Quasi-Random Graphs With Given Degree Sequences

Transcription

1 Quas-Ranom Graphs Wth Gven Degree Sequences Fan Chung,* Ron Graham Unversty of Calforna, San Dego, La Jolla, Calforna 9093; e-mal: Receve 3 May 005; accepte 0 June 006; receve n fnal form 4 December 006 Publshe onlne 4 August 007 n Wley InterScence DOI 0.00/rsa.088 ABSTRACT: It s now known that many propertes of the objects n certan combnatoral structures are equvalent, n the sense that any object possessng any of the propertes must of necessty possess them all. These propertes, terme quasranom, have been escrbe for a varety of structures such as graphs, hypergraphs, tournaments, Boolean functons, an subsets of Z n, an most recently, sparse graphs. In ths artcle, we exten these eas to the more complex case of graphs whch have a gven egree sequence. 007 Wley Perocals, Inc. Ranom Struct. Alg., 3, 9, 008 Keywors: quas-ranom; graph; egree sequence. INTRODUCTION Durng recent years there has been ncreasng nterest n nvestgatng the followng phenomenon. For a gven fnte collecton C of objects, suppose we have some probablty strbuton gven on C. Typcally, there are many propertes whch are satsfe by most or almost all) of the objects n C as seen n [4]. It turns out, however, that n many cases there s a large subclass Q of these propertes whch are strongly correlate, n the sense that any object n C whch satsfes any of the propertes n Q must n fact necessarly satsfy all the propertes n Q. Such propertes are calle quas-ranom. Specfc cases where ths behavor s nvestgate can be foun n [4, 5, ] for graphs), [, 3, 6, 7] for Corresponence to: Fan Chung *Supporte by NSF Grants DMS 04575, ITR 00506, ITR ). Supporte by NSF Grant CCR 03099). 007 Wley Perocals, Inc.

2 CHUNG AND GRAHAM hypergraphs), [9] for tournaments), [8] for sequences), [5] for permutatons), an [0] for sparse graphs), for example. In ths artcle we wll take C to be the class G n ) of all graphs on n vertces havng some gven egree sequence. Ths s rather fferent from the classcal moel of a ranom graph, n whch all vertces have the same expecte egree. Specal cases of such graph famles nclue the so-calle power law graphs n whch the number of vertces of egree k s proportonal to k β for some postve real β. Such graphs arse n a varety of applcatons such as Web connectvty [,5,6,9,4,6,8,9], communcaton networks [,3], bologcal networks [], collaboraton graphs [7], etc. In ths artcle, we wll ntrouce a class of quas-ranom propertes for G n ) an establsh quanttatve bouns on the strength of correlaton between these propertes. In partcular, these results generalze an strengthen those n [0, ].. NOTATION We wll conser graphs G = V, E) where V enotes the set of vertces of G an E enotes the set of eges of G. For unefne graph theory termnology, see [33].) Our graphs wll be unrecte, havng no loops or multple eges. We wll let V, the carnalty of V, be enote by n. If {x, y} E s an ege of G, we say that x an y are ajacent, an wrte ths as x y. The neghborhoo nx) of a vertex x V s efne by nx) := {y V : y x n G}. For x V, the egree x of x, enotes nx). The egree sequence = G of G s gven by = x : x V), or equvalently, can be vewe as a mappng : V Z + {0}. ForX, Y V, efne ex, Y) := {x, y) : x X, y Y an x y}. For X V, efne volx), the volume of X, by volx) = x X x. A walk P = P t x, y) from x to y s a sequence P = x 0, x,..., x t ), where x 0 = x, x t = y an x x + for 0 < t. Such a walk s sa to have length t. Here we o not requre all x s to be stnct. If all x s are fferent, we say the walk s a path. In ths artcle, we conser graphs wth every vertex havng postve egree. The weght wp) of such a walk P s efne to be wp) = 0<<t x thus, both enponts are exclue n the prouct). If P has length an therefore s an ege of G), then wp) s efne to be. A crcut C of length t s a sequence of t vertces x, x,..., x t ) where x x +, < t, an x t x. We remark that n ths efnton, a crcut can be vewe as a roote close Ranom Structures an Algorthms DOI 0.00/rsa

3 QUASI-RANDOM GRAPHS WITH GIVEN DEGREE SEQUENCES 3 walk.) The weght wc) of such a crcut s efne by wc) =. x The weghte ajacency matrx M = MG) s an n n matrx wth rows an columns nexe by V, efne by { f x y, Mx, y) = xy ) 0 otherwse. Note that M can be wrtten as M = I L where I s the entty matrx an L enotes the normalze) Laplacan see []). The egenvalues of M are enote by ρ,0 n, nexe so that = ρ 0 ρ ρ... ρ n usng the Perron-Frobenus theorem. Note that ρ 0 = has as ts egenvector x ) x V. Fnally, efne for X, Y V, an t, e t X, Y) = wp) t P PtX,Y) where P t X, Y) enotes the set of all walks of length t between x X an y Y. Ths s a weghte verson of the number of walks of length t between X an Y. Note that e X, Y) = ex, Y). In partcular, e V, V) = x x =. It s not ffcult to check that for t, we have e t V, V) =. 3. THE QUASI-RANDOM PROPERTIES In ths secton we wll state varous propertes that the G G n ) mght satsfy. Each of these propertes wll epen on a parameter ε, whch we wll always assume to satsfy 0 <ε<. The closer ε s to 0, the more the graph n queston behaves lke a ranom graph wth respect to the property n queston, that s, the more the value of the corresponng parameter s closer to ts expecte value for a ranom graph n G n ). DISCε): For all X, Y V, volx)voly) ex, Y) ε. DISC t ε): For all X, Y V, e tx, Y) volx)voly) ε. Note that DISC ε) s just DISCε). EIGε): Wth the matrx M = MG) = Mx, y)) x,y V as efne n ) an wth egenvalues satsfyng = ρ 0 ρ ρ... ρ n, Ranom Structures an Algorthms DOI 0.00/rsa

4 4 CHUNG AND GRAHAM we have ρ <ε for all. TRACE t ε): The egenvalues of M satsfy ρ t ε. CIRCUIT t ε): The weghte sum of the t-crcuts C t n G satsfes wc t ) ε. Ct:t crcut We wll prove the followng mplcatons n Secton 4: Theorem. For t, the followng mplcatons hol. Fg.. Implcatons of several propertes of G n ). Here the notaton A δ B s shorthan for Aε) Bδ). We say A mples B, enote by A B, f for every β>0, there exsts α>0 such that Aα) Bβ). There are several one-way mplcatons n the above Fg.. A natural queston s whch, f any, of the reverse rectons hol for any of these mplcatons. In Secton 5, we wll gve counterexamples whch show that EIG Trace t for any t. In Secton 6, we ntrouce an atonal property U t. Then we show that f a graph satsfes U t for some t, then DISC CIRCUIT t. Usng property U t, we wll prove the followng result. Theorem. If G satsfes U t for some t, then CIRCUIT t, TRACE t, EIG, DISC, DISC, DISC t are all equvalent. 4. THE IMPLICATIONS Lemma. EIGε) = DISCε). Proof. For S V, efne Then, for X, Y V, where f, g = x V f S x) = { x f x S, 0 otherwse. ex, Y) = f X, Mf Y f x)gx) enotes the usual nner prouct. Ranom Structures an Algorthms DOI 0.00/rsa

5 QUASI-RANDOM GRAPHS WITH GIVEN DEGREE SEQUENCES 5 Now, wrte f X = a φ where the φ s form an orthonormal bass of egenvectors wth v φ 0 v) =, for all v V. Hence, a 0 = f X, φ 0 = x x X = volx). Smlarly, we wrte Thus, f Y = b φ. f X, Mf Y =a 0 b 0 + = volx)voly) ρ a b + ρ a b Therefore, volx)voly) ex, Y) = ρ a b ) / ) / max ρ a b ε f X f Y = ε volx)voly) ε by usng EIG an the Cauchy-Schwarz nequalty where enotes the L -norm. Therefore, the proof s complete. In a smlar way, we prove Lemma. EIGε) = DISC t ε t ) for any t. Proof. In ths case we observe that for X, Y V, e t X, Y) = f X, M t f Y Ranom Structures an Algorthms DOI 0.00/rsa

6 6 CHUNG AND GRAHAM usng the notaton of Lemma ). Thus, wrtng f X = a φ, f Y = b φ, we fn f X, M t f Y volx)voly) = f X, M t f Y ρ t 0 a 0b 0 max ρ t a b max ρ t f X f Y ε t volx)voly) ε t an Lemma s prove. Lemma 3. CIRCUIT t ε) TRACE t ε). Proof. Let Ct u) enote a roote t-crcut wth startng an enng pont u. Then, M t u, u) = w C t u)). C t u) Thus, the trace of the matrx M t can be expresse as: TrM t ) = C t u)) u C t u) w = Ct wc t ). On the other han, the same trace can be evaluate usng egenvalues: TrM t ) = ρ t = + ρ t. Thus, we have wc t ) C t = TrMt ) = ρ t an Lemma 3 s prove. Ranom Structures an Algorthms DOI 0.00/rsa

7 QUASI-RANDOM GRAPHS WITH GIVEN DEGREE SEQUENCES 7 Lemma 4. TRACE t ε) = EIGε /t ), for any t. Proof. By hypothess, we have = ρ t ρ t ε. Therefore, max ρ ε /t. Lemma 5. TRACE t ε) = TRACE t+ ε). Proof. Snce ρ for all,wehave ρ t+ ρ t ε by hypothess. Lemma 6. For t, DISC t ε) = DISC t ε). Proof. For X V, e t X, X) = = y x,x X y e t y, X) y. e t x, y)e t y, x ) y ) By applyng DISC t ε) to e t X, X),wehave e t y, X) volx) + ε. y y Note that Therefore, e t y, X) = e t V, X) = x = volx). y x V y = y e t y, X) ) yvolx) y e t y, X) y e t V, X) volx) + volx) = e t y, X) y y ε. volx) Ranom Structures an Algorthms DOI 0.00/rsa

8 8 CHUNG AND GRAHAM by ) an DISC t ε). But y y Y = e t y, X) ) yvolx) y e t y, X) ) yvolx) y Y volx) e t y, X) y e t Y, X) voly)volx) y ) ) y Y y ) /voly) by applyng the Cauchy-Schwarz nequalty. Thus, e ty, X) voly)volx) ε voly) ε. Ths s exactly DISC t ε). Lemma 7. For any t, DISC t ε) = DISC t+ 6 ε). Proof. For X, Y V, Conser Defne S := e tx, Y) volx)voly) ε. 3) e t+ X, Y) = v ex, v)e t v, Y) v. { z V : e t z, Y) > z voly) + } ε ). Thus, e t z, Y) = e t S, Y) > vols )voly) z S Hence, by 3) apple to X = S an Y, In the same way, f we efne S :={z V : e t z, Y) < + ε vols ). vols )< ε. 4) z voly) ε )}, then vols )< ε. 5) Now, e t+ X, Y) = v S S + v S S ex, v)e tv, Y). v Ranom Structures an Algorthms DOI 0.00/rsa

9 QUASI-RANDOM GRAPHS WITH GIVEN DEGREE SEQUENCES 9 For the frst sum, we have ex, v)e t v, Y) ex, v) v v S S v v S S v voly) + ε ) an v ex, v) voly) + ε ) volx)voly) + ε ex, v)e t v, Y) ex, v) v v S S v v S S v voly) ε ) ex, v) voly) ε ) v S S by usng 4) an 5). Thus, ex, v)e t v, Y) v S S v For the secon sum, we have ex, v)e t v, Y) v e t v, Y) v S S v v S S v volx) vols ) vols )) voly) ε ) volx) ε ) voly) ε ) volx)voly) 3 ε, volx)voly) 3 ε. = e t S S, Y) vols ) + vols ))voly) + ε by DISC t ε) ε voly) + ε = ε voly) + ε 3 ε. Puttng these two estmates together, we obtan e t+x, Y) volx)voly) 6 ε whch s DISC t+ 6 ε). Ranom Structures an Algorthms DOI 0.00/rsa

10 0 CHUNG AND GRAHAM Lemma 8. For any ntegers s an t, DISC s an DISC t are relate as follows:. If s < t, then DISC s ε) = DISC t 36ε /t s ). As a specal case, DISCε) = DISC t 36ε /t ).. If t < s k t for some k, then DISC s ε) = DISC t 36 / t+k s) k ε /k. Proof. ) follows from Lemma 7,.e., DISC s ε) DISC s 36ε) DISC s+ 36ε / )... DISC t 36ε /t s ) To prove ), we have, from ) that DISC s ε) DISC k t 36ε / k t s) Now apply Lemma 6 k tmes to get the esre mplcaton. By combnng Lemmas to 8, we have prove all the mplcatons n Theorem. 5. SEPARATION OF PROPERTIES In ths secton, we gve an example showng that at least one of the mplcatons n Theorem cannot be reverse. Whether ths s true of the others s not known at ths pont. Fact. For any t, for any δ = δε). EIGε) = TRACE t δ) Proof. Choose t an let G = Gn) be a ranom regular graph wth n vertces an vertex egree n /t. Thus, M = MG) has { /n /t f u v, Mu, v) = 0 otherwse. It was shown n [3] that the egenvalue strbuton of MG) for a ranom graph G wth a gven expecte egree strbuton satsfes the sem-crcle law f the mnmum egree s greater than a power of log n. As a consequence, f = ρ 0 ρ ρ... ρ n are the egenvalues of M, then. ρ = + o))/n /t,. If Nx) enotes the number of ρ wth ρ x/n /t, then Nx) n = + o)) π Ranom Structures an Algorthms DOI 0.00/rsa x u u.

11 QUASI-RANDOM GRAPHS WITH GIVEN DEGREE SEQUENCES In partcular, for x = /, we have Thus, N/) n ρ t = + o)) 3 + ) π ) t n N/)) n /t Hence, for any ε > 0, G satsfes EIGε), prove n n 0,butG oes not satsfy TRACE t 0.39). It woul be nterestng to know f some of the other possble mplcatons hol. For example, oes DISC EIG? Recently, Blu an Lnal [7] prove the followng partal mplcaton for regular graphs: For a -regular graph G on n vertces, f for all X, Y V, ex, Y) n X Y α X Y, 6) then ρ =Oαlog/α) + )). The above property n 6) was ntrouce by Thomason [3] n the context of what he calle p, α)-jumble graphs. Of course, ths property s qute a bt stronger than DISC. Propertes of ranom graphs base on ths concept wthout the equvalence relatons) are often referre as the pseuo-ranom propertes. The reaer s referre to [30, 3] for scussons on pseuo-ranom graphs. Butler [0] combnes the methos n [7] an [8] to prove the followng: For a graph G wth no solate vertces, f for all X, Y V, e tx, Y) volx) voly) α volx)voly), then ρ t 8α 5 log α). For t =, ths s the best possble up to a constant) by conserng a class of regular graphs constructe by Bollobás an Nkforov [8]. In ther example, the graphs have α = Cn /6 an ρ cαlog n for some constants c an C. 6. REVERSING THE IMPLICATIONS It s clear from the examples n the preceng secton that n orer to establsh some of the reverse mplcatons, e.g., DISC CIRCUIT t, we wll have to make further assumptons for the G G n ). One such conton s the followng: For t, a graph satsfes U t C) f for all x, y V, e t x, y) C xy. We wll thnk of C as a large postve real. We note that for t = an for G wth mnmum egree αn, the property U C) s automatcally satsfe for C /α. Note that for a -regular graph, U t mples that n C t or, equvalently, the volume of the graph s of orer at least n +/t. Ranom Structures an Algorthms DOI 0.00/rsa

12 CHUNG AND GRAHAM Lemma 9. For any t, U t C) = U t+ C). Proof. Observe that e t+ x, y) = z z ex, z)e t z, y) z ex, z) z = C y = C x y. C z y ex, z) z The lemma s prove. Theorem 3. If G satsfes U t C) for some t, then DISCε) = CIRCUIT t η) where η = C C ε/δ + C C + ) δ + 0 δ + δ + 6C 4 δ 3/ + 8C δ 3/, wth C = C/δ /4, an δ = max{ ε,36ε /t }. Note that η 0 as ε 0.) Proof. We are gong to conser the sum u,v V u v e t u, v) ) u v where, as usual, V = VG). Snce G satsfes DISCε) by hypothess, then by Lemma 8, G also satsfes DISC t δ) where δ 36ε /t,.e., e t X, Y) volx)voly) δ for all X, Y V. We here choose δ = max{ ε,36ε /t }. For a fxe vertex u, we partton the vertex set V nto the sets W = W u),0 < C, as follows. To smplfy the notaton, we use W nstea of W u) below.) { } W 0 = v :0 e t u, v) <δ /4 u v, { } W = v : δ /4 u v e t u, v) <δ /4 u v, { } W = v :δ /4 u v e t u, v) <3δ /4 u v, Ranom Structures an Algorthms DOI 0.00/rsa

13 QUASI-RANDOM GRAPHS WITH GIVEN DEGREE SEQUENCES 3 an, n general, { } W = v : δ /4 u v e t u, v) < + )δ /4 u v for 0 < C = C/δ /4. Snce e t u, v) C u v / by U t C), the W form a partton of V. Snce then W { v : e t u, v) δ /4 u v <δ/4 u v e t u, v) δ /4 u v ) v W < δ /4 u v v W an e t u, W ) δ /4 uvolw ) uvolw ) <δ/4. 7) Snce e t u, W ) = e t u, V) = u, then δ /4 uvolw ) u = δ /4 uvolw ) e t u, W ) δ /4 u volw ) = δ /4 u. 8) Now, for each,fvolw ) δ, then efne X = X u) an X = X u) as follows: { X = v : ev, W )> vvolw ) + } δ), { X = v : ev, W )< vvolw ) } δ). If volw )< δ then efne X = X =. Also efne W u = {W :volw )< δ }. Thus, vol ) W u C δ snce there are just C possble values of. By DISCε),wehave ew, X ) volw )volx ) ε, Ranom Structures an Algorthms DOI 0.00/rsa },

14 4 CHUNG AND GRAHAM but from the efnton of X,wehave ew, X ) volw )volx ) δ volx )volw ) δvolx ) δ = δ volx ). Therefore, A smlar argument shows that volx ) ε/δ. vol X ) ε/δ as well. Consequently, for each u, we conser X u := { } X X : W W u an we have volx u ) C ε/δ. 9) For v X u, we have, from the efnton of X u, e W u, v) = v ew, v) W W u v δ) vvolw ) W Wu = v δ) )) v vol W u = δ v + δ) vol ) Wu v δ v + δ δ)c v C + ) δ v. 0) We now begn conserng the sum, e t u, v) ) u v = + ) e t u, v) ) u v. u v u v u u v v Xu For the frst sum, we use property U t C) an Lemma 9 to obtan the followng estmate: e t u, v) ) u v ) C u v u v u v u v Xu Ranom Structures an Algorthms DOI 0.00/rsa u v Xu = C volx u) C C ε/δ )

15 QUASI-RANDOM GRAPHS WITH GIVEN DEGREE SEQUENCES 5 by 9). For the secon sum we have e t u, v) u v u v u = u = u u u v z u v u v ) e t u, z)ez, v) u v z + z Wu z Wu z W u e t u, z)ez, v) z ) e t u, z)ez, v) u v z + z Wu e t u, z)ez, v) u v z. For the frst sum just above wthout a factor of ), we have e t u, z)ez, v) u u v z Wu z C u z ez, v) by U t C), u u v z Wu z = C u e W u, v )) u u v C C + ) δ u v by 0) u C C + ) δ. ) For the secon sum above, we have u u v z Wu = u u v W Wu u u v W Wu + δ/4 u z ez, v) z W z W W u e t u, z)ez, v) u v z e t u, z)ez, v) u v z W z δ/4 u z ez, v) z W z u v snce A a b A B) a B) + b ) an nequaltes n 7). Ranom Structures an Algorthms DOI 0.00/rsa

16 6 CHUNG AND GRAHAM For the secon sum we have δ/4 u ez, v) u u v W Wu z W = δ u ew, v) u u v W Wu δ u u u v + δ) vvolw ) W Wu δ + δ) ) u v u u v δ + 3δ by ef. of X u 3) upper boune by the sum over all u an v). Fnally, for the frst sum we have δ/4 u ez, v) u v u u v W Wu z W = δ /4 u ew, v) u v u u v W Wu δ /4 u v volw ) u v u u v W Wu + δ /4 u v volw ) δ by the ef. of X u, W Wu ) δ /4 u v volw ) u v u u v ) W + Wu δ/4 u v volw ) + δ /4 u v volw ) δ W Wu ) u v δ /4 ) C δ /4 u v δ / ) C u v δ 3/4 ) u,v u v by 8), ef. of Wu an the fact that < C, 4 δ + 4C 4 δ 3/ + C δ 3/. 4) Now, we have to put everythng together. Ranom Structures an Algorthms DOI 0.00/rsa

17 QUASI-RANDOM GRAPHS WITH GIVEN DEGREE SEQUENCES 7 Frst observe that u,v V u v e t u, v) ) u v = e t u, v) u,v V u v u,v V = C t t crcut = C t t crcut wc t ) e tv, V) + wc t ) e t u, v) + u,v V u v usng e t V, V) = ) so that the preceng results, nclung nequaltes ), ), 3) an 4), gve wc t ) = u,v V u v C t t crcut e t u, v) u v ) C C ε/δ + C C + ) δ + 4 δ + 3δ) + 44 δ + 4C 4 δ 3/ + C δ 3/ ) C C ε/δ + C C + ) δ + 0 δ + δ + 6C 4 δ 3/ + 8C δ 3/. Ths proves Theorem 3. Corollary. If G has mnmum egree αn, then where η epens only on ε, α an t. DISCε) = CIRCUIT t η) Theorem 4. If G has mnmum egree αn for some constant α, then CIRCUIT t, TRACE t, EIG, DISC, DISC, DISC t are all equvalent for t. 7. CONCLUDING REMARKS We can summarze the man theorems n the followng: Fg.. Quas-ranom propertes for G n ). Ranom Structures an Algorthms DOI 0.00/rsa

18 8 CHUNG AND GRAHAM We shoul note that f for our egree sequence, we choose all to be approxmately) equal, so that the G G) are approxmately) regular, then our results specalze to the case of sparse ranom graphs consere n [0], except that here we get explct functons of ε as oppose to the expressons wth o) terms occurrng n [0]). What are other propertes whch mght be nclue n Theorem? Can conton U t be replace by a weaker conton to allow DISC CIRCUIT t to be prove Fg. )? We hope to return to ths n the future. ACKNOWLEDGEMENTS The authors thank Steven Butler for careful reangs of the paper. Hs many comments mae substantal mprovements to the proofs. REFERENCES [] W. Aello, F. Chung, an L. Lu, A ranom graph moel for massve graphs, In Proceengs of the 3n Annual ACM Symposum on Theory of Computng, New York, 000, pp [] W. Aello, F. Chung, an L. Lu, A ranom graph moel for power law graphs, Exp Math 0 00), [3] W. Aello, F. Chung, an L. Lu, Ranom evoluton n massve graphs, In Es. J. Abello, et al., Massve Comput. 4, Kluwer Acaemc Publshers, 00, pp. 97. [4] N. Alon an J. H. Spencer, The probablstc metho, Wley, New York, 99. [5] A.-L. Barabás an R. Albert, Emergence of scalng n ranom networks, Scence ), [6] A.-L. Barabás, R. Albert, an H. Jeong, Scale-free characterstcs of ranom networks: The topology of the worl we web, Physca a 8 000), [7] Y. Blu an N. Lnal, Lfts, screpancy an nearly optmal spectral gap n press). [8] B. Bollobás an V. Nkforov, Hermtan matrces an graphs: Sngular values an screpancy, Dscrete Math ), 7 3. [9] A. Broer, R. Kumar, F. Maghoul, P. Raghavan, S. Rajagopalan, R. Stata, A. Tompkns, an J. Wener, Graph Structure n the Web, In Proceengs of the WWW9 Conference, May, Amsteram, 000. [0] S. Butler, Usng escrepancy to control sngular values for nonnegatve matrces, Lnear Algebra Appl ), [] F. Chung, Spectral graph theory, AMS Publcatons, 997, x+07 pp. [] F. R. K. Chung, The regularty lemma for hypergraphs an quas-ranomness, Ranom Struct Algorthms 99), 4 5. [3] F. R. K. Chung an R. L. Graham, Cohomologcal aspects of hypergraphs, Trans Am Math Soc ), [4] F. R. K. Chung an R. L. Graham, Maxmum cuts an quas-ranom graphs, In Es. Alan Freze an Tomasz Luczak, Ranom graphs, Wley, New York, 99, pp [5] F. R. K. Chung an R. L. Graham, On graphs not contanng prescrbe nuce subgraphs, In Es. A. Baker, et al., A trbute to Paul Erö, Cambrge Unversty Press, 990, pp. 0. [6] F. R. K. Chung an R. L. Graham, Quas-ranom hypergraphs, Ranom Struct Algorthms 990), Ranom Structures an Algorthms DOI 0.00/rsa

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