Discrepancy of random graphs and hypergraphs

Transcription

1 Discrepacy of radom graphs ad hypergraphs Jie Ma Humberto aves Bey Sudaov Abstract Aswerig i a strog form a questio posed by Bollobás ad Scott, i this paper we determie the discrepacy betwee two radom -uiform hypergraphs, up to a costat factor depedig solely o. Itroductio A hypergraph H is a ordered pair H = V, E, where V is a fiite set the vertex set, ad E is a family of distict subsets of V the edge set. The hypergraph H is -uiform if all its edges are of size. I this paper we cosider oly -uiform hypergraphs. The edge desity of a -uiform hypergraph H with vertices is ρ H = eh/. We defie the discrepacy of H to be disch = max S S V H es ρ H, where es = eh[s] is the umber of edges i the sub-hypergraph iduced by S. The discrepacy ca be viewed as a measure of how uiformly the edges of H are distributed amog the vertices. This importat cocept appears aturally i various braches of combiatorics ad has bee studied by may researchers i recet years. The discrepacy is closely related to the theory of quasi-radom graphs see [6], as the property discg = o V G implies the quasi-radomess of the graph G. Erdős ad Specer [8] proved that for, ay -uiform hypergraph H with vertices has a subset S satisfyig es S + c, which implies the boud disch c + for -uiform hypergraphs H of edge desity. Erdős, Goldberg, Pach ad Specer [7] obtaied a similar lower boud for graphs of edge desity smaller tha. These results were later geeralized by Bollobás ad + Scott i [3], who proved the iequality disch c r for -uiform hypergraphs H, wheever r = ρ H ρ H /. The radom hypergraphs show that all the aforemetioed lower bouds are optimal up to costat factors. For more discussio ad geeral accouts of discrepacy, we refer the iterested reader to Bec ad Sós [], Bollobás ad Scott [3], Chazelle [5], Matouše [0] ad Sós []. A similar otio is the relative discrepacy of two hypergraphs. Let G ad H be two -uiform hypergraphs over the same vertex set V, with V =. For a bijectio π : V V, let G π be obtaied from G by permutig all edges accordig to π, i.e., EG π = πeg. The overlap of G ad H Departmet of Mathematics, UCLA, Los Ageles, CA jiema@math.ucla.edu. Research supported i part by AMS-Simos travel grat. Departmet of Mathematics, UCLA, Los Ageles, CA haves@math.ucla.edu. Departmet of Mathematics, UCLA, Los Ageles, CA bsudaov@math.ucla.edu. Research supported i part by SF grat DMS-085, by AFOSR MURI grat FA ad by a USA-Israel BSF grat.

2 with respect to π, deoted by G π H, is a hypergraph with the same vertex set V ad with edge set EG π EH. The discrepacy of G with respect to H is discg, H = max π eg π H ρ G ρ H, where the maximum is tae over all bijectios π : V V. For radom bijectios π, the expected size of EG π EH is ρ G ρ H, thus discg, H measures how much the overlap ca deviate from its average. I a certai sese, the defiitio is more geeral tha, because oe ca write disch = max i discg i, H, where G i is obtaied from the complete i-vertex -uiform hypergraph by addig i isolated vertices. Bollobás ad Scott itroduced the otio of relative discrepacy i [4] ad showed that for ay two -vertex graphs G ad H, if 6 ρ G, ρ H 6, the discg, H c fρ G, ρ H 3, where c is a absolute costat ad fx, y = x x y y. As a corollary, they proved a cojecture i [7] regardig the bipartite discrepacy discg, K,. Moreover, they also cojectured that a similar boud holds for -uiform hypergraphs, amely, there exists c = c, ρ G, ρ H for which discg, H c + holds for ay -uiform hypergraphs G ad H satisfyig ρ G, ρ H. I their paper, Bollobás ad Scott also ased the followig questio see Problem i [4]. Give two radom -vertex graphs G, H with costat edge probability p, what is the expected value of discg, H? I this paper, we solve this questio completely for geeral -uiform hypergraphs. Let H, p deote the radom -uiform hypergraph o vertices, i which every edge is icluded idepedetly with probability p. We say that a evet happes with high probability, or w.h.p. for brevity, if it happes with probability at least w, where here ad later w > 0 deotes a arbitrary fuctio tedig to ifiity together with. Theorem.. For positive itegers ad, let =. Let G ad H be two radom hypergraphs distributed accordig to H, p ad H, q respectively, where w p q. dese case If pq > 30 log, the w.h.p. discg, H = Θ sparse case If pq log 30 log, let γ = pq, the. if p log 5 log γ, the w.h.p. discg, H = Θ. if p < log 5 log γ, the w.h.p. discg, H = Θ log log γ p.. pq log ; The previous theorem also provides tight bouds whe p ad/or q, as we shall see i the cocludig remars. The result of Theorem. i the sparse rage is closely related to the recet wor of the third author with Lee ad Loh [9]. Amog other results, the authors of [9] show that two idepedet copies G, H of the radom graph G, p with p log / w.h.p. have overlap of order Θ log log γ, where γ = log p. Hece discg, H = Θ log log γ holds, sice i this rage of edge probability, log log γ is larger tha the average overlap p. Our proof i the sparse case borrows some ideas from [9]. O the other had, oe ca ot use their approach for all cases, hece some ew ideas were eeded to prove Theorem.. It will become evidet from our proof that the problem of determiig the discrepacy ca be essetially reduced to the followig questio. Let K > 0 ad let X be a biomial radom variable with

3 parameters m ad ρ. What is the maximum value of Λ = Λm, ρ, K satisfyig P [ X mρ > Λ ] e K? This questio is related to the rate fuctio of biomial distributio. I all cases, the discrepacy i the statemet of Theorem. is w.h.p. discg, H = Θ Λ p, q, log. 3 ote that p is roughly the size of the eighborhood of a vertex i the hypergraph G. The rest of this paper is orgaized as follows. Sectio cotais a list of iequalities ad techical lemmas used throughout the paper. I sectio 3, we defie the probabilistic discrepacy disc P G, H ad prove that w.h.p. it does ot deviate too much from discg, H. Additioally, we establish the upper bouds for discg, H based o aalogous bouds for disc P G, H. I sectio 4, we give a detailed proof of the lower bouds. The fial sectio cotais some cocludig remars ad ope problems. I this paper, the fuctio log refers to the atural logarithm ad all asymptotic otatio symbols Ω, O, o ad Θ are with respect to the variable. Furthermore, the -subscripts i these symbols idicate the depedece o i the relevat costats. Auxiliary results I this sectio we list ad prove some useful cocetratio iequalities about the biomial ad hypergeometric distributios ad also prove a corollary from the well-ow Vizig s Theorem which asserts the existece of a liear-size matchig i early regular graphs i.e., the maximum degree is close to the average degree. We will ot attempt to optimize our costats, preferrig rather to choose values which provide a simpler presetatio. Let us start with classical Cheroff-type estimates for the tail of the biomial distributio see, e.g., []. Lemma.. Let X = l i= X i be the sum of idepedet zero-oe radom variables with average µ = E[X]. The for all o-egative λ µ, we have P[ X µ > λ] e λ The followig lower tail iequality see [] is due to Jaso. Lemma.. Let A, A,..., A l be subsets of a fiite set Ω, ad let R be a radom subset of Ω for which the evets r R are mutually idepedet over r Ω. Defie X j to be the idicator radom variable of A j R. Let X = l j= X j, µ = E[X], ad = i j E[X i X j ], where i j meas that X i ad X j are depedet i.e., A i itersects A j. The for ay λ > 0, P[X µ λ] < e λ µ+. I the proof of the dese case of the mai theorem we will eed a lower boud for the tail of the hypergeometric distributio. To prove it we use the followig well-ow estimates for the biomial coefficiet. Propositio.3. Let Hp = p log p+ p log p, the for ay iteger m > 0 ad real p 0, satisfyig pm Z we have π m mp e pe mhp e pm π. 4µ. 3

4 Proof. This ca be derived from Stirlig s formula πm m m e m! e m m m. e Lemma.4. Let d, d, ad be itegers ad K be a real parameter such that d, d 3, K d d 00 ad = d d K. The d d t d t e 40K. d t d d + Proof. For coveiece, we write ft = d d t d t / d. I order to show the desired lower boud of the hypergeometric sum, it suffices to prove that ft 4e 40K, d d + for every iteger t = d d + θ with θ. Ideed, to see this, ote that there are at least itegers betwee d d + ad d d + ad > + d d +. ext we prove the boud for ft. For our choice of, the iequality d 5 is true sice d d K = = d d K d d d d 00 = d 0 d d 5. Similarly d 5. Let x = d, y = θ d ad z = θ But 0 < x + y <, because 0 < x 3 z x = θ d d 3θ d 5 ad x 3 d. The t = x+yd ad d t = x z d. ad 0 < y d < 3. Furthermore, 0 < x z <, because. By Propositio.3, we have d d x+yd ft = x z d 4π Re L e 5, x where L = d Hx + y + d Hx z Hx ad R = x x x z x + zx + y x yd d x + yd d d +. Here we used the iequality θ ad the idetity x + yd = t = d d + θ. Because d y = d z = θ ad log + s s, we obtai 4

5 L = d [x + y log + d d [ x + yy x + y x [ x z log x yy x = θ y + z x + = x Thus we always have ft 4π e 5 e 36K d d + x y log y x z + x + z log x ] + d 4e 40K + d d Lemma.5. For positive itegers ad, let = pq ] + z x [ x zz + x θ 3 d d d d 36K. +, completig the proof. ] x + zz x The ext lemma will be used to prove the lower boud i the sparse case of Theorem., ad was ispired by a aalogous result i [9]., w p q ad suppose that 30 log. Defie γ = log pq. Let,..., s B be s /3 disjoit sets of size + op, ad cosider the radom set B q, obtaied by taig each elemet of B idepedetly with probability q. The w.h.p., there is a idex i for which B q i log 6 log γ i B q if p < log 5 log γ. log if p log 5 log γ ; Proof. Let t = 6 log γ. Clearly q e 3q/ whe q /. For a fixed idex i, the probability that B q i t is at least i t q t q i t. Usig the bouds a b a b b for a b, ad log 30 log pq = γ, we obtai i + opq t q t q i t t /6 / t 5 log γ e pq γ ] log 6 log γ /5 Hece the expected umber of idices i such that B q i t is at least s 0.3 /30. Sice the sets i are disjoit, these evets are idepedet for differet choices of i. Therefore by Lemma. w.h.p. we ca fid such a idex actually may. If p < log 5 log γ log, the q = γp 5 log γ γ ad we ca complete the proof as i the first case. γ. Therefore the probability that some i B q is q i γ +op γ log 4 log γ = /4, The last lemma i this sectio, which ca be easily derived from Vizig s Theorem, will be used to fid a liear-size matchig i early regular graphs. Lemma.6. Every graph G with maximum degree G, cotais a matchig of size at least eg G+. 5

6 Proof. By Vizig s Theorem, the graph G has a proper edge colorig f : EG {,,..., G+}. For each color c G +, the edges f c form a matchig i G. By the pigeohole priciple, there is a color c such that f c has at least 3 Upper bouds eg G+ edges. I this sectio we prove the upper boud for the discrepacy i Theorem.. Let G ad H be two radom hypergraphs over the same vertex set V, distributed accordig to H, p ad H, q, respectively. The probabilistic discrepacy of G ad H is defied by disc P G, H = max π eg π H pq, where the maximum is tae over all bijectios π : V V. We will show that w.h.p. the differece betwee discg, H ad disc P G, H is very small. Before we proceed, we state the followig fact whose proof is fairly trivial. Propositio 3.. If AB A 0 B 0 ɛ ɛ + A 0 ɛ + B 0 ɛ, the either A A 0 ɛ or B B 0 ɛ. Lemma 3.. With probability at least 4e, the iequality discg, H disc P G, H ε holds, where ε = 4 4 pq. Proof. Sice p = Ω, applyig Lemma. to the radom variable eg for λ = 4 yields [ eg ] P p 4 p e. [ Similarly, we have P eh q 4 p p q ] e. Therefore, with probability at least 4e, ρ G p 4 p/ / ad ρh q 4 q/ /. These iequalities, together with Propositio 3., imply ρ Gρ H pq 4 pq + p 4 q + q 4 p ε, completig the proof of the lemma. It is easy to chec that the error term ε is much smaller tha the bouds i Theorem.. Therefore, i order to prove Theorem. for discg, H, it suffices to prove the correspodig bouds for disc P G, H istead. Lemma 3.3. Let G ad H be as i Theorem.. The with high probability disc P G, H satisfies the stated upper bouds of this theorem. Proof. Sice the umber of edges of G is distributed biomially ad p = Ω, by Lemma., we have eg < p with probability at least e Θ. Sice disc P G, H is bouded by max { eg, pq }, this implies the assertio i the case. of Theorem.. 6

7 For ay fixed bijectio π : V V, the umber of edges i G π H is distributed biomially with parameters ad pq. If pq > 4 log let λ = pq log pq. The by Lemma., the probability that eg π H pq > λ is at most e log. O the other had, if pq 4 log, let γ = 4e pq log e ad λ = 4e log log γ 4e log γ = epq. Sice a b ea b, b the probability that eg π H > λ is at most e λ pq λ pq 4e λ log γ 4e log log γ = λ λ γ = λ log γ < e log. Sice there are! possible bijectios π : V V, by the uio boud P [disc P G, H > λ ]! e log e /. To fiish the proof of the lemma ote that γ, defied i Theorem., satisfies γ = Θ γ. Also observe that for p, q satisfyig both pq 4 log ad pq 30 log, where =, we have pq log = 4e log Θ log γ. 4 Lower bouds I this sectio we prove the lower bouds i Theorem.. As we previously explaied, it is eough to obtai these bouds for disc P G, H. We divide the proof ito two cases. The first dese case will be discussed i the ext subsectio. The secod sparse case will be discussed i subsectio 4.. Throughout the proofs, we assume that is fixed ad is tedig to ifiity. 4. Dese Case Let = ad let p, q be such that pq > 30 log. Select a arbitrary set L V of size L =. We prove that w.h.p. there exists a L-bijectio π : V V with overlap eg π H pq + Θ pq log = pq + Θ pq log, 4 where a L-bijectio π : V V is a bijectio from V to V which oly permutes the elemets of L, i.e., πx = x for all x L. From the radom hypergraph G we costruct a radom bipartite graph G with vertex set L G R, where L G = L ad R is the set of all -tuples i V \ L. ote that R =. The vertices v L G ad {v, v 3,..., v } R are adjacet if {v, v,..., v } forms a edge i the hypergraph G. With slight abuse of otatio, we view G as a sub-hypergraph of G, cotaiig all edges e havig exactly oe vertex i L, i.e. e L =. Similarly, from the radom hypergraph H we costruct a radom bipartite graph H with vertex set L H R. Figure shows the resultig bipartite graphs. Give a L-bijectio π : V V, we divide the edge set of G π H ito two subsets: the edge set of G π H ad its complemet. To prove our result we first expose the radom edges i G ad H, ad show how to fid a L-bijectio π havig overlap at least Θ pq log more tha the expectatio. The we fix such π ad expose all the remaiig edges i G ad H showig that the cotributio of these edges to G π H does ot deviate much from the expected cotributio. More precisely, let 7

8 R L G L H u v p q Figure : Radom bipartite graphs G ad H. e π = EG G π EH H, the eg π H = e G π H + e π. Moreover, e π is distributed accordig to Bim, pq, where m =. Thus w.h.p. eπ pqm < pqm log, as Lemma. shows. Sice pqm log pq log, i order to obtai 4, it is eough to show that w.h.p. there exists a L-bijectio π such that e G π H pq + Θ pq log. 5 We defie a auxiliary bipartite graph Γ = Γ G, H as follows. A vertex u L G survives if deg Gu p p ad similarly, a vertex v L H survives if deg Hv q q. Let S G ad S H be the sets of all survivig vertices of G ad H, respectively. Let s G = S G ad s H = S H. The set of vertices of Γ is the uio of S G ad S H. The edges of Γ are defied by the property u Γ v codegu, v deg Gu deg Hv + 0 pq log, where codegu, v deotes the codegree of u L G ad v L H, i.e. codegu, v = Gu Hv. The graph Γ has may vertices i both parts, as the followig simple lemma demostrates Lemma 4.. W.h.p. each part of Γ has size at least 4. Proof. Let α be the probability that some vertex u survives i L G. Sice p w 8, we have that p p. Thus Lemma. applied to deg Gu implies α e /. Sice the evets that vertices survive are idepedet, s G stochastically domiates the biomial distributio with parameters / ad /. Thus, agai by Lemma., w.h.p. s G /4 ad a similar estimate holds for s H. To prove 5, we will show that the followig two statemets hold w.h.p. a Γ has a matchig M = {u, v,..., u l, v l } of size l = 50 ; b there exists a L-bijectio π such that πu i = v i for all i =,,..., l, ad, codegu, πu l pq pq. u L G \{u,u,...,u l } 8

9 Ideed, for ay two adjacet vertices u, v i Γ, we have deg Gu deg Hv Thus usig a, b ad l = 50 p 8pq 8q we obtai e G π H = l codegu, πu u L G i= l [ deg Gu i deg Hv i + 0 ] pq log i= l i= [ pq 6 ] pq + pq pq log pq 6 pq. codegu i, v i + l pq pq 50 0 pq log + We eed the followig lemma i order to prove that b holds. + l pq pq l pq pq Lemma 4.. Let 0 < α < be ay absolute costat. The with probability at least e, ay two subsets A L G ad B L H with A = B = α satisfy X A,B := α codegu, v pq α pq. u A,v B Proof. Let X w,u,v be the idicator of wu E G ad wv E H for w R, u A, v B. So X A,B = w R,u A,v B X w,u,v ad E[X w,u,v ] = pq. Moreover, X w,u,v ad X w,u,v are depedet if ad oly if wu = w u or wv = w v. Thus, µ = E[X A,B ] = α pq ad = w R,u A v,v B E[X w,u,v X w,u,v ] + A L G α,b L H α w R,v B u,u A E[X w,u,v X w,u,v] = α α pq p + q, where µ ad are defied as i Lemma.. Let F be the evet that there exists at least oe pair of subsets A L G, B L H with A = B = α satisfyig X A,B < α pq α pq. By the uio boud ad by Lemma., we have [ ] P[F ] P X A,B < µ α pq sice µ e αα e 3 e, α 3 pq, α < ad α loge/α for all such α. α e α pq µ+ Let M = {u, v,..., u l, v l } be a matchig satisfyig a ad let A = L G \ {u, u,..., u l } ad B = L H \ {v, v,..., v l }. Oe ca write A = B = l = α 49, where α = 50. Cosider X A,B = u A,v B codegu, v. The, by Lemma 4., with probability at least e, we have codegu, v l pq l pq. u A,v B 9

10 Sice the complete bipartite graph with parts A, B is a disjoit uio of l perfect matchigs, by the pigeohole priciple, there exists a matchig M betwee A ad B such that u,v M codegu, v u A,v B codegu, v l l pq pq. The the matchig M M betwee L G ad L H gives the desired L-bijectio π ad proves b. To fiish the proof we eed to establish a. If Γ is early regular, the by Lemma.6, Γ would cotai a liear-size matchig. Ufortuately this is ot the case. However, we will show that it is possible to delete some edges of Γ at radom ad obtai a prued graph Γ, which is early regular. Let fd, d := P [ u Γ v deg Gu = d, deg Hv = d ], where d p p ad d q q. Let f 0 be the miimum of fd, d over all pairs d, d i the domai of f. Suppose that f 0, which we shall prove later. We eep each edge uv of Γ i Γ f idepedetly with probability 0 fd,d, where d = deg Gu ad d = deg Hv. The, we claim that for ay vertex u S G, deg Γ u is biomially distributed with parameters s H ad f 0. Ideed, by defiitio, P [ ] u Γ v deg Gu = d, deg Hv = d = f0 for all possible d, d. Moreover, coditioig o the eighbors of u i G ad o the values of the degrees deg Hv, deg Hv,..., deg Hv m, the evets u Γ v, u Γ v,..., ad u Γ v m are all idepedet. Therefore, by defiitio of Γ, it is easy to see that u Γ v, u Γ v,..., ad u Γ v m are idepedet as well. Thus for ay u S G, deg Γ u Bis H, f 0 ad similarly, deg Γ v Bis G, f 0 for all v S H. Coditioig o the degrees of all vertices i G, H, we obtai sets S G ad S H, which w.h.p. satisfy the assertio of Lemma 4., i.e., S G = s G 4 ad S H = s H 4. Thus both s Gf 0 ad s H f 0 are Ω. Sice all degrees i Γ are biomially distributed, Lemma 4. together with the uio boud imply that w.h.p. all vertices u S G, v S H satisfy s H f 0 deg Γ u 3s Hf 0 ad s Gf 0 } { Therefore, the max-degree Γ max 3sH f 0, 3s Gf 0 Lemma.6, Γ has a matchig of size at least eγ Γ + deg Γ v 3s Gf 0. 3f 0 ad eγ s Gs H f 0 f 0 3. Thus by 50, completig the proof of a.. Sice p teds to ifiity, p q / It remais to prove the boud f 0. Let K = log 5000 ad d p p, we have d = +op 3. Similarly d = +oq 3. Also recall that pq 30 log, which implies d d 00 = + opq 00 d d K Therefore we ca apply Lemma.4 with = > have d d fd, d = t d t d This completes the proof. t d d + pq log 00 + olog 3000 > K. pq log t d d By the defiitio of fd, d, we d d t d t d e 40K >. 0

11 4. Sparse case I this subsectio, we prove the lower boud i the sparse case pq 30 log. ote that, sice p q i this case, we have p /+o. The proof rus alog the same lies as that of the dese case differig oly i the applicatio of Lemma.5 to obtai a L-bijectio π : V V whose sum of codegrees u L G codegu, πu is large. Suppose first that p log log 5 log γ. Recall that γ = pq 30 ad thus log 6 log γ log 4 log γ + log γ = log 4 log γ + pq. Therefore it is eough to fid a bijectio π betwee L G ad L H such that u L G codegu, πu + o log 6 log γ. Partitio the vertices of L G ito r = s disjoit sets S,..., S r each of size s = /5. We will costruct π by applyig the followig greedy algorithm to each set. Let us start with S. The algorithm will reveal the edges emaatig from S to R i G by repeatedly exposig the eighborhood of a vertex i S, oe at a time. Throughout this process, we costruct a subset S S of size o S ad a family of disjoit sets u R, such that each u has size + op ad is cotaied i the eighborhood of u, for all u S. At each step, we pic a fresh vertex u i S ad expose its eighborhood. If u has a set of + op eighbors which is disjoit from w for all w i the curret S, deote this particular set by u ad put u i the set S ; otherwise move to the ext step. At every step, the uio X = w S w has size at most Op s 0.9+o. Moreover, every vertex i R \ X is adjacet to u idepedetly with probability p. Sice p w teds to ifiity with, the set of eighbors of u outside X has size + o R \ X p = + op with probability o. Furthermore, for differet vertices such evets are idepedet. Therefore, by Lemma., w.h.p. S = o S. ow we will costruct the partial matchig for S. Cosider the disjoit sets u, for u S, each of size +op. Pic a arbitrary vertex v i L H ad expose its eighbors i H. This is a radom subset v of R, obtaied by taig each elemet idepedetly with probability q. Therefore by case of Lemma.5, w.h.p there is a vertex u S such that codegu, v u v log 6 log γ. Defie πu = v, remove u from S, remove v from L H ad cotiue. ote that, as log as there are at least /3 vertices remaiig i S, we ca match oe of them with a ewly exposed vertex from L H such that the codegree of this pair is at least log 6 log γ. Oce the umber of vertices i S drops below /3, leave the remaiig vertices umatched. W.h.p. we ca match a o fractio of the vertices i S. Cotiue the above procedure for S,..., S r as well. At the ed of the process, we will have matched a o fractio of all the vertices i L G with distict vertices i L H such that codegree of every matched pair is at least log 6 log γ. Therefore the sum of the codegrees of this partial matchig is at least + o log 6 log γ. To obtai the bijectio π, oe ca match the remaiig vertices i L G ad L H arbitrarily. Whe p < log 5 log γ the same proof as above together with case of Lemma.5 yields a bijectio π such that u L G codegu, πu + o p. Sice q, this is at least + o p more tha the expectatio, fiishig the aalysis of the sparse case. 5 Cocludig remars As we stated i the itroductio, Theorem. also yields tight bouds whe p ad/or q >. For ay G ad H, oe ca chec that discg, H = discg, H, where H is the complemet of H. Moreover, H is distributed accordig to H, q, hece we ca reduce the case q > to the case q = q ;

12 the same holds whe we tae the complemet of G istead. We remar that oe ca determie the discrepacy whe p is smaller tha ω, but we chose ot to discuss this rage here, sice the proof is similar to the sparse case ad it would t provide ay ew isight. The defiitio of discrepacy ca be rephrased as discg, H = max {disc + G, H, disc G, H}, where disc + G, H = max π eg π H ρ G ρ H ad disc G, H = ρ G ρ H miπ eg π H are the oe-sided relative discrepacies. I fact, all the lower bouds we obtaied are for disc + G, H, ad some of them are ot true for disc G, H. This is because disc G, H ρ G ρ H pq ad i the sparse case, pq could be much smaller tha discg, H. Uder the same hypothesis ad usig similar ideas as i Theorem., oe ca show that disc G, H = { Θ pq log Θ pq if pq > 30 log ; otherwise. The last equatio is related to the lower tail of the biomial distributio. Lastly, we would lie to metio that there are a substatial umber of ope problems about discg, H ad its related topics i [4]. Refereces []. Alo ad J. Specer, The Probabilistic Method, Joh Wiley Ic., ew Yor 008. [] J. Bec ad V.T. Sós, Discrepacy theory, i Hadboo of Combiatorics, Vol., , Elsevier, Amsterdam, 995. [3] B. Bollobás ad A. Scott, Discrepacy i graphs ad hypergraphs, i More sets, graphs ad umbers, Ervi Gyori, Gyula O.H. Katoa ad Laszlo Lovász, eds, pp , Bolyai Soc. Math. Stud. 5, Spriger, Berli, 006. [4] B. Bollobás ad A. Scott, Itersectio of graphs, J. Graph Theory 66 0, 6 8. [5] B. Chazelle, The discrepacy method, Cambridge Uiversity Press, Cambridge, 000, xviii+463 pp. [6] F.R.K. Chug, R.L. Graham ad R.M. Wilso, Quasi-radom graphs, Combiatorica 9 989, [7] P. Erdős, M. Goldberg, J. Pach ad J. Specer, Cuttig a graph ito two dissimilar halves, J. Graph Theory 988, 3. [8] P. Erdős ad J. Specer, Imbalaces i -coloratios, etwors 97/, [9] C. Lee, P. Loh ad B. Sudaov, Self-similarity of graphs, SIAM J. of Discrete Math., to appear. [0] J. Matouše, Geometric discrepacy, Algorithms ad Combiatorics 8, Spriger-Verlag, Berli, 999, xii+88 pp. [] V.T. Sós, Irregularities of partitios: Rasey theory, uiform distributio, i Surveys i Combiatorics Southampto, 983, 0 46, Lodo Math. Soc. Lecture ote Ser., 8, Cambridge Uiv. Press, Cambridge-ew Yor, 983.