Dense H-free graphs are almost (χ(h) 1)-partite

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1 Dese H-free graphs are almost χh) 1)-partite Peter Alle arxiv: v1 [math.co] 22 Jul 2009 July 22, 2009 Abstract By usig the Szemerédi Regularity Lemma [9], Alo ad Sudakov [1] recetly exteded the classical Adrásfai-Erdős-Sós theorem [2] to cover geeral graphs. We prove, without usig the Regularity Lemma, that the followig stroger statemet is true. Give ay r + 1)-partite graph H whose smallest part has t vertices, ad ay fixed ε > 0, there exists a costat C such that wheever G is a -vertex graph with miimum degree δg) 1 ) r 1 + ε, either G cotais H, or we ca delete C 2 1 t graph. 1 Itroductio edges from G to obtai a r-partite We recall the classical theorem of Turá [10] we give oly the miimum degree case): Theorem 1. If the -vertex graph G has miimum degree exceedig 1 1 the G cotais K r+1. As is well kow, this theorem is best possible, the extremal example beig the r-partite Turá graph. A old result of Adrásfai, Erdős ad Sós [2], which amouts to a very strog) stability result for Turá s theorem, is the followig. Theorem 2. Suppose r 2. If the -vertex graph G has miimum degree exceedig 1 r 1) ad G does ot cotai Kr+1, the G is r-partite. DIMAP ad Mathematics Istitute, Uiversity of Warwick, Covetry, CV4 7AL, U.K. p.d.alle@warwick.ac.uk. Research supported by the Cetre for Discrete Mathematics ad its Applicatios, EPSRC award EP/D06191/1. 1

2 This theorem is best possible; however the extremal example is a little more complex tha the Turá graph. We costruct a graph E r ) as follows: we partitio vertices ito r 2 sets X 1,..., X r 2 each cotaiig vertices ad five sets Y r 1 1,...,Y 5 each cotaiig vertices. Each of these sets is idepedet; we set every vertex i each X r 1 i adjacet to all vertices outside X i, ad we make Y i, Y i+1 mod 5 ) a complete bipartite graph for each i so that the five sets form a blow-up of C 5 ). It is straightforward to check that each vertex has degree 1 r 1) ; sice χc5 ) = the chromatic umber of E r ) is r + 1, but E r ) does ot cotai K r+1. Erdős ad Stoe [6] exteded Turá s theorem, showig that for ay fixed graph H, the chromatic umber of H govers the miimum degree threshold at which H appears i a large graph G: Theorem. Let H be ay fixed graph with chromatic umber r + 1. If the -vertex graph G has miimum degree exceedig 1 1 r + o1)) the G cotais H. Although the extremal graphs for this theorem are ot ecessarily r-partite, it is true that oe may delete o 2 ) edges from ay extremal graph to obtai a r-partite graph. Ideed, it is ot hard to show that there exists = H) > 0 such that deletio of oly O 2 ) edges from a extremal graph yields a r-partite graph. Quite recetly, Alo ad Sudakov [1] gave a extesio of Adrásfai, Erdős ad Sós result to cover all fixed graphs H Erdős ad Simoovits [5] had previously cosidered the case whe H is critical, i.e. whe there is a edge of H whose removal decreases the chromatic umbe: Theorem 4. Let ay fixed graph H with chromatic umber r + 1 ad costat ε > 0 be give. The there exist = H) > 0 ad 0 = 0 H, ε) such that the followig holds. If 0 ad G is a -vertex graph with miimum degree exceedig 1 + ε) which r 1 does ot cotai H, the oe ca delete at most O 2 ) edges from G to yield a r-partite graph. Alo ad Sudakov gave a explicit formula for : if H K r+1 t) the we may take = 1. Ufortuately, their proof relies upo the Szemerédi Regularity Lemma [9], as a 4r 2/ t result of which the costat 0 has a exceptioally upleasat depedece o ε, r ad t. The purpose of this paper is to give a simpler proof, avoidig the use of the Regularity Lemma. We also improve H) to a value which we believe to be best possible. Theorem 5. Let ε > 0 ad ay fixed graph H with chromatic umber r+1 ad smallest part i a r+1)-partitio of size t be give. The there exists C = Cε, H) such that the followig holds. Wheever G is a -vertex graph with miimum degree exceedig 1 + ε) r 1 which does ot cotai H, the oe ca delete at most C 2 1 t edges from G to yield a r-partite graph. We ote that the case r = 1 is the Zarakiewicz problem. The classical theorem of Kövári, Sós ad Turá [7] o this problem is the followig. 2

3 Theorem 6. Let 1 t s be fixed itegers. If G is ay -vertex graph with Ω 2 1 t ) edges, the G cotais K t,s. I fact, it is the Kövári-Sós-Turá theorem which yields our value of : ay improvemet i their boud would immediately improve ours similarly. We ote that for t = 1, 2, there exist lower boud costructios matchig the upper boud of the Kövári-Sós-Turá see [8, ]); for t 4 the best kow lower boud is Ω 2 2 t+1 ), but it is cojectured that the correct boud is Ω 2 1 t ). We give three costructios which demostrate the tightess of our theorem. First, for t = 1, cosider the graph T r,1,s ) obtaied from the r-partite Turá graph T r ) by replacig oe part with a graph of maximum degree s 1. This graph satisfies the miimum degree requiremet of Theorem 5. Sice every copy of K r+1 i T r,1,s ) must use oe of the iserted edges, it does ot cotai the r + 1-partite graph K 1,s,s,...,s ; ad to make it r-partite we must remove Ω) edges, matchig the expoet i Theorem 5. Secod, for t 2, we let the graph T r,t ) be obtaied similarly, this time replacig oe part of T r ) with a extremal K t,t -free graph. Agai this satisfies the miimum degree requiremets of Theorem 5 ad does ot cotai K r+1 t), ad to make it r-partite we must remove all the edges of the K t,t -free graph; for t = 2, this is Ω 2 1 t ) edges, matchig Theorem 5, while for t 4 it is Ω 2 2 t+1 ) edges. Agai, a improvemet to the lower boud costructios for the Zarakiewicz problem would automatically improve our lower boud. Alo ad Sudakov asked whether it is possible to replace the term ε i the miimum degree of their theorem with a O1) term. It is ot possible; ideed, for ay µ > 0 there are graphs H such that the correspodig term must be larger tha 1 µ. Cosider the followig modificatio of E r ). We replace the idepedet sets Y 1 ad Y with K t,t -free graphs of largest possible miimum degree. We icrease the sizes of these two sets, ad simultaeously decrease the sizes of Y 2, Y 4, Y 5 ad each X i, by Ω 1 2 t+1 ). This yields a graph E r,t ) whose miimum degree is ) 1 r 1 + Ω 1 2 t+1 ). Every copy of K r+1 i E r,t) must use a edge of either Y 1 or Y ; there are o edges betwee Y 1 ad Y, ad thus a copy of K r+1 t) would have to use a copy of K t,t i either Y 1 or Y, which does ot exist. The graph E r,t) caot be made r-partite without removig Ω 2 ) edges. A careful aalysis of our proof will yield the fact that for every H there exists µh) > 0 such that we ca take ε = O µ ). However the value of µh) obtaied seems certai to be far from optimal; we leave its computatio as a exercise for the reader. 2 Costructig complete r + 1)-partite graphs We describe a costructio based o the Kövári-Sós-Turá theorem which allows us to fid the r+1)-partite graph K t,s,...,s from a suitably well-structured) set of copies of K r+1. This costructio improves that used by Alo ad Sudakov: specifically, their costructio as its first step chooses oe specific complete bipartite graph which will cotai two of the parts

4 of their fial K r+1 t). Because this object is fixed i the later stages of their costructio, it is ecessary that the part sizes of this first bipartite graph are ot t but much larger i particular, the part sizes must deped o r ad this is the cause of the relatively small value of that they obtai. Our costructio avoids this choice, relyig istead o coutig objects of the desired size util the fial step. We give first a coutig variat of a lemma of Erdős [4]; this is essetially a statemet about hypergraphs. Lemma 7. For every r ad s there exists a fuctio δ = δ r,s ε) > 0 such that the followig holds for sufficietly large. If the -vertex graph G cotais at least ε copies of Kr, the G cotais δ r,s ε) rs) copies of Kr s). Proof. For r = 1 the statemet holds trivially. We complete the proof by iductio. Let G be a -vertex graph cotaiig ε copies of Kr : the there are some ε vertices D of G which are each cotaied i ε r 1) copies of Kr i G. Give v V G), let the eighbourhood graph G v be the subgraph of G obtaied by discardig all edges which are ot cotaied i Γv). By costructio, for each d D, G d cotais ε r 1) copies of Kr 1 ; by iductio it cotais δ r 1,s ε) r 1)s) copies of Kr 1 s). For a give copy S of K r 1 s), let d S be the umber of vertices of D whose eighbourhoods cotai S. The we have usig the covetio that ) a b = 0 whe a < b) at least 1 ds ) r S s copies of K r s) cotaied i G. Sice the average value of d S is δ r 1,s ε) D, applyig Jese s iequality the umber of copies of K r s) i G is at least 1 ) ) ) ) ds δr 1,s D δ r 1,s ε) /r δ r,s ε), r s r 1)s s rs as required. S It is immediately clear that if we have a large) graph G which does ot cotai the r- partite graph K t,s,...,s, the it certaily does ot cotai ε r+1) copies of Kr+1. Furthermore, there are at most tδr,s 1ε) vertices whose eighbourhood graphs cotai ε copies of Kr. To complete our costructio, we give the followig simple result. Corollary 8. Give ε > 0, r, s ad t there exists C such that if is sufficietly large the followig is true. Every -vertex graph G i which there are C 2 1 t edges E of G each cotaied i ε r 1) copies of Kr+1 cotais K t,s,...,s. Proof. Give such a graph, by Lemma 7 each edge of E is cotaied i δ r 1,s ε) r 1)s) copies of the r + 1-partite graph K 1,1,s,...,s. It follows that at least δ r 1,s ε) E edges E E form K 1,1,s,...,s with the same set S of r 1)s vertices. Provided that C is sufficietly large, by Theorem 6 this set E cotais K t,s, ad hece G cotais K t,s,...,s as desired. Note that the value of δ r,s ε) give by Lemma 7 is clearly far smaller tha the truth; but this affects oly the costat C i this corollary ad ot the expoet 2 1 t. 4

5 Proof of Theorem 5 We first prove a desity versio of Theorem 2. Lemma 9. Give ε > 0 there exist positive costats η 0, η 1,... such that, give r, wheever is sufficietly large, the followig is true. Ay -vertex graph G with δg) > 1 r 1 + 2r ε ) either cotais more tha η r r+1 copies of K r+1, or has a partitio ito D V 1... V r, with the properties that G[V i ]) 2 r 2 ε for each i, each vertex of D is cotaied i at least η r 1 r copies of K r+1, ad D 2 r 2 ε. Proof. We set η 0 = ε ad η 4 1 = ε2. The the statemet for r = 1 is equivalet to the fact that 2 a graph with more tha ε vertices of degree at least η 4 0 = ε ε2 has more tha = η 1 2 edges. We complete the proof by iductio; we will give several upper bouds for η r ad fially choose ay sufficietly small value at the ed. Suppose r 2. We isist that η r ε r+1 /r + 1)!. Let G be a -vertex graph with miimum degree 1 r 1 + 2r ε ). Let D V G) be a maximum set of vertices such that for each d D there are at least η r 1 r copies of K r i Γd). We isist that η r 2r εη r 1 4r+1). We have 1 r+1 D η r 1 r copies of K r+1 i G; so we will be doe uless D 2 r 2 ε. Let G = G[V G) D]. This graph has miimum degree r 4 r 1 + ε2r 2) ; oe of its vertices are cotaied i more tha η r 1 r copies of K r. Fact 1. For every v V G ), by iductio, we ca partitio G [Γv)] ito sets B X 1... X r 1 with B 2 r ε ad G [X i ]) 2 r ε for each i. We geerate a partitio of V G ) as follows. First, we apply the iductio Fact with ay vertex i V G ). Secod, we forget all but the largest of the sets X i, which we take to be X 1, ad apply the Fact agai with oe of the vertices of X 1 to partitio its eighbourhood graph i G. This yields r 1 further sets X 2,...,X r, which have i total at most 2r ε vertices i commo with X 1. We let X i = X i X 1 for each 2 i r. If ow X i > X 1 for some i we iterate this secod step, forgettig all sets but X i. Whe the process termiates, we let L be the vertices of G which are ot i ay of the X i. By costructio, we have X X r ) r 4 r r 1 + 2r 1 ε r 1 r 4)r r 1)r 1) + 2r 1 ε, ad so L r 1)r 1) 2r 1 ε. 2 We have also that X r 1 i for each i by costructio. For coveiece, we r 1 presume that the X i are i order of decreasig size. Now, if we have ay two adjacet vertices u ad v of G whose codegree exceeds r 6 r 1 +ε, the we may costruct a clique K r+1 extedig uv greedily by simply pickig ay commo eighbour of the so far chose vertices at each step. At the fial step ad therefore at all 5

6 steps) we have at least ε choices. If furthermore there were 2 r 2 ε eighbours of u possessig similarly large codegrees with u, the sice r 2) we have i total ε r r /r! η r 1 r copies of K r+1 cotaiig u, which cotradicts u ot beig i D. Sice G[X i ]) 2 r ε, if a vertex u outside X i has less tha X i eighbours i r 1 X i, the the codegree of u ad ay eighbour v X i exceeds r 6 +ε. It follows that ay r 1 vertex outside X i has either fewer tha 2 r 2 ε eighbours i X i or more tha X i r 1 eighbours i X i. Cosider the set L i of vertices of L which all have less tha 2 r 2 ε eighbours i X i. Ay oe of these vertices has codegree exceedig r 6 + ε with ay other, ad with ay r 1 vertex of X i. It follows that L i X i has maximum degree 2 r 2 ε. Let this set be V i. Let the vertices of G ot i ay X i be L. If L = the we have V G) = D V 1... V r is the desired partitio. So we may assume there is a vertex l L. This vertex must be adjacet to all but at most vertices r 1 of each set V i. We costruct a K r+1 extedig l greedily; we first choose ay eighbour v 1 of l i V 1, v 2 of l i V 2, ad so o. We have V 1 choices for v r 1 1, V 2 V r r 2 ε) r 4 r 1 2r ε) = V 1 + V r 1 2r 1 ε choices for v 2, ad so o. Fially, for v r we have V V r 1+r 1) +r 1)2 r 1 ε choices. But ote 1+r 1) = r 2 < V r 1 r 1 r V r, sice the sets V i exted the X i. Sice the V i are i order of decreasig size, at each step we certaily had at least 2 r 1 ε choices; so there are more tha ε r /r! r η r 1 r copies of K r+1 cotaiig l, cotradictig its ot beig i D. This completes the proof. Note that oe could set ε = η 0 =... = η r i the above to obtai a proof of the Adrásfai- Erdős-Sós theorem. Proof of Theorem 5. Give ε > 0, let G be a sufficietly large -vertex graph with δg) 1 + ε) which does ot cotai the r + 1-partite graph K r 1 t,s,...,s. By Lemma 9 there exist positive costats η, δ such that either G cotais η r+1) copies of Kr+1 or V G) may be partitioed as V G) = D V 1... V r, where G[V i ]) ε/4 for each i, where each vertex of D is cotaied i at least δ copies of Kr+1, ad where D ε/4. Whe is sufficietly large, by Lemma 7 every graph G with η r+1) copies of Kr+1 cotais K r+1 s) ad thus K t,s,...,s. It follows that V G) possesses the give partitio. Whe is sufficietly large, by Lemma 7 there is δ > 0 such that every -vertex graph with δ ) r copies of Kr cotais δ rs) copies of Kr s). Applyig this to the eighbourhood graphs G d for each d D, we have that if D t/δ the there exists some copy S of K r s) which is i the eighbourhood of t distict vertices of D; the S together with these t vertices form K t,s,...,s i G. It follows that D < t/δ. Fially, let E be the set of edges of G which are cotaied i ay oe of the sets V i. For ay edge uv E, the commo eighbourhood of u ad v i V G) D cotais at least 2 r 4 V r 1 i ) r 5 vertices, sice both u ad v are adjacet to at most ε/4 vertices r 1 of V i. As before, we ca exted uv to a clique K r+1 by choosig vertices greedily; at each 6

7 stage we have at least choices, ad hece uv is cotaied i at least r 1 of K r+1. By Lemma 8, sice G does ot cotai K t,s,...,s, there exists C such that G cotais fewer tha C 2 1 t such edges. r 1 r 1) r 1 r 1)! copies Upo deletig from G all edges icidet to D or cotaied i E, oe obtais a r-partite graph. The total umber of such edges is at most t/δ + C 2 1 t C 2 1 t as required. Ackowledgemet The author would like to thak Daiela Küh ad Deryk Osthus for suggestig this ice problem. Refereces [1] N. Alo ad B. Sudakov, H-free graphs of large miimum degree, Elec. J. Combi ), R19. [2] B. Adrásfai, P. Erdős ad V. T. Sós, O the coectio betwee chromatic umber, maximal clique ad miimal degree of a graph, Discrete Math ), [] W. G. Brow, O graphs that do ot cotai a Thomse graph, Caad. Math. Bull ), [4] P. Erdős, O extremal problems of graphs ad geeralized graphs, Israel J. Math ), [5] P. Erdős ad M. Simoovits, O a valece problem i extremal graph theory, Discrete Math ) 2 4. [6] P. Erdős ad A. Stoe, O the structure of liear graphs, Bull. Amer. Math. Soc ), [7] T. Kövári, V. Sós ad P. Turá, O a problem of K. Zarakiewicz, Colloquium Math. 1954), [8] I. Reima, Über ei Problem vo K. Zarakiewicz, Acta. Math. Acad. Sci. Hugar ), [9] E. Szemerédi, Regular partitios of graphs, Colloques Iteratioaux C.N.R.S. Vol. 260, i Problèmes Combiatoires et Théorie des Graphes, Orsay 1976), [10] P. Turá, Eie Extremalaufgabe aus der Graphetheorie, Mat. Fiz. Lapok ),