Quasi-Randomness and the Distribution of Copies of a Fixed Graph

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1 Quasi-Randomness and the Distibution of Copies of a Fixed Gaph Asaf Shapia Abstact We show that if a gaph G has the popety that all subsets of vetices of size n/4 contain the coect numbe of tiangles one would expect to find in a andom gaph G(n, 1 2 ), then G behaves like a andom gaph, that is, it is quasi-andom in the sense of Chung, Gaham, and Wilson [6]. This answes positively an open poblem of Simonovits and Sós [10], who showed that in ode to deduce that G is quasi-andom one needs to assume that all sets of vetices have the coect numbe of tiangles. A simila impovement of [10] is also obtained fo any fixed gaph othe than the tiangle, and fo any edge density othe than 1 2. The poof elies on a theoem of Gottlieb [7] in algebaic combinatoics, concening the ank of set inclusion matices. 1 Intoduction The theoy of quasi-andom gaphs deals with the following fundamental question: which popeties P of gaphs have the popety, that any gaph that satisfies P behaves like an appopiate andom gaph. When we say behaves, we mean that it satisfies the popeties that a andom gaph would satisfy with high pobability. Such popeties ae called quasi-andom. The study of quasi-andom gaphs was initiated by Thomason [11, 12] and then followed by Chung, Gaham and Wilson [6]. Following the esults on quasi-andom gaphs, quasi-andom popeties wee also studied in vaious othe contexts such as set systems [2], tounaments [3] and hypegaphs [4]. Thee ae also some vey ecent esults on quasi-andom goups [8] and genealized quasi-andom gaphs [9]. Let us state the fundamental theoem of quasi-andom gaphs. A labeled copy of a gaph H in a gaph G is an injective mapping φ : V (H) V (G), that maps edges to edges, that is (i, j) E(H) (φ(i), φ(j)) E(G). Fo a set of vetices U V we denote by H[U] the numbe of labeled copies of H spanned by U, and by e(u) the numbe of edges spanned by U. A gaph sequence (G n ) is an infinite set of gaphs {G 1, G 2,...} of inceasing size, whee we denote by G n the n th gaph in the sequence whose size is n. The following is (pat of 1 ) the main esult of [6]: Theoem 1 (Chung, Gaham and Wilson [6]) Fix any 0 < p < 1. (G n ) the following popeties ae equivalent: Fo any gaph sequence P 1 (t): Fo an even t 4, let C t denote the cycle of length t. Then e(g n ) = ( 1 2 p + o(1))n2 and C t [G n ] = (p t + o(1))n t. Micosoft Reseach. asafico@tau.ac.il. 1 The esult of [6] also contains quasi-andom popeties involving the numbe of induced copies of gaphs of a fixed size, quasi-andom popeties elated to the spectum of the adjacency matix of a gaph, and moe. 1

2 P 2 : Fo any subset of vetices U V (G n ) we have e(u) = 1 2 p U 2 + o(n 2 ). P 3 : Fo any subset of vetices U V (G n ) of size 1 2 n we have e(u) = 1 2 p U 2 + o(n 2 ). P 4 (α): Fix an α (0, 1 2 ). Fo any subset of vetices U V (G n) of size αn we have that the numbe of edges connecting U and V (G n ) \ U satisfies e(u, V (G n ) \ U) = (pα(1 α) + o(1))n 2. The meaning of the o(1) tems in Theoem 1 is that fo any δ > 0 and lage enough n > n 0 (δ), G n satisfies the appopiate condition up to δ. So a sequence of gaphs (G n ) satisfies 2 popety P 3 if fo any δ > 0 and lage enough n > n 0 (δ), any subset of vetices U V (G n ) of size 1 2 n satisfies e(u) = 1 2 p U 2 ± δn 2. Also, the meaning of the fact that P 3 implies P 2 is that fo any δ > 0 thee is an ɛ = ɛ(δ) > 0 such that if G is an n-vetex gaph with the popety that any subset of vetices U V (G) of size 1 2 n satisfies e(u) = 1 2 p U 2 ± δn 2, then in fact any subset of vetices satisfies this condition. In what follows, let us say that a gaph popety is quasi-andom if it is equivalent to any (and theefoe all) of the 4 popeties defined above. Note, that each of the fou items in Theoem 1 is a popety we would expect G(n, p) to satisfy with high pobability. Thus a quasi-andom popety assets that G behaves like G(n, p) (fo the ight p). Given Theoem 1 (and its omitted pats) one may stipulate that any popety that holds fo andom gaphs with high pobability is quasi-andom. That howeve, is fa fom tue. Fo example, it is easy to see that having the coect vetex degees is not a quasi-andom popety (conside K n/2,n/2 ). As anothe example, note that in P 4 we equie α < 1 2, because when α = 1 2 the popety is not quasi-andom (see [6]). A moe elevant family of non quasi-andom popeties ae those equiing the gaphs in the sequence to have the coect numbe of copies of a fixed gaph H. Note that P 1 (t) guaantees that fo any even t, if a gaph sequence has the coect numbe of edges as well as the coect numbe of copies of C t then the sequence is quasi-andom. As obseved in [6] this is not tue fo all gaphs. Simonovits and Sós obseved that the counte-examples showing that fo some gaphs H, having the coect numbe of copies of H is not enough to guaantee quasi-andomness, all have the popety that some of the induced subgaphs of these counte-examples have significantly moe/less copies of H than should be. As quasi-andomness is a heeditay popety, in the sense that we expect a sub-stuctue of a andom-like object to be andom-like as well, they intoduced the following vaiant of popety P 1 of Theoem 1, whee now we equie all subsets of vetices to contains the coect numbe of copies of H. Definition 1.1 (P H ) Fo a fixed gaph H on h vetices, we say that a gaph sequence (G n ) satisfies P H if all subsets of vetices U V (G n ) satisfy H[U] = p e(h) U h + o(n h ). Note that the above condition does not impose any estiction on the numbe of edges of G, while in popety P 1 thee is. As opposed to P 1, which is quasi-andom only fo even-cycles, Simonovits and Sós [10] showed that P H is quasi-andom fo any gaph H. Theoem 2 (Simonovits and Sós [10]) Fo any fixed H with e(h) > 0, popety P H is quasiandom. 2 Hee and thoughout the pape we use the notation x = y ± z to denote that fact that y z x y + z. 2

3 Note that popety P H is clealy not quasi-andom if H is an edgeless gaph, so one needs the assumption that e(h) > 0 in Theoem 2. We will hencefoth assume that this is the case without explicitly mentioning this. As we have agued above, the popeties P H can be thought of as a vaiant of popety P 1 in Theoem 1. But one can also think of the popeties P H as a genealization of popety P 2 in Theoem 1, namely, popety P 2 is just P K1,1. Popety P 3 in Theoem 1 guaantees that in ode to infe that a sequence is quasi-andom, and thus satisfies P 2, it is enough to equie only the sets of vetices of size n/2 to contain the coect numbe of edges. At this point it is natual to ask if an analogous weake condition also holds fo the popeties P H when H is not an edge, that is, if it is enough to equie that only sets of vetices of size n/c will have the coect numbe of copies of H. This question was fist aised by Simonovits and Sós in [10]. Ou main esult hee is a positive answe to this question. Fist let us define the following weake vaiant of the popeties P H : Definition 1.2 (P H ) Fo a fixed gaph H on h vetices, we say that a gaph sequence (G n) satisfies P H if all subsets of vetices U V (G n) of size n/(h + 1) satisfy H[U] = p e(h) U h + o(n h ). Theoem 3 Fo any fixed H, popety P H is quasi-andom. A natual open poblem is if one can stengthen Theoem 3 by showing that it is in fact enough to conside sets of vetices of size n/c, fo some absolute constant c that is independent of H. We conjectue that the answe is yes. In fact, it seems easonable to conjectue that one can take c to be any constant lage than 1. The main idea of the poof of Theoem 3 is to show that any gaph that satisfies P H also satisfies P H. To this end we fist show in Section 2 a simple equivalence between two types of popeties egading the numbe of copies of H in subsets of vetices. In Section 3 we apply an old algebaic esult of Gottlieb [7] on the ank of Inclusion Matices to show that cetain sets of linea equations on hypegaphs ae linealy independent. Finally, in Section 4 we apply the esults of the fist two sections in ode to pove Theoem 3. 2 Copies of H vs Patite Copies of H Recall that fo a set of vetices U V we denote by H[U] the numbe of labeled copies of H spanned by U. Fo an h-tuple of vetex sets U 1,..., U h let us denote by H[U 1,..., U h ] the numbe of copies of H spanned by U 1,..., U h that have pecisely one vetex in each of the sets U 1,..., U h. We call such copies of H, patite copies of H with espect to U 1,..., U h. Ou main goal in this section is to pove that the popety of having the coect numbe of copies of a gaph H in sets of vetices of size at least m is equivalent to the popety of having the coect numbe of patite copies of H in h-tuples of sets of vetices of size at least m. This equivalence is poved in the following two lemmas. Lemma 2.1 Fo any δ > 0 and any gaph H on h vetices, thee exists a γ = γ 2.1 (δ, > 0 such that the following holds: Suppose a gaph G has the popety that evey set of vetices U of size k m satisfies H[U] = k h (p e(h) ± γ). Then, any h-tuple of sets of vetices U 1,..., U h of size k m satisfies H[U 1,..., U h ] = h!k h (p e(h) ± δ). 3

4 Poof: By the assumed popeties of G we know that fo any 1 t h and any subset S [h] of size t we have H[ U i ] = (tk) h (p e(h) ± γ). (1) i S By Inclusion-Exclusion we have that H[U 1,..., U h ] = 1 ( 1) h t t=h S [h]: S =t H[ i S U i ] 1 ( ) h = ( 1) h t (tk) h (p e(h) ± γ) t t=h 1 ( ) h = k h p e(h) ( 1) h t t h ± γ(2hk) h t t=h = k h h!p e(h) ± γ(2hk) h whee we have used (1) when moving fom the fist line to the second line, and the combinatoial identity h! = 1 ( h t=h t) ( 1) h t t h when moving fom the thid line to the fouth. Theefoe, it is enough to take γ = γ 2.1 (δ, = δ/(2 h. Lemma 2.2 Fo any δ > 0 and any gaph H on h vetices, thee exists a γ = γ 2.2 (δ) > 0 such that the following holds fo all lage enough k k 2.2 (δ): Suppose a gaph G has the popety that evey h-tuple of vetex sets U 1,..., U h of size k/h ae such that H[U 1,..., U h ] = h!(k/ h (p e(h) ±γ). Then evey set of vetices U of size k is such that H[U] = k h (p e(h) ± δ). Poof: Suppose G has a set of vetices U of size k fo which H[U] < k h p e(h) δk h (the case H[U] > k h p e(h) +δk h is identical). Define a patition of U into h sets U 1,..., U h by taking a andom pemutation of the vetices of U, and defining U 1 as the fist k/h vetices in the odeing, U 2 as the following k/h vetices, and so on. It is easy to see that fo any fixed labelled copy of H in U the numbe of pemutations fo which the vetices of H belong to distinct sets U i is pecisely h!(k!(k/ h. Hence, the pobability that the vetices of H belong to distinct sets U i is h! ( ) k h (k! = h! h k! h h k h k (k 1) (k h + 1) h! h h ( δ), whee the inequality is valid if k k 2.2 (δ) fo an appopiate k 2.2 (δ). As we assumed that U spans less than k h p e(h) δk h labelled copies of H, we conclude by lineaity of expectation that the expected numbe of labelled copies of H in such a patition is less than h! h h ( δ) (kh p e(h) δk h ) < h!(k/ h (p e(h) γ), if we choose γ = γ 2.2 (δ) = 1 2δ. So thee must be at least one pemutation that defines a patition U 1,..., U h with at most this many labelled copies of H, contadicting ou assumption on G. 4

5 3 Inclusion Matices, Linea Equations and Hypegaphs Ou main goal in this section is to pove Lemma 3.2 that will be used in the following section in the poof of Theoem 3. The poof of Lemma 3.2 will be a consequence of a esult on the ank of inclusion matices which we tun to define. As usual, let us denote the set of integes {1,..., } by []. Fo integes h d let Ih,d be the ( ( d) matix whose ows ae indexed by d-element subsets of [] and whose columns ae indexed by h-element subsets of []. With this convention, enty (S, S ) of Ih,d is 1 if S S and 0 othewise. The following theoem of Gottlieb [7] (see also [1] and [13]) will be a cental tool fo us. Theoem 4 (Gottlieb [7]) Fo any h d and h + d the matix I h,d has ank (. Ou main goal in this section is to deive a hypegaph intepetation of the above esult. We stat with some basic definitions. An h-unifom hypegaph (h-gaph fo shot) H = (V, E) has a set of vetices V = V (H) and a set of edges E = E(H) whee evey edge contains h distinct vetices fom V. In what follows, we will identify V with the set of integes [] and denote by H,h the complete h-gaph on vetices. Let 2 h d and h + d be integes and let D be any h-gaph on d vetices. Define Ch (D) as the incidence matix between the edges of H,h and the copies of D in H,h. Moe pecisely, let c D denote the numbe of distinct copies of D that ae spanned by d vetices of H,h (that is, by H d,h ), and let Ch (D) be a c D ( ( d) ) h matix whose cd ( d) ows ae indexed by the copies of D in H,h, and whose ( columns ae indexed by edges of H,h. Then, the enty of Ch (D) coesponding to a copy of D and an edge e E(H,h ) is 1 if e E(D) and is 0 othewise. Lemma 3.1 Fo any tiple of integes 2 h d and h + d, and any h-gaph D on d vetices, the matix C h (D) has ank (. Poof: Obseve that when D = H d,h (i.e. the complete h-gaph on d vetices) then c D = 1 and theefoe Ch (H d, is an ( ( d) matix. Futhemoe, if we identify a copy of Hd,h with its set of vetices then one can easily see that Ch (H d, is in fact identical to the matix Ih,d. Theoem 4 thus implies that Ch (H d, has ank (. Conside now an abitay h-gaph D on d vetices. Let vs be the ow of Ch (H d,) coesponding to the copy of H d,h on the set of vetices S. Fo each S [] of size d, let u S be the ow vecto obtained by taking the sum of the c D ows of Ch (D) coesponding to the copies of D spanned by S. By symmety, evey edge of H,h in S is contained in the same numbe of copies of D that ae spanned by S. Theefoe, u S = c D v S fo some constant c D. Hence, the ows of Ch (D) span the ows of C h (H d,, so the ank of Ch (D) must be (. Fo ou puposes we will in fact be inteested in one type of h-gaph D. Let D h,b be the complete h-patite h-gaph on bh vetices whose evey patition class contains b vetices. In othe wods, we think of the vetex set of D h,b as being composed of h sets of vetices S 1,..., S h of size b each. The edge set of D h,b has all sets of h vetices that contain one vetex fom each of the sets S i (so D h,b has b h edges). It will be convenient to efe to a copy of D h,b by efeing to its h vetex sets S 1,..., S h. Note that by Lemma 3.1 we have that fo any h 2, b 1 the matix C h (D h,b ) has ank (. The following is the main esult of this section. 5

6 Lemma 3.2 Let H = H,h be the complete h-gaph on bh + h vetices, let D = D h,b be the complete h-patite h-gaph on bh vetices. Assign to each edge e = {v 1,..., v h } of H an unknown vaiable x v1,...,v h. Then H contains t = ( ) h copies of D, denoted D1,..., D t, with the following popety: suppose that fo evey copy D j, which is spanned by S j 1,..., Sj h, we wite a linea equation x v1,...,v h = b j, (2) v 1 S j 1,...,v h S j h whee b j is an abitay eal. Then this system of linea equations has a unique solution. Poof: This system of linea equations has ( ) ( h unknowns (one pe edge of H) and t = ) h equations (one pe copy D j ). If we wite it as Ax = b, then we only need to show that we can choose the copies of D in such a way that A is non singula. The impotant obsevation now is that the ow of C h (D h,b ) coesponding to a copy of D is the same as the ow of A coesponding to the same copy of D. This is because fo a copy D j of D, the linea equation in (2) involves the unknowns x i1,...,i h that coespond to the edges {i 1,..., i h } of D j. By Lemma 3.1 we know that C h (D h,b ) has ank ( ) ( h, hence, it has a set of ) h ows that ae linealy independent. Theefoe, if we use the ( ) h copies of D that coespond to these ows we get that the matix A is non-singula. 4 Poof of Main Result In this section we combine the esults fom the pevious sections and pove Theoem 3. One additional ingedient we will need, is the following simple claim. Claim 4.1 Fo any intege p thee is a C = C 4.1 (p) with the following popety: Let A be any p p non-singula 0/1 matix, let b be any vecto in R p and let x R p be the unique solution of the system of linea equations Ax = b. Then if b satisfies l (b, b) γ then the unique solution x of Ax = b satisfies l (x, x) Cγ. Poof: Fix any p p non-singula matix A with 0/1 enties. Then the solution of Ax = b is given by x = A 1 b. As x i = p j=1 A 1 i,j b j is a linea function of b it is clea that thee is a C = C(A) such that if l (b, b) γ then the unique solution x of Ax = b satisfies l (x, x) Cγ. Now, as thee ae finitely many 0/1 p p matices, we can set C = C 4.1 (p) = max A C(A), whee the maximum is taken ove all 0/1 p p matices. Poof of Theoem 3: Fix any gaph H on h vetices. Ou goal is to show that any gaph sequence that satisfies P H also satisfies P H and is thus quasi-andom by Theoem 2. Recall that P H is equivalent to equiing that fo all δ > 0 and fo all lage enough n > n 0 (δ), all subsets of vetices U V (G n ) must satisfy H[U] = p e(h) U h ± δn h. As this condition vacuously holds fo sets of size at most δn, it is enough to show that fo all δ > 0 and lage enough n > n 0 (δ), all subsets of vetices U V (G n ) of size at leats δn satisfy the stonge condition H[U] = (p e(h) ± δ) U h. (3) 6

7 If fact, by aveaging, if (3) holds fo sets of size δn then it holds also fo all sets of size at least δn (with a slightly lage δ). Fix then any δ > 0, and assume wlog that δ < 1 h+1. Define the following constants γ 2.2 = γ 2.2 (δ), γ 4.1 = γ 2.2 /((1/δ 1) h C 4.1 ( ( h/δ) h )) and γ3 = γ 2.1 (γ 4.1,. As (G n ) is assumed to satisfy P H, we know that fo any γ > 0, any lage enough gaph in the sequence, has the popety that all subsets of vetices U of size V (G n ) /(h + 1) satisfy H[U] = p e(h) U h ± γn h. This is equivalent to saying that (fo a slightly smalle γ) such sets satisfy H[U] = (p e(h) ± γ) U h. (4) We claim that any gaph G (G n ) that satisfies (4) with γ 3 (defined above) also satisfies (3) and thus by the above discussion (G n ) satisfies P H. By Lemma 2.2 we know that in ode to deive that any subset U of size δn satisfies (3) it is enough to pove that evey h-tuple of vetex sets U 1,..., U h of size δn/h satisfies H[U 1,..., U h ] = h!(δn/ h (p e(h) ± γ 2.2 ). So conside any h-tuples of sets U 1,..., U h of size δn/h each and let us patition the est of the vetices of G into sets U h+1,..., U h/δ of size δn/h each (we assume wlog that h/δ is an intege). Fo the est of the poof, let us denote the size of the sets U i by m = δn h. We will show that in fact all h-tuples U i 1,..., U ih in this patition have h!m h (p e(h) ± γ 2.2 ) patite copies of H, that is, that H(U i1,..., U ih ) = h!m h (p e(h) ± γ 2.2 ). (5) Ou assumption is that evey set of vetices U V (G) of size n/(h + 1) satisfies (4) with γ = γ 3. Hence 3, we have though Lemma 2.1 that evey h tuple of vetices, each of size k n/(h + 1), spans h!k h (p e(h) ± γ 4.1 ) copies of H. We now tun to use this fact to deive (5). Recall that we have a patition of G into h/δ sets U i each of size m = δn/h. Fo convenience, let us define b as an intege fo which bh + h = h/δ, that is b = 1 δ 1. Define a supe patition of the bh + h sets U i as a patition that clustes bh of the sets into h supesets S 1,..., S h of size b each (the othe h sets ae ignoed in this supe patition). Given a supe patition S 1,..., S h of the bh + b sets U i, let us define the coesponding supe patition of V (G) into h sets V 1,..., V h by setting V i = t S i U t. As each of the supesets S i contains b of the sets U i, we get that each of the sets V i is of size bm n/(h + 1) (hee we use δ < 1 h+1 ). Thus, the popeties of G guaantee that fo any supe patition V 1,..., V h H(V 1,..., V h ) = h!(bm) h (p e(h) ± γ 4.1 ). (6) We can also count the numbe of copies of H with one vetex in each set V i by consideing the numbe of copies of H with one vetex in one of the sets U t V i. That is, we can ewite (6) as H(V 1,..., V h ) = H(U i1,..., U ih ) = h!(bm) h (p e(h) ± γ 4.1 ). (7) i 1 S 1,...,i h S h Let us set x i1,...,i h = 1 (bm) h h! H(U i 1,..., U ih ). (8) 3 Hee again we use the simple obsevation that if all the sets U of size n/(h + 1) satisfy (4) then by aveaging the same applies fo lage sets. 7

8 Then, fo any supe patition S 1,..., S h we can ewite (7) as follows i 1 S 1,...,i h S h x i1,...,i h = 1 (bm) h h! H(V 1,..., V h ) = p e(h) ± γ 4.1. (9) Recall that we have a patition of G into bh+b sets U i, a supe patition of these bh+b sets into h clustes S 1,..., S h and that we have assigned a numbe x i1,...,i h to each subset of h of these clustes. Let H be the complete h-gaph on bh + b vetices, and think of each x i1,...,i h as an unknown assigned to the hype edge containing the vetices {i 1,..., i h }. The impotant obsevation at this point is that each of the linea equations in (9) is exactly the linea equation in (2) when taking the copy 1 of D in H to be the one spanned by S 1,..., S h, and b j to be (bm) h h! H(V 1,..., V h ) = p e(h) ± γ 4.1. Lemma 3.2 thus guaantees that thee ae ( ) h supe patitions of the bh + h sets Ui such that the coesponding ( ) h linea equations as in (9) have a unique solution. Note that each of the equations in (9) involves b h tems x i1,...,i h. Theefoe, if we had γ 4.1 = 0 in all the RHSs of the linea equations, then the unique solution would have been x i1,...,i h = p e(h) /b h fo all {i 1,..., i h }. Now, as the system involves ( ) ( h = h/δ ) h linea equations, and γ4.1 = γ 2.2 /(b h C 4.1 ( ( h/δ) h )) we have though Claim 4.1 that the unique solution to the system of linea equations satisfies x i1,...,i h = p e(h) /b h ± γ 2.2 /b h fo all {i 1,..., i h }. Recalling (8), this means that fo all i 1,..., i h, the numbe of patite copies of H spanned by U i1,..., U ih satisfies thus completing the poof. H(U i1,..., U ih ) = h!(bm) h x i1,...,i h = h!m h (p e(h) ± γ 2.2 ), Acknowledgements: I would to thank Miki Simonovits, Vea T. Sós and Asaf Nachmias fo helpful discussions. Special thanks to Raphy Yuste fo a caeful eading of this pape and fo many helpful comments. Refeences [1] D. de Caen, A note on the anks of set-inclusion matices, Elect. J. Comb. 8 (2001) No 5. [2] F. R. K. Chung and R. L. Gaham, Quasi-andom set systems, J. of the AMS, 4 (1991), [3] F. R. K. Chung and R. L. Gaham, Quasi-andom tounaments, J. of Gaph Theoy 15 (1991), [4] F. R. K. Chung and R. L. Gaham, Quasi-andom hypegaphs, Random Stuctues and Algoithms 1 (1990), [5] F. R. K. Chung and R. L. Gaham, Maximum cuts and quasi-andom gaphs, Random Gaphs, (Poznan Conf., 1989) Wiley-Intesci, Publ. vol 2, [6] F. R. K. Chung, R. L. Gaham and R. M. Wilson, Quasi-andom gaphs, Combinatoica 9 (1989), [7] D. H. Gottlieb, A class of incidence matices, Poc. Ame. Math. Soc. 17 (1966),

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