4, while the secod stateet follows fro Lea 3.7. With the otatio of Lea 3.4, ad usig the asyptotic Propositio 3. for large eough, qh < p C M T + ε = p C M T + ε T T C M a!+ε. 4 T T T We clai that, uder the assuptio that b = C M =, 4 takes its axiu whe C M =... = C M b =. Let us assue that the b C M s are ot all equal the there is a

12 ad such that C M T T T C M = < b < C M. Rewritig the first factor i 4, we have T {, } =0 T [b]\{, } a T T {, } = {}}{{}}{ C M + C M + C C M M T {, } = {}}{ + C M C C M M. T [b]\{, } a T T [b]\{, } a T If we replace C M with, b adc M with C M b +C M, 4 does ot decrease: the first two ters do ot chage, while i the third ter, C M C M is replaced by b C M b +C M = C M C M + C M b b C M > C M C M. Repeatig this step, we ca icrease the uber of C M s which equal without decreasig 4, util all of the equal. b b So qh T T T a!+ε = b b a! a b a+ε. 4 Subgraphs of C k -free graphs Proof of Theores.3 ad.4 Proof of Theore.3. Fix ε > 0. By Theore.7, there exists a k -uifor hypergraph H with girth ore tha k which does ot cotai a -colorable subhypergraph havig ore tha eh+ε hyperedges. We produce a graph k GH fro the hypergraph H by replacig each hyperedge of H with a coplete graph i.e. a clique o k vertices. We refer to these coplete graphs as k -cliques. It is easy to check that the resultig graph G H is C k -free. Notice that sice the girth of H is ore tha k 4, o two hyperedges of H itersect i ore thavertex. Therefore, thek -cliques ofg H are edge-disoit, ad by defiitio every edge of G H is i soe k -clique. We show that G H does ot have a bipartite subgraph with girth ore tha k which has ore tha k k k eg H+ε = eg k k H+ε edges. Assue that B is a bipartite subgraph of G H with girth ore tha k. Notice that ay set of ore tha k edges fro a clique o k vertices ust cotai a cycle of legth at ost k. Therefore B ca cotai at ost k edges fro each k -clique of G H. Furtherore, sice B is bipartite, there is a -colorig of the vertices so that the edges of B are properly colored. If a edge of B is cotaied i a k -clique of G H, the the correspodig hyperedge of H cotais two vertices with differet colors i this -colorig. By our assuptio o H, at ost +ε fractio of the hyperedges are ot oochroatic i this k -colorig of the vertices. So B has at ost k eh+ε edges. Sice k eg H = k eh, B has at ost eg k k H+ε edges, as desired.

13 I the proof of Theore.4, we use the followig propositio. For a proof, see the proof of Propositio 5 i [9]. Note that the boud ca be attaied whe w u. Propositio 4.. I the coplete bipartite graph K u,w, a C 4 -free subgraph has at ost w + u edges. Proof of Theore.4. Let l be a large iteger. By Propositio 3.7, there exists a k +l-uifor orieted hypergraph O with girth ore tha k for which 3 i Lea ε 3.5 holds with i place of ε. Let be the uber of vertices of O. We produce a 4 k +l graph G O fro the orieted hypergraph O by replacig each hyperedge of O with a copy of K k, the followig way: i a hyperedge v,...,v k +l, we coect every vertex i {v,...,v k } with every vertex i {v k,...,v k +l } with a edge. The resultig graph G O is C k -free. Sice the girth of O is ore tha k 4, o two hyperedges of O itersect i ore tha vertex. Therefore the copies of K k,l i G O are edge-disoit, ad by defiitio every edge of G O is i oe of the copies of K k,l. We show that G O does ot have a bipartite ad C 4 -free subgraph which has ore tha eg k k O + ε edges. Assue that B is a bipartite ad C 4 -free subgraph of G O, its classes beig p red vertices ad p blue vertices. Now cosider a rado hyperedge e = v,...,v k,v k,...,v k +l of O. How ay edges of B ca there be betwee the vertices of e? Each such edge has a red ad a blue edpoit; also, each such edge has a edpoit i {v,...,v k } ad a edpoit i {v k,...,v k +l }. Let u ad w be the uber of red vertices aog {v,...,v k } ad {v k,...,v k +l } respectively. The restrictio of B to the vertices of e which we will deote B e is thus a C 4 -free subgraph of the uio of a K u,l w ad a K k u,w o disoit vertex sets. We have three possibilities: u / {0,k }. The, by Propositio 4., B e cosists of at ost l w+ u +w+ k u < l+ k edges. u = k. The K k u,w is degeerate as k u = 0, ad B e has at ost l w+ k edges. u = 0. The K u,l w is degeerate, adb e has at ost w+ k = l+ k l w edges. Let C,...,C k +l be the color sequece of e with C i {red,blue}. For ay color sequece c,...,c k +l with c i {red,blue}, the probability that C,...,C k +l = c,...,c k +l is p {i:ci=red} p {i:ci=blue} ε ± sice 3 i Lea 3.5 holds for O 4 k +l ε with. Note that e was chose as a rado hyperedge of O. Let C 4 k +l,..., C k +l be idepedet rado variables which take the value red with probability p ad the value blue with probability p. Let fc,...,c k +l be a real valued fuctio of a color sequece. We clai that EfC,...,C k +l Ef C,..., C k +l ε ax f. 4 5 Ideed, Ef C,..., C k +l = p {i:ci=red} p {i:ci=blue} fc,...,c k +l, ad c,...,c k +l {red,blue} k +l 3

14 EfC,...,C k +l = c,...,c k +l {red,blue} k +l p {i:c i=red} p {i:c i=blue} ± fc,...,c k +l = Ef C,..., C k +l ε + ± fc 4 k +l,...,c k +l c,...,c k +l {red,blue} k +l ε 4 k +l = Ef C,..., C k +l ± ε 4 ax f. { if C =... = C k = red Usig 5 with fc,...,c k +l =, we have Pu = k 0 otherwise { = EI u=k = p k ± ε ; with fc if C =... = C k = blue 4,...,C k +l =, 0 otherwise we have Pu = 0 = EI u=0 = p k ± ε ; ad with fc 4,...,C k +l = {i {k,...,k +l} : C i = red}, we have Ew = pl± ε l. So 4 k k EeB e = Pu / {0,k } l+ +Pu = k E l w+ k +Pu = 0E l + l w k l+ p k ± ε p± ε l p k ± ε p± ε l k l+ p k l p k l+ ε4 l k l+ + ε k 4 l assuig ε. That is, if O has hyperedges, eb = l + k k + ε l, while eg 4 O = k l. Let l kk, the ε eb ε + eg O k k k + k l k + ε k eg O 4k k 5 Pastig C 6 s to produce a C 8 -free graph k eg O+ε. We will ake use of the followig propositio of Nešetřil ad Rödl [] i the secod exaple, ad i the geeral versio of the first exaple. Propositio 5. Nešetřil, Rödl []. For ay positive itegers r ad s 3, there exists a 0 N such that for ay iteger 0 there is a r-uifor hypergraph with girth at least s ad ore tha +/s hyperedges. 4

15 a a a 3 a 4 a 5 a 6 a a b b b 3 b 4 b 5 b 6 b b b b b 3 b 4 b 5 b 6 b b a a a 3 a 4 a 5 a 6 a a Figure : First pastig 5. First exaple For our costructio here we will eed a bipartite graph of girth at least 0 with ay edges ad with degree at least i every vertex. We will derive such a graph fro the followig costructio of Beso []. Theore 5. Beso []. Let q be a odd prie power. There is a q + -regular, bipartite, girth graph Q with q 5 +q 4 +q 3 +q +q + vertices. First, let us otice that sice Q is a regular bipartite graph, it has color classes of equal size. Moreover, we ay assue that Q is coected, for otherwise we ay add soe edges to ake it coected without creatig cycles. So we have the followig corollary. Corollary 5.3. There exists a coected bipartite graph of girth at least 0 with / vertices i each color class such that every vertex has degree at least / /5 o. So it cotais at least o/ 6/5 edges. Theore 5.4. There exists a C 8 -free graph G o 4 vertices with average degree at least 4 /5 which is pasted together fro C 6 s. To prove Theore 5.4, let us take a coected, bipartite graph G of girth at least 0 o vertices such that each vertex has degree at least /5 o ad havig vertices i each color class. The existece of such a graph is guarateed by Corollary 5.3. Let a,a,...,a ad b,b,...,b be the two color classes of G. Now let G be a copy of G with vertices a,a,...,a ad b,b,...,b ad edge set EG = {a i b a ib EG }. Fially, the graph G is defied to have the vertex set VG = VG VG ad the edge set EG = EG EG {b i b i i } see Figure. So G has 4 vertices ad 6/5 o+ edges. To show thatgis pasted together froc 6 s, we have to show that every edge is cotaied i a C 6, ad for ay two edges e,e EG, there is a sequece O,O,...,O of C 6 s i G such that for ay i, O i ad O i+ share at least oe edge, ad e ad 5

16 e are edges of O ad O respectively. The graph ca the be built startig fro a arbitrary fixed edge. It is easy to see that every edge is cotaied i soe C 6 of the for a i b b a i b k b k, ad so we ca assue that both e ad e are of the for a i b. Let a i0 b i a i b i3 a i4 b i5...a it b it a it+ be a path startig with e ad edig with e, with e = a i0 b i or e = b i a i, ad e = a it b it or e = b it a it+ such a path exists sice G is coected. The the path b i a i b i 3 a i 4 b i 5...a i t b i t is cotaied i G. These two paths together with the edges b i b i,b i b i,...,b it b i t give the desired sequece of C 6 s together with a arbitrary C 6 of the for a i0 b i b i a i 0 b b if e = a i0 b i, ad a C 6 of the for a it+ b it b i t a i t+ b k b k if e = b it a it+. It reais to show that G is C 8 -free. Suppose for a cotradictio that it has a C 8. Sice the graph G is of girth at least 0, this C 8 caot be copletely i G or G. So it has to cotai at least oe edge fro each of the three sets EG, EG ad {b i b i i }. Moreover, it is easy to see that it cotais exactly two edges fro {b i b i i }, say b i b i ad b b. So there is a path of q edges betwee b i ad b i G ad a path of r edges betwee b i ad b i G such that q +r = 6. Let these paths be b i a i b i...a iq b ad b i a b...a r b respectively. By costructio, the secod path i G iplies that G cotais the path b i a b...a r b, which, together with b i a i b i...a iq b, would produce a cycle of legth 4 or 6 i G, a cotradictio. Reark 5.5. We ay odify the above costructio as described below to fid a pastig of C l s to produce a C k -free graph G for ay give itegers k > l 3 ad havig average degree at least Ω /k+. Proof. A graph of girth k+ ad havig Ω +/k+ edges exists by applyig Propositio 5. with r =. So it has average degree Ω /k+. It is easy to fid a bipartite subgraph of such a graph, with equal color classes ad havig a costat fractio of all the edges. The we ca delete vertices of degree without decreasig its average degree, so we ca assue it has iiu degree at least, ad as usual, we ca assue it is coected, because otherwise we ca add edges without creatig a cycle to ake it coected. Let G be this bipartite, coected graph of girth greater tha k o vertices with average degree Ω /k+. The let G be defied i the sae way as i the above proof based o G. However, ow, for each i we coect the vertices b i VG ad b i VG by a path cotaiig l edges ad let the resultig graph be G. Usig the sae arguet as i the above proof, we ca see that this gives a pastig of C l s ad that G is C k -free. 5. Secod exaple A hypergraph H is coected if for ay two vertices u,v VH, there exist hyperedges h i EH, i, such that u h,v h ad h i h i+ for all i. A iial collectio of such hyperedges is called a path betwee u ad v i H. We ay assue that the hypergraph provided by Propositio 5. is coected, for otherwise we ca siply take a coected copoet of it cotaiig the biggest uber of hyperedges. Theore 5.6. There exists a C 8 -free graph G o vertices with average degree at least 6 /9 which is pasted together fro C 6 s. 6

17 To prove Theore 5.6, we apply Propositio 5. to fid a coected 3-uifor hypergraph H o vertices with girth at least 9 ad ore tha +/9 hyperedges. Let VH = {u,u,...,u }. Replace each vertex u i VH with a pair of vertices u i,u i so that every hyperedge cotaiig u i ow cotais both u i ad u i. This produces a 6-uifor hypergraph which we deote by H. Now we costruct the desired graph G fro H i the followig fashio. If {u i,u i,u,u, u k,u k } is a hyperedge i H with i k, the we add the edges u i u i,u i u,u u,u u k,u k u k,u k u i to EG. We repeat this procedure for every hyperedge of H. Let us call the edges u i u i EG i fat edges ad the rest of the edges of G thi edges. Note that two fat edges ever share a vertex. We clai that a thi edge caot be added by two differet hyperedges of H. Suppose by cotradictio that u i u is a thi edge added by two differet hyperedges h,h of H. The sice a hyperedge of H either cotais both vertices u r,u r or either of the for ay give r, it follows that {u i,u i,u,u } h ad {u i,u i,u,u } h. Cosider the two hyperedges i H which correspod to h ad h. They both cotai u i ad u ; so they itersect i at least two vertices, which is a cotradictio sice H is a liear hypergraph. Notice, o the other had, that a fat edge ay have bee added by several hyperedges. So each hyperedge i H adds precisely 3 ew thi edges to EG. Therefore the uber of thi edges i G is three ties the uber of hyperedges i H. Sice the uber of fat edges is, we have EG = 3 +/9 +. Thus it has the desired average degree. Sice H is coected, we ca costruct it by addig hyperedges oe by oe, i such a way that each hyperedge itersects oe of the previous hyperedges i at least oe vertex. We ca costruct H by addig the C 6 s correspodig to the hyperedges of H i the sae order; this shows that G is pasted together fro C 6 s. It oly reais to show that G is C 8 -free. We say a edge is betwee two edges e, e if oe of its ed vertices is i e ad the other is i e. Clai 5.7. There is at ost oe thi edge betwee ay two fat edges of G. Proof. Cosider ay two fat edges u i u i ad u u of G. As oted earlier, ay thi edge betwee the is added by a uique hyperedge h of H, ad h cotais all four vertices u i,u i,u,u. Because of the liearity of H, o hyperedge of H other tha h ay cotai all four vertices u i,u i,u,u Now ote that i our procedure, ay hyperedge of H adds at ost oe thi edge betwee ay two fat edges cotaied i it, provig the clai. Now suppose for a cotradictio that G cotais a C 8. Sice o two fat edges i G share a vertex, there ca be at ost four fat edges i this C 8. Cotract every pair of vertices u i,u i i G ito a sigle vertex u i. The this C 8 would becoe a closed walk C of legth at ost 8 ad at least 4 oly thi edges reai after cotractio. While this closed walk ay have repeated vertices, we show that it caot have repeated edges i.e., it is actually a circuit. Suppose that after cotractig every pair of vertices u i,u i to u i, soe two thi edges xy ad zw coicide. The, for soe i ad, we have x,z {u i,u i} ad y,w {u,u }. Betwee the fat edges u iu i ad u u, there are two thi edges aely xy ad zw, cotradictig Clai 5.7. The -shadow of a hypergraph H is the graph which cotais a edge uv if ad oly if there is a hyperedge of H which cotais u ad v. C ust be cotaied i the -shadow 7

18 of H. Sice H has girth at least 9, it is ot difficult to see that the oly possible legth of a circuit i its -shadow that is betwee 4 ad 8 is 6 ad it ust be of the for ab,bc,ce,ed,dc,ca otice that c is a repeated vertex. Therefore, the origial C 8 i G ust be cotaied i the set of edges added by the hyperedges {a,a,b,b,c,c } ad {c,c,d,d,e,e } of H, but this is ipossible as these edges cosist of two C 6 s sharig exactly oe edge. Ackowledgets We are grateful to Mariaa Bolla, Ervi Győri ad Döötör Pálvölgyi for poitig us to helpful refereces ad for useful discussios. The authors Methuku ad Topkis were supported by the Natioal Research, Developet ad Iovatio Office NKFIH uder the grat K6769. Refereces [] Clark Beso. Miial regular graphs of girth eight ad twelve. Caad. J. Math, 8:94, 966. [] Pál Erdős. Gráfok páros körülárású részgráfairól O bipartite subgraphs of graphs, i Hugaria. Mat. Lapok, 8:83 88, 967. [3] Pál Erdős ad Adrás Haal. O chroatic uber of graphs ad set-systes. Acta Matheatica Hugarica, 7-:6 99, [4] Pál Erdős ad Miklós Sioovits. Copactess results i extreal graph theory. Cobiatorica, 3:75 88, 98. [5] Zoltá Füredi, Assaf Naor, ad Jacques Verstraëte. O the Turá uber for the hexago. Adv. Math., 03: , 006. [6] Ervi Győri. C 6 -free bipartite graphs ad product represetatio of squares. Discrete Math., 65/66:37 375, 997. Graphs ad cobiatorics Marseille, 995. [7] Ervi Győri, Scott Kesell, ad Casey Topkis. Makig a C 6 -free graph C 4 -free ad bipartite. Discrete Applied Matheatics, 05. [8] Wassily Hoeffdig. Probability iequalities for sus of bouded rado variables. Joural of the Aerica Statistical Associatio, 5830:3 30, 963. [9] Daiela Küh ad Deryk Osthus. Four-cycles i graphs without a give eve cycle. J. Graph Theory, 48:47 56, 005. [0] László Lovász. O chroatic uber of fiite set-systes. Acta Matheatica Hugarica, 9-:59 67, [] Leo Mirsky. A dual of Dilworth s decopositio theore. The Aerica Matheatical Mothly, 788: , 97. 8

19 [] Jaroslav Nešetřil ad Votěch Rödl. O a probabilistic graph-theoretical ethod. Proceedigs of the Aerica Matheatical Society, 7:47 4,