Modular orientations of random and quasi-random regular graphs

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1 Moular orietatios of raom a quasi-raom regular graphs Noga Alo Pawe l Pra lat Abstract Exteig a ol cojecture of Tutte, Jaeger cojecture i 1988 that for ay fixe iteger p 1, the eges of ay 4p-ege coecte graph ca be oriete so that the ifferece betwee the outegree a the iegree of each vertex is ivisible by 2p+1. It is kow that it suffices to prove this cojecture for (4p + 1)-regular, 4p-ege coecte graphs. Here we show that there exists a fiite p 0 so that for every p > p 0 the assertio of the cojecture hols for all (4p + 1)-regular graphs that satisfy some mil quasi-raom properties, amely, the absolute value of each of their otrivial eigevalues is at most c 1 p 2/3 a the eighborhoo of each vertex cotais at most c 2 p 3/2 eges, where c 1, c 2 > 0 are two absolute costats. I particular, this implies that for p > p 0 the assertio of the cojecture hols asymptotically almost surely for raom (4p + 1)-regular graphs. 1 Itrouctio A owhere-zero 3-flow i a uirecte graph G = (V, E) is a orietatio of its eges a a fuctio f assigig a umber f(e) {1, 2} to ay oriete ege e such that for ay vertex v V, f(e) f(e) = 0, e D + (v) e D (v) where D + (v) is the set of all eges emaatig from v, a D (v) is the set of all eges eterig v. A well kow cojecture of Tutte, raise i 1966 i [19], asserts that ay 4-ege coecte graph amits a owhere-zero 3-flow. This cojecture is still wie ope, a it is ot eve kow whether or ot there is a fiite k so that ay k-ege coecte graph has a owhere-zero 3-flow, although it is kow that if the ege coectivity of a -vertex graph is at least c log 2, the it oes have a owhere-zero 3-flow. This is prove i [4] (i a somewhat implicit, stroger form, with c = 2), a i [14] (with c = 4). It is kow (see, e.g., [17]) that a graph amits a owhere-zero 3-flow if a oly if it has a owhere-zero flow over Z 3, or equivaletly, a ege orietatio i which the ifferece betwee the outegree a the iegree of ay vertex is ivisible by 3. It is also kow (see, e.g., [8]) that it is eough to prove the cojecture for 5-regular graphs. Thus, Tutte s cojecture has the followig equivalet form. Sackler School of Mathematics a Blavatik School of Computer Sciece, Tel Aviv Uiversity, Tel Aviv 69978, Israel a Istitute for Avace Stuy, Priceto, New Jersey, 08540, USA. aress: ogaa@tau.ac.il. Research supporte i part by a ERC Avace grat, by a USA-Israeli BSF grat, by the Oswal Veble Fu a by the Bell Compaies Fellowship. Departmet of Mathematics, West Virgiia Uiversity, Morgatow, WV , USA. 1

2 Cojecture 1.1 (Tutte) Every 4-ege coecte 5-regular graph has a ege orietatio i which every outegree is either 4 or 1. Jaeger [12] extee this statemet a cojecture that for ay iteger p 1, the eges of ay 4p ege-coecte graph ca be oriete so that the ifferece betwee the outegree a the iegree of ay vertex is ivisible by 2p + 1. Such a orietatio is calle a mo (2p + 1)- orietatio. Similarly as before, it is kow that the geeral case ca be reuce to the (4p+1)- regular oe, a thus the cojecture has the followig equivalet form. Cojecture 1.2 (Jaeger s moular orietatio cojecture) For ay fixe iteger p 1, every 4p-ege coecte, (4p + 1)-regular graph has a mo (2p + 1)-orietatio, that is, a ege orietatio i which every outegree is either 3p + 1 or p. This cojecture is still ope, a appears to be ifficult. It is thus atural to try a prove that its assertio hols for almost all (4p+1)-regular graphs. (It is kow that a typical (4p+1)- regular graph is (4p + 1)-ege coecte.) Our mai result i this ote is that the assertio of the cojecture hols for all (4p + 1)-regular graphs with a sufficietly large eigevalue gap a with o ese eighborhoos, for all sufficietly large p. As a special case this implies that the assertio hols for almost all (4p + 1)-regular graphs. I orer to state the mai result we ee the otio of a (,, λ)-graph. A (,, λ)-graph is a -regular graph o vertices i which the absolute value of ay otrivial eigevalue of the ajacecy matrix is at most λ. This otatio was itrouce by the first author, motivate by several results showig that if λ is sigificatly smaller tha the the graph exhibits some strog pseuo-raom properties. Theorem 1.3 There are absolute positive costats 0, c 1, c 2 so that if λ < c 1 2/3, the ay (,, λ)-graph G = (V, E), where = 4p+1 > 0, i which o eighborhoo of a vertex cotais more tha c 2 3/2 eges, has a mo (2p + 1)-orietatio, that is, a orietatio i which every outegree is either 3p + 1 or p. I orer to prove the mai result it is coveiet to cosier a equivalet formulatio of Cojecture 1.2, prove i [8] for p = 1 a i [13] for geeral p. The equivalece is a cosequece of a ol result of Hakimi [10] which follows from Hall s Theorem, or from the maxflow micut Theorem. For two isjoit sets of vertices S a T i a graph G = (V, E), let E(S, T ) eote the set of all eges with a e i S a a e i T, a let S c = V \ S eote the complemet of S. Theorem 1.4 ([13]) Let p > 0 be a iteger, a let G be a (4p + 1)-regular graph. The G = (V, E) has a orietatio i which every outegree is either 3p + 1 or p if a oly if there is a partitio V = V + V with V + = V = V /2 such that for all S V, E(S, S c ) (2p + 1) S V + S V. (1) I view of the above, the followig result implies the assertio of Theorem 1.3. Theorem 1.5 There are absolute positive costats 0, c 1, c 2, c 3 so that if > 0 a λ < c 1 2/3, the ay (,, λ)-graph G = (V, E) with a eve umber of vertices i which o eighborhoo of a vertex cotais more tha c 2 3/2 eges, has a vertex partitio V = V + V with V + = V = V /2 such that for all S V (G), ( ) E(S, S c ) 2 + c 3 S V + S V. (2) 2

3 The above theorem implies, as a special case, that the assertio of Cojecture 1.2 hols for almost all = (4p + 1)-regular graphs. This refers to the probability space of raom = (4p + 1)-regular graphs with uiform probability istributio. This space is eote G,, where is a fixe iteger. We say that a property hols i this space asymptotically almost surely (or a.a.s., for short) if the probability that a member G G, satisfies the property tes to 1 as tes to ( is eve sice is o). See, e.g., [7], [20] for more etails about G,. Theorem 1.6 There exists a fiite p 0 so that for ay fixe iteger p > p 0, a raom (4p + 1)- regular graph G amits, a.a.s., a mo (2p + 1)-orietatio, that is, a orietatio i which every outegree is either 3p + 1 or p. The rest of this ote is orgaize as follows. I Sectio 2 we preset a few useful lemmas. The mai result, Theorem 1.5 (which implies Theorem 1.3), is prove i Sectio 3. Sectio 4 cotais the simple erivatio of Theorem 1.6 from the mai result, a the fial sectio cotais some cocluig remarks a ope problems. Throughout the ote we assume, wheever this is eee, that the umber of vertices of the graphs cosiere is sufficietly large as a fuctio of their egree of regularity. 2 Prelimiaries To prove the result we use the expasio properties of raom -regular graphs that follow from their eigevalues. The ajacecy matrix A = A(G) of a give -regular graph G o vertices, is a real symmetric matrix. Thus, the matrix A has real eigevalues which we eote by λ 1 λ 2 λ. It is kow that several structural properties of a -regular graph are reflecte i its spectrum. Sice we focus o expasio properties, we are particularly itereste i the followig quatity: λ = λ(g) = max( λ 2, λ ). I wors, λ is the largest absolute value of a eigevalue other tha λ 1 = (for more etails, see the geeral survey [11] about expaers, or [6], Chapter 9). The umber of eges E(S, T ) betwee two sets S a T i a raom -regular graph o vertices is expecte to be close to S T /. A small λ (that is, a large spectral gap) implies that the eviatio is small. The followig useful bou is essetially prove i [2] (see also [6]): Lemma 2.1 (The Expaer Mixig Lemma) Let G be a -regular graph with vertices a set λ = λ(g). The for all S, T V S T E(S, T ) λ S T. Whe T = S c is the complemet of S, it will be sometimes coveiet to apply the followig lower estimate for E(S, S c ), E(S, S c ) ( λ) S Sc (3) for all S V. This is prove i [5] (see also [6]). We also ee the well kow fact (see [1], [15]) that for fixe a large, ay (,, λ)- regular graph satisfies λ (2 o(1)) 1. (4) 3

4 For a partitio (A, A c ) of the vertex set, efie δ(a, A c ) = E(A, A c ) A Ac, that is, δ(a, A c ) measures the ifferece betwee the actual umber of eges betwee A a A c a the expecte value of this umber i a graph of ege esity /. The followig simple lemma shows that for a small λ, if two partitios are ot too far from each other, the the sizes of the two correspoig cuts are similar. Lemma 2.2 Let G be a -regular graph with vertices a set λ = λ(g). For ay two partitios (A, A c ), (B, B c ) of the vertex set with A \ B + B \ A = x, we have δ(a, A c ) δ(b, B c ) 4λ x. Proof: For ay two partitios (A, A c ), (B, B c ), δ(a, A c ) δ(b, B c ) E(A B, Ac B) A B Ac B + E(A B, A Bc ) A B A Bc + E(Ac B c, A B c ) Ac B c A B c + E(Ac B c, A c B) Ac B c A c B 4λ x, where the last iequality follows from Lemma The proof of the mai result I this sectio we prove Theorem 1.5, that is, we show that a -regular graph G = (V, E) with a large spectral gap a o ese eighborhoos, with 0 for some positive iteger 0, has a partitio (V +, V ) of V with V + = V = /2, where = V is eve, such that the coitio (2) hols for ay S V. Note that for S = V + (or S = V ) this gives E(V +, V ) ( 2 + c 3 ) V + = 4 + Ω( ). Therefore, it is atural to start with a proof that there is such a ese bisectio. We ee the followig result prove i [3]. Lemma 3.1 ([3]) There are two absolute costats b 1, b 2 > 0 such that the followig hols. Ay -regular graph i which the eighborhoo of ay vertex cotais at most b 1 3/2 eges, has a cut of size at least 4 + b 2. 4

5 Note that, i particular, the coitio of the theorem hols for ay graph i which o ege is cotaie i more tha b 1 triagles. Usig this lemma, we prove that i fact oe ca always esure a large bisectio, that is, a cut i which the two vertex classes are of equal size. Theorem 3.2 There are absolute costats 0, b 1, b 3 > 0 so that the followig hols. Let G = (V, E) be a -regular graph o a eve umber of vertices, where 0, i which the eighborhoo of ay vertex cotais at most b 1 3/2 eges. The V has a cut (V +, V ) such that V + = V = /2 a ( ) E(V +, V ) 4 + b 3. Proof: By Lemma 3.1 there is a cut (A, B) of G of size E(A, B) 4 + b 2. Without loss of geerality assume that A B. Defie b 2 = mi{ b 2 4, 1 4 } a b 3 = b 2 2. If A = B, there is othig to prove. Otherwise, we prove the existece of the require bisectio by shiftig vertices from A to B util they have equal sizes. For each vertex v A, let C (v) eote the egree of the vertex v i the cut (A, B), that is, its umber of eighbors i B. Startig with the cut (A, B) cosier, first, the case A ( ). I this case, if for every v A, C (v) 2, the after shiftig ay vertex from A to B the size of the ew cut is still at least ( ) b. Otherwise, there is a vertex v A with C (v) < 2, a we ca shift it to B a icrease the size of the cut. Keepig this process we obtai a cut (A, B) (with the moifie sets A, B geerate), which is of size at least b, i which B A ( ). If, ow, for ay vertex v A, C (v) 2 + b 2, the after shiftig a arbitray vertex from A to B we obtai a ew cut of size at least 2 ( 2 + b 2 ) > 4 + b 2. Else, we ca shift a vertex v with C (v) < 2 + b 2 from A to B, ecreasig the size of the cut by less tha 2b 2. As there are at most require steps util A a B are of the same size, a i the e of each step either the size of the cut is above 4 + b 2 or the size ecreases by at most 2b 2, we coclue that there is a bisectio of size at least 4 + b 2 2b 2 = 4 + b 2 2b2 > 4 + b 3, where here we use the fact that > 0 a b 3 = b 2 2. This completes the proof. We ca ow prove the mai result of this ote. Proof of Theorem 1.5: Fix a sufficietly large positive iteger 0, a cosier a (,, λ) graph G = (V, E) with > 0, λ < c 1 2/3, a o eighborhoo with more tha c 2 3/2 eges, where c 1, c 2 > 0 are small absolute costats to be chose later, a is eve. By Theorem 3.2 there is a ese bisectio cut (V +, V ) of G with E(V +, V ) 4 + b 3. 5

6 Fix such a partitio (V +, V ). We procee to show that the coitio (2) hols for all S V. Without loss of geerality, we may assume that S /2. Iee, if (2) hols for S, the it hols for S c as well, as both sies of the iequality o ot chage whe replacig S by S c. Moreover, we may assume that S ( 1 2 λ ) sice otherwise it follows from (3), (4) a the facts that λ < c 1 2/3 a > 0, that E(S, S c ) ( λ) S Sc ( λ)( λ ) S = 2 (1 λ )(1 + 2λ ) S = 2 (1 + λ 2λ2 2 ) S > 2 (1 + λ 2 ) S 2 ( ) S = ( ) S ( ) S V + S V, supplyig the esire iequality. Hece, it suffices to cosier sets S with ( 1 2 λ ) S /2. Without loss of geerality, we may assume that S V + S V. Suppose, first, that S V λ. The by (3) E(S, S c ) ( λ) S Sc 2 S λ 2 S ( 2 + λ ) S λ S 2 ( 2 + λ 2 ) S λ 2 > ( 2 + λ 2 )( S 2 S V ) = ( 2 + λ 2 )( S V + S V ) ( )( S V + S V ), where the last iequality follows from (4). Thus coitio (2) hols i this case. It therefore remais to show that the coitio hols for sets S with S V + ( 1 2 2λ ), S V λ. For such sets V + \ S + S \ V + 3λ a hece oe ca apply Theorem 3.2 a Lemma 2.2 with x = 3λ to get that E(S, S c ) = S Sc + δ(s, S c ) 2 S + δ(v +, V ) 4λ x 2 S + b 3 4λ 2 S + b 3 4 3(c 1 2/3 ) 3/2 = 2 S + b 3 4 3c 3/2 1 > 2 S + b 3 2, 3λ where the last iequality hols for a appropriate choice of c 1 > 0. Takig c 3 = b 3 we coclue that the last quatity is at least completig the proof. ( 2 + c 3 ) S ( 2 + c 3 ) S V + S V, 4 Moular orietatio of raom regular graphs The value of λ for raom -regular graphs has bee stuie extesively. A major result ue to Friema [9] is the followig: 6

7 Lemma 4.1 ([9]) For every fixe ε > 0 a for G G,, a.a.s. λ(g) ε. Sice it is easy a well kow that for ay fixe, a.a.s., the raom -regular graph o vertices oes ot cotai two triagles sharig a ege (a hece certaily oes ot cotai a eighborhoo with c 2 3/2 eges), the assertio of Theorem 1.6 follows from Theorem 1.5 a Lemma Cocluig remarks a ope problems The assertio of Theorem 1.5 shows that there is a absolute positive costat a so that for all sufficietly large p, a -regular graph with (4p a p) satisfyig the coitios of the theorem has a mo (2p + 1) orietatio. I particular this hols, a.a.s., for a raom regular graph of this egree. Note that such a graph is ot 4p-ege coecte, as its miimum egree is smaller tha 4p. This is similar to the mai result of Suakov i [18] that asserts that as soo as the (o-regular) raom graph G(, p) has miimum egree 2, it has, a.a.s., a owhere-zero 3-flow (although it is obviously ot 4-ege coecte.) The proof of Theorem 1.3 here hols oly for p > p 0 for some fixe p 0, a we have mae o serious attempts to optimize its value (or optimize the costats i Theorem 1.5). This ca be oe but will make the computatio more teious, a will ot lea to a proof that works for all values of p. It will be iterestig to formulate a prove a versio of the theorem for p = 1, which correspos to the Cojecture of Tutte metioe as Cojecture 1.1 here. For the special case of raom 5-regular graphs this has bee prove very recetly by the seco author a Wormal [16]. Ackowlegmet: We thak a aoymous referee for suggestios that improve the presetatio. Refereces [1] N. Alo, Eigevalues a expaers, Combiatorica 6 (1986), [2] N. Alo a F.R.K. Chug, Explicit costructio of liear size tolerat etworks, Discrete Math. 72 (1988), [3] N. Alo, M. Krivelevich, a B. Suakov, MaxCut i H-free graphs, Combiatorics, Probability a Computig 14 (2005), [4] N. Alo, N. Liial, a R. Meshulam, Aitive bases of vector spaces over prime fiels, J. Combiatorial Theory, Ser. A 57 (1991), [5] N. Alo a V.D. Milma, λ 1, isoperimetric iequalities for graphs a supercocetrators, J. Combiatorial Theory, Ser. B 38 (1985), [6] N. Alo a J.H. Specer, The Probabilistic Metho, Wiley, 1992 (Thir Eitio, 2008). 7

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