Large holes i quasi-radom graphs Joaa Polcy Departmet of Discrete Mathematics Adam Mickiewicz Uiversity Pozań, Polad joaska@amuedupl Submitted: Nov 23, 2006; Accepted: Apr 10, 2008; Published: Apr 18, 2008 Abstract Quasi-radom graphs have the property that the desities of almost all pairs of large subsets of vertices are similar, ad therefore we caot expect too large empty or complete bipartite iduced subgraphs i these graphs I this paper we aswer the questio what is the largest possible size of such subgraphs As a applicatio, a degree coditio that guaratees the coectio by short paths i quasi-radom pairs is stated 1 Itroductio Szemerédi s Regularity Lemma for graphs [6] has become oe of the most importat tools i the moder graph theory Whe solvig certai problems, this Lemma allows us to cocetrate o quasi-radom subgraphs (called also ε-regular pairs) istead of cosiderig the whole graph Notable examples of this method ca be foud i [2, 3] This approach is very coveiet sice such regular pairs have a lot of ice properties I particular, i quasi-radom graphs the desities of almost all pairs of large subsets of vertices are similar, ad therefore we caot expect too large empty iduced subgraphs (holes) i these graphs The problem of holes i ε-regular pairs was already studied i [4] Let h(ε, d, ) be defied as the largest iteger h such that every balaced bipartite graph G with 2 vertices ad desity at least d, cotais a subgraph H o h + h vertices ad with o hole with at least ε vertices o each side of the bipartitio The authors, havig give 0 < ε, d < 1 ad a positive iteger, estimate the umber h(ε, d, ) I this paper we study a similar problem With give d ad ε we determie the maximal size of a hole that ca be cotaied i some, sufficietly large, (d; ε)-regular graph As a corollary, the size of a largest complete bipartite graph that ca be cotaied i a (d; ε)-regular pair is also give the electroic joural of combiatorics 15 (2008), #R60 1
We start with some prelimiary facts ad defiitios Let G = (V, E) be a graph with a vertex set V = V (G) ad a edge set E = E(G) [V ] 2 For U, W V defie e G (U, W ) = {(x, y) : x U, y W, {x, y} E} Moreover, for oempty ad disjoit U ad W let d G (U, W ) = e G(U, W ) U W be the desity of the graph G betwee U ad W, or simply, the desity of the pair (U, W ) I the rest of this paper we assume that G is a bipartite graph with bipartitio V = V 1 V 2 A stadard averagig argumet yields the followig fact Fact 11 If d G (V 1, V 2 ) < d [> d], the for all atural umbers l 1 V 1 ad l 2 V 2 there exist subsets U V 1, U = l 1 ad W V 2, W = l 2 with d G (U, W ) < d [> d] Defiitio 12 Give ε 1, > 0, a bipartite graph G with bipartitio (V 1, V 2 ), where V 1 = ad V 2 = m, is called (ε 1, )-regular if for each pair of subsets U V 1 ad W V 2, U ε 1, W ε 1 m, the iequalities d < d G (U, W ) < d + (1) hold for some real umber d > 0 We may the also say that G, or the pair (V 1, V 2 ), is (d; ε 1, )-regular Moreover, if ε 1 = = ε, we will use the ames (d; ε)-regular ad ε-regular For example, accordig to the above defiitio, a complete bipartite graph has its desity equal to 1 Therefore it is ε-regular for all ε > 0 Remark 13 Each (ε 1, )-regular graph is ε-regular for all ε max{ε 1, } Note also that checkig if the give graph is (d; ε 1, )-regular we eed to cosider oly sets of the size ε 1 V i, i = 1, 2 I the followig sectio we state our mai results proved i Sectios 3 I Sectio 4, as a applicatios, we preset a degree coditio that guaratees the coectio by short paths i quasi-radom pairs 2 Mai results From the defiitio of a (d; ε)-regular pair it follows that the desities of most pairs of subsets of vertices are close to d However, it turs out that eve i such highly regular graphs, some pairs of small subsets may have their desities far from d I particular, there exist (d; ε)-regular graphs which cotai relatively large empty bipartite subgraphs (holes) Clearly these holes caot be too large The goal of this sectio is to fid the the electroic joural of combiatorics 15 (2008), #R60 2
maximal size of them As a corollary, the size of a largest complete bipartite graph that ca be cotaied i a (d; ε)-regular pair is also give Let us begi with some defiitios for a bipartite graph G = (V 1 V 2, E) Set K(U, W ) for the complete bipartite graph with vertex sets U ad W ad defie the bipartite complemet G = (V 1 V 2, K(V 1, V 2 ) \ E(G)) of G The largest iteger r such that K r,r G is the bipartite clique umber ω bip (G) of G, ad the largest iteger r such that K r,r G is the bipartite idepedece umber α bip (G) of G Clearly, α bip (G) = ω bip (G) We also set α bip (; d, ε) = max {α bip (G) : G = (V 1 V 2, E) is (d; ε)-regular with V 1 = V 2 = }, ω bip (; d, ε) = max {ω bip (G) : G = (V 1 V 2, E) is (d; ε)-regular with V 1 = V 2 = } Our mai results determie these parameters asymptotically whe goes to ifiity With give real umbers d ad ε we set α 0 = 2ε( εd ε)/(d ε) ad ω 0 = 2ε( ε(1 d) ε)/(1 d ε) Theorem 21 For all real umbers 0 < d < 1 there exists ε 0 > 0 such that for all ε < ε 0 α bip (; d, ε) lim = α 0 Corollary 22 For all real umbers 0 < d < 1 there exists ε 0 > 0 such that for all ε < ε 0 ω bip (; d, ε) lim = ω 0 Proof To prove Corollary 22 it is eough to observe that a graph G is (d; ε)-regular if ad oly if its bipartite complemet G is (1 d, ε)-regular Remark 23 With ε 0 we have α 0 2ε 3/2 / d ad ω 0 2ε 3/2 / 1 d I fact, oe ca prove a stroger result tha Theorem 21 We o loger assume that the bipartitio is balaced Before we make this precise, let us state the formal defiitio of a (α, β)-hole Defiitio 24 Let G = (V 1 V 2, E) be a bipartite graph ad 0 < α, β 1 A (α, β)- hole is a iduced subgraph (A B, ) of G with A V 1, B V 2, A α V 1 ad B β V 2 If α = β the we are simply talkig about a α-hole Note that with give d ad ε, the size of oe set of the bipartitio of a largest hole that ca be cotaied i a (d; ε)-regular pair depeds o the size of the other oe Hece, our task is to fid the value of the fuctio β 0 (α; d, ε) defied as follows: Defiitio 25 Let β 0 = β 0 (α; d, ε) be a real umber satisfyig the property that for all δ > 0 there exists 0 = 0 (d, ε, α, δ) such that: (a) o (d; ε)-regular graph G with V 1, V 2 0 cotais a (α, β 0 + δ)-hole, (b) for all 0 there exists a (d; ε)-regular graph with V 1 = V 2 = cotaiig a (α, β 0 δ)-hole the electroic joural of combiatorics 15 (2008), #R60 3
It is easy to see that if the umber β 0 (α; d, ε) exists the it is uique Note that for ε > d, the empty graph (a (1, 1)-hole) is (d; ε)-regular Oe ca also show that for ε = d, a (d; ε)-regular graph may cotai ay (α, β)-hole with α < ε or β < ε Therefore for the rest of the paper we will be assumig that 0 < ε < d < 1 To prove that a give β 0 is the value of β 0 (α; d, ε) at first we show that for all β > β 0 there exists 0 such that o (d; ε)-regular graph with at least 0 vertices o each side of the bipartitio cotais a (α, β)-hole The, for ay give β < β 0 we costruct a (d; ε)- regular graph cotaiig a (α, β)-hole I these costructios the desities of some pairs of small subsets ca exceed d + ε, but surely ca ot be larger tha oe Therefore for large d ad ε these costructios, ad hece the formula of the fuctio β 0 (α; d, ε), are differet tha for small oes It turs out that i most cases the value of β 0 (α; d, ε) is give by oe of the followig fuctios: f(α) = 2ε2 (2ε α) α(d ε) + 2, 2ε3 g(α) = α + ε(1 d ε), h(α) = 2ε 3 α ε(1 d ε) All these fuctios are decreasig Note that for ε < d the equatio β = f(α) is equivalet to ( ) ) β + (α 2ε2 + 2ε2 = 4ε3 d d ε d ε (d ε) 2 Hece the fuctio f is symmetric with respect to the lie α = β, which meas that f = f 1 Note also, that equatios β = g(α) ad β = h(α) are equivalet to α (β ε(1 d ε)) = 2ε 3 ad (α ε(1 d ε)) β = 2ε 3, respectively, ad therefore g = h 1 Now let us state without the proof results givig the values of the fuctio β 0 (α; d, ε) Ufortuately, we do ot kow this value for α = 2 /(d + ε) We set c = c(d, ε) = (ε/2)(1 (d + ε) + 1 + d 2 + + 2εd 2d + 6ε) for the positive solutio of the equatio g(α) = h(α) = α Theorem 26 For d 1/2 we have 1 for 0 < α < 2 /(d + ε), β 0 (α; d, ε) = f(α) for 2 /(d + ε) < α < ε, 2 /(d + ε) for ε α 1, for d > 1/2 ad ε < (1 d) 2 /d < 1 d we have 1 for 0 < α < 2 /(d + ε), g(α) for 2 /(d + ε) < α < 2 /(1 d + ε), β 0 (α; d, ε) = f(α) for 2 /(1 d + ε) α < 2ε(1 d), h(α) for 2ε(1 d) α < ε, 2 /(d + ε) for ε α 1, the electroic joural of combiatorics 15 (2008), #R60 4
for d > 1/2 ad (1 d) 2 /d ε 1 d we have 1 for 0 < α < 2ε 2 /(d + ε), g(α) for 2ε β 0 (α; d, ε) = 2 /(d + ε) < α < c, h(α) for c α < ε, 2 /(d + ε) for ε α 1, ad for d > 1/2 ad ε > 1 d we have 1 for 0 < α < ε(1 d + ε), β 0 (α; d, ε) = ( /α)(1 d + ε) for ε(1 d + ε) α < ε, ε(1 d + ε) for ε α 1 β 0 1 ε 2 d+ε 2 d+ε ε 1 α Figure 1: A sketch of the graph of β 0 = β 0 (α; d, ε) as a fuctio of α for d = 05 ad ε = 02 Note that sice a bipartite graph is (d; ε)-regular if ad oly if its bipartite complemet is (1 d; ε)-regular, we ca simply replace d by 1 d i the above results to get the size of a largest complete bipartite subgraph that ca be cotaied i a (d; ε)-regular graph 3 The Proof of Theorem 21 Before we prove Theorem 21, we state a result showig how, by usig radom graphs, oe ca fid a (d; ε)-regular bipartite graph for ay give real umbers d, ε (0, 1) For a later applicatio, we give it here i a more geeral form Fact 31 For all real umbers d, ε (0, 1) ad γ > 0, there exists 0 = 0 (d, ε, γ) such that for all 0, there exists a (d; ε)-regular bipartite graph G = (V 1 V 2, E) with V 1 = ad V 2 = γ the electroic joural of combiatorics 15 (2008), #R60 5
Proof Without loss of geerality we may assume that γ is iteger Let G = G(, γ, d) = (V 1 V 2, E) be a radom bipartite graph with V 1 =, V 2 = γ, ad edge probability d Moreover, for each pair of subsets U, W, U V 1, W V 2, U = ε, W = εγ, let X U,W = e G (U, W ) deote a radom variable coutig edges betwee sets U ad W Note, that each of these radom variables has the same biomial distributio with expected value µ = U W d = γ 2 d Applyig Cheroff s iequality (see iequality (29) i [1]) with ɛ = 1 3 we get IP ( U, W : X U,W µ 1 3 µ) 2 2 γ IP ( X µ 1 3 µ) { } { } (2 1+γ ) 2 exp 2 3 3 µ = (2 1+γ ) 2 exp ε2 γ 4 3 d = o(1), 3 where X has the same distributio as all radom variables X U,W Therefore there exists a graph G with vertex set V 1 V 2 such that for each pair of subsets U, W like above we have d G (U, W ) d < d = ε 0, thus G is (d; ε, ε 0 )-regular Now we are ready to prove Theorem 21 Proof of Theorem 21 To prove Theorem 21 we have to show that 1 3 0<d<1 ε 0 >0 ε<ε 0 δ>0 N IN >N the followig to statemets are true: (i) There exists a (d; ε)-regular bipartite graph G = (V 1 V 2, E), V 1 = V 2 =, cotaiig a (2ε( εd ε)/(d ε) δ) -hole (ii) No (d; ε)-regular graph G = (V 1 V 2, E), V 1 = V 2 =, cotais a (2ε( εd ε)/(d ε) + δ) -hole We start with the proof of the part (i) For ay 0 < d < 1 let { } (1 d) 2 ε 0 = mi, 1 d, d d Further for ay ε < ε 0 ad δ > 0 let N IN be as large as eeded Now for ay > N we will costruct a (d; ε)-regular graph G = (V 1 V 2, E), V 1 = V 2 =, cotaiig a (2ε( εd ε)/(d ε) δ) - hole Let α = 2ε( εd ε) d ε δ ad α = 2ε( εd ε) d ε the electroic joural of combiatorics 15 (2008), #R60 6
G 1 G 2 Next we defie ξ = 1 3 ε2 (1 α α ) 2 α, (2) ξ = 1 ε(ε α)ξ, (3) 8 d 1 = d ε + 2ξ ad d 2 = d 3 = d ε + 2 ε2 α ξ Note that α > 2 /(1 d + ε) ad therefore d 2 = d 3 < 1 ξ We costruct the desired graph as follows We take four disjoit sets of vertices A, B, A = B = α, V 1 ad V 2, V 1 = V 2 = α ad three graphs G 1 = (V 1 V 2, E(V 1, V 2 )), G 2 = (B V 1, E(B, V 1 )), G 3 = (A V 2, E(A, V 2 )), where G i is (d i ; ξ, ξ)-regular, i = 1, 2, 3, guarateed by Fact 31 We set G = G 1 G 2 G 3 = ((A V 1 ) (B V 2 ), E(V 1, V 2 ) E(B, V 1 ) E(A, V 2 )) By the costructio, G cotais a (2ε( εd ε)/(d ε) δ)-hole, to complete the proof PSfrag it remais replacemets to show that G is (d; ε)-regular To prove this, let U A V 1, W B V 2, be ay subsets of the set of vertices, U = W = ε We set A = A U, B = B W, U = U V 1, W = W V 2, ad let A = a α < α, B = b α < α (see Figure 2) V 2 W W B B G 3 V 1 U U A A The we get Figure 2: The costructio of the graph G ad the sets U ad W d G (U, W ) = + a(ε b) d G3 (A, W b(ε a) ) + d G2 (B, U ) ( 1 d G1 (U, W ) + O ) (4) the electroic joural of combiatorics 15 (2008), #R60 7
Note that for ay choice of U ad W, by (3), we have U > ξ, W > ξ, ad thus we may use the (d 1 ; ξ, ξ)-regularity of G 1 to boud the desity d G1 (U, W ) as follows: d ε + ξ < d G1 (U, W ) < d ε + 3ξ (5) Ufortuately, both sets, A ad B, ca be smaller the ξ I these cases we will be assumig that 0 d G3 (A, W ) 1 ad 0 d G2 (B, U ) 1 respectively Otherwise, we will use the (d i ; ξ, ξ)-regularity of the graphs G i, i = 2, 3, to get d ε + 2 ε2 α 2ξ < d G 2 (B, U ) < d ε + 2 ε2 α, (6) d ε + 2 ε2 α 2ξ < d G 3 (A, W ) < d ε + 2 ε2 α (7) Therefore, i order to prove that d ε < d G (U, W ) < d + ε, by (4), (5), (6) ad (7), we have to show the validity of the followig iequalities: a(ε b) b(ε a) + + (d ε + 3ξ) + O (d ε + ξ) + O ( ) 1 > d ε, ( ) 1 < d + ε, where A < ξ ad B < ξ This follows from (2) ad (3) I the case, whe A < ξ, B ξ (or similarly, whe A ξ, B < ξ ) we get a(ε b) b(ε a) + b(ε a) ) (d ε + 2 ε2 + α ) (d ε + 2 ε2 α 2ξ + (d ε + 3ξ) + O (d ε + ξ) + O ( ) 1 < d + ε, ( ) 1 > d ε Here, to prove the last iequality, apart from (2) ad (3), we use also the fact that ε 1/2 Fially, if A ξ ad B ξ, by (2), we have ) ) a(ε b) f 1 (a, b) = (d ε + 2 ε2 b(ε a) + (d ε + 2 ε2 + α α ( ) 1 (d ε + 3ξ) + O < ( b d ε + 2ε α 2 ab α + a ) + 3ξ < d + ε, 2 α the electroic joural of combiatorics 15 (2008), #R60 8
) a(ε b) f 2 (a, b) = (d ε + 2 ε2 α 2ξ + ( ) 1 (d ε + ξ) + O = ( b d ε + 2ε α 2 ab α + a ) 2 α b(ε a) + ξ ε2 3εa 3εb + 5ab ) (d ε + 2 ε2 α 2ξ + + O ( ) 1 > d ε Sice both, f 1 (a, b) ad f 2 (a, b), are double liear fuctios, they achieve their extreme values i the corers of the rectagle, o which they are defied Therefore, to fiish the proof of the part (i) of Theorem 21, we eed to check the validity of the last iequality oly at poits (a, b) equal to (0, 0), (0, α + 1/), (α + 1/, 0) ad (α + 1/, α + 1/) Now we ca move to part (ii) For ay real umber d (0, 1) we set ε 0 = mi{d, 1 d} Now, for ay ε < ε 0 ad δ > 0 we defie 2( εd ε) N = δ(d ε) Take ay > N ad let G = (V 1 V 2, E), V 1 = V 2 =, be a (d; ε)-regular bipartite graph Suppose, for a cotradictio, that G cotais a (2ε( εd ε)/(d ε) + δ) -hole betwee sets A V 1 ad B V 2 Without loss of the geerality we may assume, that ( 2ε( ) εd ε) A = B = + δ = r, d ε ad also that r < ε, sice otherwise we would get a cotradictio with the (d; ε)- regularity of G Note also that sice > N, we have r ε > 2( εd ε) d ε We take two sets U V 1 \ A, U = ε r, ad W V 2 \ B, W = ε i such a way that d G (U, W ) > d ε Sice V 1 \ A > ε, we have d G (V 1 \ A, W ) > d ε, ad therefore, by Fact 11, this choice is possible Note that by the (d; ε)-regularity of G we get d G (A U, W ) < d + ε ad thus Hece (d ε) U W + d G (A, W ) A W < e G (U, W ) + e G (A, W ) = e G (A U, W ) < (d + ε) ε W d G (A, W ) < (d + ε) ε (d ε)( ε A ) A = d ε + 2ε ε r the electroic joural of combiatorics 15 (2008), #R60 9
Therefore, by Fact 11, we may choose a set W W, W = ε r, i such a way that d G (A, W ) < d ε + 2ε ε /r Next we take U V 1 \ A, U = ε r, with d G (U, B W ) < d + ε We will show that d G (A U, B W ) < d ε gettig a cotradictio with the (d; ε)-regularity of G Ideed, we have d G (A U, B W ) = d G(U, B W ) U ε + d G(A, W ) A W < ε 2 ε ( 2 (d + ε) 1 r ) ( + d ε + 2ε ε ) ( r 1 r ) = ε r ε ε d + 3ε 4ε r ( ) 2 r (d ε) < ε ε ( d + 3ε 4( εd ε) 2( ) 2 εd ε) (d ε) = d ε d ε d ε 4 Applicatios I this sectio we preset the degree coditio of vertices i (d; ε)-regular graphs that guaratees their coectio by a path More studies about this problem ca be foud i [5] By dist G (x, y) we deote the distace of vertices x, y V, that is, the legth of a shortest path coectig them, if such a path exists Otherwise we set dist G (x, y) = By the diameter of G we mea diam(g) = max x,y V dist G (x, y) I particular, if G is ot coected, the diam(g) = Theorem 41 I ay (d; ε)-regular bipartite graph G, where 0 < ε d 1 ε, if deg G (v), deg G (w) > 2 /(d + ε), the { 5 if v Vi, w V dist G (v, w) j, 4 if v, w V i Proof Let 0 < ε d 1 ε ad let a (d; ε)-regular bipartite graph G be give Furthermore let v, w V, deg G (v) > 2 /(d + ε), deg G (w) > 2 /(d + ε) We set A = N G (v), B = N G (w) Without loss of geerality, we may assume that v V 1 As the first oe we will cosider the case where w V 1 We let C V 1 be the set of all vertices adjacet to some vertex of B The C ε ε, sice otherwise the sets B ad V 1 \ C would provide a (α, ε)-hole, where α > 2 /(d + ε), which cotradicts Theorem 26 Therefore e G (A, C) > 0 ad so the vertices v ad w are coected i G by a path of legth at most 4 Now we tur to the situatio where w V 2 Similarly to the above, the set of vertices C V 2 adjacet to some vertex of B has cardiality C ε ε Now we repeat the reasoig from the first part of the proof to the sets A ad C, gettig a path of legth at most 5 (see Figure 3) the electroic joural of combiatorics 15 (2008), #R60 10
PSfrag replacemets V 2 A C w V 1 u v B Figure 3: The path from v to w, where w V 2 Corollary 42 For each (d; ε)-regular bipartite graph G, where 0 < ε d 1 ε, if δ G > 2 /(d + ε), the diam(g) 5 It turs out that the above result is the best possible, amely that there exist (d, ε)- regular graphs cotaiig a vertex with degree slightly smaller the 2 /(d + ε), which is ot coected by a path with ay other vertices except of its eighbors Theorem 43 For all real umbers ε, d, α (0, 1) where 0 < ε d 1 ε ad α < 2 /(d + ε), there exists a (d; ε)-regular graph G cotaiig a isolated star with α edges, ad therefore there exists a vertex of degree α, which is coected (by a path) oly with its eighbors Sketch of the proof Accordig to Theorem 26, there exists a (d; ε)-regular graph G cotaiig a (α, 1)-hole spaed betwee sets A V 1 ad V 2 We add to V 2 oe vertex w ad coect it with all vertices of A This additio has a very small impact o the regularity of G (for details see [5]) So we have gotte a isolated star with α edges, as required (see Figure 4) V PSfrag replacemets 2 w V 1 Figure 4: The graph G with a ew vertex w A Ackowledgemets I would like to thak Adrzej Ruciński for suggestig this problem ad for his help ad advice i preparig this mauscript I also would like to thak Referee for his valuable commets o the presetatio of this paper the electroic joural of combiatorics 15 (2008), #R60 11
Refereces [1] S Jaso, T Luczak & A Ruciński, Radom Graphs, Joh Wiley ad Sos, New York 2000 [2] J Komlós, GN Sárközy & E Szemerédi, O the square of a Hamiltoia cycle i dese graphs, Radom Structures Algorithms 9 (1996), o 1-2, 193-211 [3] J Komlós, GN Sárközy & E Szemerédi, O the Pósa-Seymour cojecture, J Graph Theory 29 (1998), 167-176 [4] Y Peg, V Rödl & A Ruciński, Holes i graphs, Electro J Combi 8 (2001), #R13 [5] J Polcy & A Ruciński, Short paths i ε-regular pairs ad small diameter decompositios of dese graphs, submitted [6] E Szemerédi, Regular partitios of graphs, Problèmes combiatoires et théorie des graphes (Colloq Iterat CNRS, Uiv Orsay, Orsay, 1976) (Bermod, J-C, Fourier, J-C, Las Vergas, M, Sotteau, D, eds),(1976), pp 399-401 the electroic joural of combiatorics 15 (2008), #R60 12