Clique-factors in sparse pseudorandom graphs
aa r X i v : . [ m a t h . C O ] J un CLIQUE-FACTORS IN SPARSE PSEUDORANDOM GRAPHS
JIE HAN, YOSHIHARU KOHAYAKAWA, PATRICK MORRIS, AND YURY PERSON
Abstract.
We prove that for any t ≥ c > n such that any d -regular n -vertex graph G with t ∣ n ≥ n and second largest eigenvalue in absolute value λ satisfying λ ≤ cd t / n t − contains a K t -factor, that is, vertex-disjoint copies of K t covering everyvertex of G . Introduction An ( n, d, λ ) -graph is an n -vertex d -regular graph whose second largest eigenvalue in absolutevalue is at most λ . These graphs are central objects in extremal, random and algebraic graphtheory. The interest in these graphs lies in the fact that various pseudorandom properties canbe inferred from the value of λ , in terms of the other parameters. For example, if λ ≪ d thensuch a graph has the property that its edges are ‘distributed’ uniformly, which is one of theessential properties exploited in random graphs and the regularity method from extremal graphtheory. More precisely, the following inequality, called the expander mixing lemma (see e.g. [4]),makes this quantitative: ∣ e ( A, B ) − dn ∣ A ∣∣ B ∣∣ < λ √∣ A ∣∣ B ∣ , (1)whenever A and B are vertex subsets of an ( n, d, λ ) -graph G , where e ( A, B ) denotes the numberof edges between A and B (edges in A ∩ B are counted twice). An excellent introduction to thestudy of ( n, d, λ ) -graphs is given in a survey of Krivelevich and Sudakov [11]. The emphasisthere and throughout the field is on the interplay between the parameters n , d and λ and graphproperties of interest; more precisely, given some property, one wishes to establish best possibleconditions on n , d and λ that ensure that any ( n, d, λ ) -graph with parameters satisfying suchconditions has the property.In this note we will only be concerned with conditions on the parameters n , d and λ thatguarantee the existence of certain spanning structures, i.e., subgraphs that occupy the wholevertex set (and have minimum degree at least one). Thus, we will be somewhat selective in ourdiscussion. In particular, we are interested in whether, for some fixed t ≥
3, our ( n, d, λ ) -graphcontains a family of vertex-disjoint copies of K t covering each vertex exactly once, which wecall a K t -factor . We remark that the case d = Θ ( n ) is well-understood since the existenceof bounded degree spanning graphs in ( n, d, λ ) -graphs follows from the celebrated Blow-uplemma of Koml´os, S´ark¨ozy and Szemer´edi [10]; see the discussion in e.g. [8]. Moreover, a sparseblow-up lemma for subgraphs of ( n, d, λ ) -graphs [2] gives general nontrivial conditions for theexistence of a given bounded degree spanning subgraph in the case d = o ( n ) . These conditions Date : 2018/06/06, 12:29am.JH is supported by FAPESP (2014/18641-5, 2013/03447-6). YK is partially supported by FAPESP (2013/03447-6) and CNPq (310974/2013-5). PM is supported by a Leverhulme Trust Study Abroad Studentship (SAS-2017-052 / This is also sometimes called a perfect K t -matching or a perfect K t -tiling in the literature. re stronger than those discussed in what follows (and hence, as expected, the general resultsin [2] are weaker than the ones below).While extremal and random graph theory provide tools to answer questions in this line ofresearch positively, one is naturally interested in the asymptotic tightness of the obtained results.This requires constructions of special pseudorandom graphs, most of the known examples ofwhich come from algebraic graph theory or geometry. For our study here, there is essentiallyone prime example of such a construction, due to Alon [5], who gave K -free ( n, d, λ ) -graphswith d = Θ ( n / ) and λ = Θ ( n / ) . Krivelevich, Sudakov and Szab´o [12] then extended theseto the whole possible range of d = d ( n ) , constructing K -free ( n, d, λ ) -graphs with λ = Θ ( d / n ) .Alon’s construction is an important milestone in the study of ( n, d, λ ) -graphs. It provides arare example of something that is reminiscent of threshold phenomena in random graphs in thecontext of ( n, d, λ ) -graphs: if λ ≤ . d / n , then any vertex of an ( n, d, λ ) -graph is containedin a copy of K , while there are K -free ( n, d, λ ) -graphs with λ = Ω ( d / n ) . Even more is true:as proved by Krivelevich, Sudakov and Szab´o [12], ( n, d, λ ) -graphs with λ ≤ . d / n contain afractional triangle-factor. A natural conjecture from the same authors [12] states the following.
Conjecture 1.1 (Conjecture 7.1 in [12]) . There exists an absolute constant c > such thatevery ( n, d, λ ) -graph G on n ∈ N vertices with λ ≤ cd / n has a triangle-factor. The first result in this direction was given by Krivelevich, Sudakov and Szab´o [12], who provedthat it suffices to impose λ ≤ cd /( n log n ) for some absolute constant c >
0. This was improvedby Allen, B¨ottcher, H`an and two of the current authors [3] to λ ≤ cd / / n / for some c >
0. Infact, the result in [3] is that this condition on λ is enough to guarantee the appearance of squaresof Hamilton cycles (the square of a Hamilton cycle is obtained by connecting distinct verticesat distance at most 2 in the cycle). Another piece of evidence in support of Conjecture 1.1 isa result in [8, 9] that states that, under the condition λ ≤ ( / ) d / n , any ( n, d, λ ) -graph G with n sufficiently large contains a ‘near-perfect K -factor’; in fact, G contains a family of vertex-disjoint copies of K covering all but at most n / vertices of G . Very recently, Nenadov [15]proved that λ ≤ cd /( n log n ) for some constant c > K -factor. With regard to K t -factors for general t ∈ N , Krivelevich, Sudakov and Szab´o [12] remark thatthe condition λ ≤ cd t − / n t − yields, for appropriate c = c ( t ) , a fractional K t -factor. Althoughthere is, alas, no known suitable generalization of Alon’s construction to K t -free graphs, thiscondition on λ may be seen as a benchmark in the study of K t -factors in ( n, d, λ ) -graphs. Thefirst nontrivial result for this study was the result in [3] showing that λ ≤ cd t / n − t / forsome c = c ( t ) > t -powers of Hamilton cycles (and thus K t + -factorswhen ( t + ) ∣ n ). The aforementioned result of Nenadov [15] generalizes to K t -factors [15],giving the condition λ ≤ cd t − /( n t − log n ) for some constant c = c ( t ) >
0. The purpose of thisnote is to present a proof that, under the condition λ ≤ cd t / n t − for some suitable c = c ( t ) > ( n, d, λ ) -graph contains a K t -factor. More precisely, we prove the following. Theorem 1.2.
Given an integer t ≥ , there exist c > and n > such that every ( n, d, λ ) -graph G with n ≥ n and λ ≤ cd t / n t − contains a K t -factor. A fractional K t -factor in a graph G is a function f ∶ K t → R + , where K t is the set of copies of K t in G , such that ∑ v ∈ K ∈K t f ( K ) = v ∈ V ( G ) . Nenadov studied a larger class of graphs, namely, the class of bijumbled graphs with sufficient minimum degree.This class contains ( n, d, λ ) -graphs as a special case. For details, see concluding remarks at the end of this note. f d ≥ cn / log n for some suitable c = c ( t ) > t ≥
4, then the condition in Theorem 1.2is the weakest that is currently known to imply the existence of K t -factors. Theorem 1.2, firstannounced in [8], was obtained independently of Nenadov’s result. There are however similaritiesin both approaches, since they both use absorption techniques. However, the structures andarguments that are used are different and Nenadov’s method is more effective in the moreinteresting sparse range.We use standard notation from graph theory, see e.g. [16]. We will omit floor and ceilingsigns in order not to clutter the arguments.2. Tools
Properties of ( n, d, λ ) -graphs. We begin by giving some basic properties of ( n, d, λ ) -graphs. Some of these are well known and used throughout the study of ( n, d, λ ) -graphs whilstothers are specifically catered to our purposes here. Theorem 2.1 (Expander mixing lemma [4]) . If G is an ( n, d, λ ) -graph and A , B ⊆ V ( G ) , then ∣ e ( A, B ) − dn ∣ A ∣∣ B ∣∣ < λ √∣ A ∣∣ B ∣ . (2) Proposition 2.2 (Proposition 2.3 in [12]) . Let G be an ( n, d, λ ) -graph with d ≤ n / . Then λ ≥ √ d / . Fact 2.3.
Let G be an ( n, d, λ ) -graph with d ≤ n / . Suppose λ ≤ d t / n t − for some t ≥ . Then d ≥ n − /( t − ) / .Proof. Proposition 2.2 tells us that λ ≥ √ d /
2. Thus λ ≤ d t / n t − implies that d t − ≥ n t − / d ≥ n − /( t − ) / /( t − ) follows. (cid:3) Fact 2.4.
Let G be an ( n, d, λ ) -graph with λ ≤ εd t / n t − . If U is a set of m ′ ≥ d / vertices, thenthere are at most εd vertices u with ∣ N G ( u ) ∩ U ∣ < dm ′ /( n ) .Proof. Let U ′ be the set of vertices u such that ∣ N G ( u ) ∩ U ∣ < dm ′ /( n ) . By Theorem 2.1, wehave dn m ′ ∣ U ′ ∣ − λ √ m ′ ∣ U ′ ∣ < e ( U, U ′ ) < ∣ U ′ ∣ dm ′ n . Together with λ ≤ εd t / n t − , we obtain that ∣ U ′ ∣ ≤ ε d t − / n t − ≤ εd . (cid:3) Write K t ( , . . . , , ) for the graph obtained by replacing one vertex of K t by an independentset of size 40. Fact 2.5.
Let G be an ( n, d, λ ) -graph with λ ≤ cd t / n t − and suppose cd t / n t − ≥ . Then any setof t cd vertices spans a copy of K t ( , . . . , , ) . Moreover, any set of t − cd / n vertices spansa copy of K t − .Proof. Let U be a set of at least 2 cd t − / n t − vertices in G . Since λ ≤ cd t / n t − ≤ d ∣ U ∣/( n ) , itfollows from Theorem 2.1 that 2 e ( U ) ≥ dn ∣ U ∣ − λ ∣ U ∣ ≥ ∣ U ∣ ⋅ d ∣ U ∣ n , which implies that U contains a vertex with degree at least d ∣ U ∣/( n ) .Thus, given a set of 2 t cd vertices, we can iteratively pick vertices with large degree in thecommon neighborhood, and get a ( t − ) -clique whose common neighborhood has size at least t cd ⋅ ( d /( n )) t − = cd t / n t − ≥
40 (the smallest set from which we pick a vertex in this process hassize 4 cd t − / n t − ). Therefore, we obtain a copy of K t ( , . . . , , ) . The proof of the ‘moreover’-part is analogous. (cid:3) Templates and absorbing structures. A template T with flexibility m ∈ N is a bipartitegraph on 7 m vertices with vertex parts X and Z ∪ Z , such that ∣ X ∣ = m , ∣ Z ∣ = ∣ Z ∣ = m ,and for any ¯ Z ⊆ Z , with ∣ ¯ Z ∣ = m , the induced graph T [ V ( T ) ∖ ¯ Z ] has a perfect matching. Wecall Z the flexible set of vertices for the template. Montgomery first introduced the use ofsuch templates when applying the absorbing method in his work on spanning trees in randomgraphs [14]. There, he used a sparse template of maximum degree 40, which we will also use. Itis not difficult to prove the existence of such templates for large enough m probabilistically; seee.g. [14, Lemma 2.8]. The use of a sparse template as part of the absorbing method was alsoapplied by Kwan in [13], where he generalized the notion to 3-uniform hypergraphs in orderto study random Steiner triple systems, and by Ferber and Nenadov [6] in their work on theuniversality of random graphs.Here, we will use a sparse template to build an absorbing structure suitable for our purposes.The absorbing structure we will use is defined as follows. Let m be a sufficiently large integer. Let T = ( X, Z ∪ Z , E ) be the bipartite template with flexibility m , maximum degree ∆ ( T ) ≤
40 andflexible set Z . Write X = { x , . . . , x m } , Z = { z , . . . , z m } , Z = { z m + , . . . , z m } and define Z ∶ = Z ∪ Z . An absorbing structure ( T, K , A, S , Z, Z ) with flexibility m contains two sets K and S consisting of vertex-disjoint ( t − ) -cliques and two vertex sets A and Z such that V (K) , V (S) , A and Z are pairwise disjoint. Furthermore, with the labelling K ∶ = { K , K , . . . , K m } , A = { a ij ∶ x i z j ∈ E ( T )} and S = { S ij ∶ x i z j ∈ E ( T )} , the following holds. For all i and j suchthat x i z j ∈ E ( T ) , ● each { a ij } ∪ K i spans a copy of K t , ● each { a ij } ∪ S ij spans a copy of K t , and ● each { z j } ∪ S ij spans a copy of K t . Fact 2.6.
The absorbing structure ( T, K , A, S , Z, Z ) has the property that, for any subset ¯ Z ⊆ Z with ∣ ¯ Z ∣ = m , the removal of ¯ Z leaves a graph with a K t -factor.Proof. By the property of the template T = ( X, Z ∪ Z , E ) , there is a perfect matching M in T that covers X and Z ∖ ¯ Z . Then for each edge x i z j ∈ M , we take the t -cliques on { a ij } ∪ K i and { z j } ∪ S ij ; for the pairs { i, j } such that x i z j ∈ E ( T ) ∖ M , we take the t -cliques on { a ij } ∪ S ij .This gives the desired K t -factor. (cid:3) The following lemma asserts that ( n, d, λ ) -graphs possess the absorbing structures above. Lemma 2.7.
There exists ε > such that for all ε ∈ ( , ε ) there is an n ∈ N such that thefollowing holds for all n ≥ n . Let G be an ( n, d, λ ) -graph with λ ≤ εd t / n t − and suppose m = εd .Then there exists an absorbing structure ( T, K , A, S , Z, Z ) with flexibility m such that, for anyvertex v in G , we have deg ( v, Z ) ≥ d ∣ Z ∣/( n ) . The following concentration result will be used in the proof of Lemma 2.7
Lemma 2.8 (Lemma 2.2 in [2]) . Let Ω be a finite probability space and let F ⊆ ⋅ ⋅ ⋅ ⊆ F n be afiltration on Ω . For each i ∈ [ n ] let Y i be a Bernoulli random variable on Ω that is constant n each atom of F i , that is, let Y i be F i -measurable. Furthermore, let p i be a real-valued F i − -measurable random variable on Ω . Let x and δ be a real numbers with δ ∈ ( , / ) , and let X = Y + ⋅ ⋅ ⋅ + Y n . If ∑ ni = p i ≥ x holds almost surely and E [ Y i ∣ F i − ] ≥ p i holds almost surely forall i ∈ [ n ] , then Pr ( X < ( − δ ) x ) < e − δ x / . Proof of Lemma 2.7.
First we choose ε = /( t ) and let ε ∈ ( , ε ) . Then we take n largeenough.Let T = ( X, Z ∪ Z , E ) be a bipartite template with flexibility m and flexible set Z such that∆ ( T ) ≤
40. Pick an arbitrary collection of 3 m vertex-disjoint copies of K t ( , . . . , , ) (usingFact 2.5). Label the copies of K t − as K ∶ = { K , K , . . . , K m } . Then label A = { a ij ∶ x i z j ∈ E ( T )} as the vertices in the classes of 40 vertices in the copies of K t ( , . . . , , ) such that each a ij together with K i forms a copy of K t (we may then discard some extra vertices, accordingto the degree of x i in T ).We will pick Z = { z , . . . , z m } and S = { S ij ∶ x i z j ∈ E ( T )} satisfying the definition of theabsorbing structure as follows. Suppose that we have picked Z ( j − ) = { z , . . . , z j − } and S( j − ) = { S ij ′ ∶ j ′ < j } with the desired properties. At step j , we pick as z j a uniform randomvertex in V ( G ) ∖ ( V (K) ∪ Z ( j − ) ∪ V (S( j − )) ∪ B j ) , where B j is the set of vertices z in G such that ∣( N G ( a ij ) ∖ ( V ( K ) ∪ Z ( j − ) ∪ V ( S ( j − )))) ∩ N G ( z )∣ < d /( n ) for some i with x i z j ∈ E ( T ) . Since ∣ V ( K )∣ + m + ( t − ) m ≤ ( t + ) εd < d / ( T ) ≤
40, Fact 2.4 with U = N G ( a ij ) ∖ ( V ( K ) ∪ Z ( j − ) ∪ V ( S ( j − ))) implies that ∣ B j ∣ ≤ εd . Next, for each i such that x i ∈ N T ( z j ) , we pick a ( t − ) -clique S ij in ( N G ( a ij ) ∖ ( V ( K ) ∪ Z ( j − ) ∪ V ( S ( j − )))) ∩ N G ( z j ) ,which is possible by Fact 2.5 because this set contains at least d /( n ) vertices of G . Moreover,we can choose these at most 40 cliques to be vertex-disjoint, because they only take up 40 ( t − ) vertices and any set in G of size d /( n ) − ( t − ) > d /( n ) still contains a ( t − ) -clique.At last we analyse the random process for Z and prove that, with positive probability, allvertices v of G are such that deg ( v, Z ) is appropriately large. Note that at step j we have fixed ∣ V ( K ) ∪ V ( S ( j − )) ∪ Z ( j − )∣ ≤ ( t + ) m + ( t − ) m + m ≤ ( t + ) εd vertices. Sincewhen we pick z j , we also need to avoid the set B j of size at most 40 εd , in total we need to avoidat most 140 tεd vertices. Let v be a vertex in G . Given a choice of V ( K ) ∪ V ( S ( j − )) ∪ Z ( j − ) ,the probability that z j ∈ N G ( v ) is at least ( − tε ) d / n . Then, by Lemma 2.8 with δ = ε , wehavePr ( deg ( v, Z ) < d n ∣ Z ∣) < Pr ( deg G ( v, Z ) < ( − ε )( − tε )∣ Z ∣ d / n ) < e − ε d / n = o ( n ) . Thus, the union bound over all vertices of G implies that the existence of Z with the desiredproperty in the lemma. (cid:3) A Hall-type result.
Another tool that we will use is the following theorem of Aharoniand Haxell [1, Corollary 1.2].
Theorem 2.9.
Let H be a family of k -uniform hypergraphs on the same vertex set. A sufficientcondition for the existence of a system of disjoint representatives for H is that for every G ⊆ H there exists a matching in ⋃ H ∈G E ( H ) of size greater than k (∣ G ∣ − ) . By this we mean a selection of edges e H ∈ H for all H ∈ H such that e H ∩ e ′ H = ∅ for all H ≠ H ′ ∈ H . . Proof of Theorem 1.2
We are now ready to prove our main result.
Proof of Theorem 1.2.
Let t ≥ ε be given by Lemma 2.7. Choose ε ∶ = min { ε , t − } and let n be given by Lemma 2.7 on input ε . Finally, set c = ε / t + . Wemay assume that dn ≤ ε t , (3)since otherwise the existence of a K t -factor is guaranteed by the Blow-up Lemma [10] (see the dis-cussion in [8]). We apply Lemma 2.7 to G and obtain an absorbing structure ( T, K , A, S , Z, Z ) with flexibility m = εd on a set W of at most 124 tεd vertices. Thus Z ⊆ W is such that ∣ Z ∣ = εd and, for any subset ¯ Z ⊆ Z with ∣ ¯ Z ∣ = εd , the absorbing structure with ¯ Z removed hasa K t -factor. Moreover, deg ( v, Z ) ≥ d ∣ Z ∣/ n for any vertex v in G .Now we greedily find vertex-disjoint copies of K t in G ∖ W as long as there are at most ε d vertices left. This is possible by Fact 2.5 because ε d > t cd . We denote the set of uncoveredvertices in V ( G ) ∖ W by U . Thus ∣ U ∣ ≤ ε d .Next we will cover U by vertex-disjoint copies of K t with one vertex in U and the othervertices from Z by applying Theorem 2.9. To that end, for each vertex v ∈ U , let H v be the setof ( t − ) -element sets of N ( v ) ∩ Z that induce copies of K t − in G and let H = { H v ∶ v ∈ U } . Weclaim that H has a system of disjoint representatives. To verify the assumption of Theorem 2.9,we first consider sets X ⊆ U of size at least d t − / n t − . Let Z ′ be any subset of Z of size εd . Notethat ∣ X ∣∣ Z ′ ∣ ≥ εd t − / n t − , which implies that λ ≤ cd t / n t − ≤ ε ( d / n )√∣ X ∣∣ Z ′ ∣ . By Theorem 2.1,we have e ( X, Z ′ ) ≥ dn ∣ X ∣∣ Z ′ ∣ − λ √∣ X ∣∣ Z ′ ∣ ≥ dn ∣ X ∣∣ Z ′ ∣ − ε dn ∣ X ∣∣ Z ′ ∣ ≥ d n ∣ X ∣∣ Z ′ ∣ . Hence there exists a vertex v ∈ X such that deg ( v, Z ′ ) ≥ d ∣ Z ′ ∣/( n ) = εd /( n ) . By Fact 2.5,we can find a copy of K t − in N ( v ) ∩ Z ′ . Thus in this case we can greedily find vertex-disjointcopies of K t − which belong to ⋃ v ∈ X H v , as long as there are εd vertices in Z left uncovered.This gives a matching in ⋃ v ∈ X H v of size εd /( t − ) > ( t − ) ε d ≥ ( t − )∣ X ∣ , for ε sufficientlysmall. It remains to consider sets X ⊆ U of size at most d t − / n t − . In this case we fix anyvertex v ∈ X and only consider matchings in H v . Indeed, by the construction of the absorbingstructure (see Lemma 2.7), we have deg ( v, Z ) ≥ d ∣ Z ∣/ n = εd / n for any v . Since by Fact 2.5there is a copy of K t − in every set of size εd /( n ) , we can find a set of εd /( ( t − ) n ) vertex-disjoint copies of K t − in H v . We are done because εd /( ( t − ) n ) ≥ d t − / n t − holds by ourinitial assumption (3). Theorem 2.9 then tells us that a system of disjoint representatives doesexist for H , whence we conclude that there are vertex-disjoint copies of K t which cover all thevertices in V ( G ) ∖ W and ( t − ) ε d vertices of Z . We can then greedily find vertex-disjointcopies of K t in the remainder of Z , which exist by Fact 2.5, until exactly εd vertices of Z remain (which will be the case due to the divisibility assumption t ∣ n ). Then the key propertyof the absorbing structure completes a full K t -factor. (cid:3) Concluding remarks
Bijumbled graphs.
A graph G on n vertices is called ( λ, p ) -bijumbled if for any two vertexsubsets A , B ⊆ V ( G ) , ∣ e ( A, B ) − p ∣ A ∣∣ B ∣∣ < λ √∣ A ∣∣ B ∣ . hus by Theorem 2.1, any ( n, d, λ ) -graph is ( λ, d / n ) -bijumbled. Nenadov’s result [15] assertsthat for p ∈ ( , ] , any ( εp t − n / log n, p ) -bijumbled graph of minimum degree at least pn / K t -factor if ε = ε t > t ∣ n . It is straightforward to generalizeTheorem 1.2 to K t -factors and ( εp t n, p ) -bijumbled graphs with minimum degree pn / ε = ε t > A condition for arbitrary -factors. In his concluding remarks, Nenadov [15] raises the ques-tion whether the condition λ = o ( p n / log n ) is sufficient to force any ( λ, p ) -bijumbled graph G ofminimum degree Ω ( pn ) to contain any given 2-factor, i.e., any n -vertex 2-regular graph. Sinceany 2-factor consists of vertex-disjoint cycles whose lengths add up to n , the problem is thus tofind any given collection of such cycles in G . We will return to this question elsewhere [7], witha positive answer to Nenadov’s question. References [1] R. Aharoni and P. Haxell,
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E-mail address : { jhan|yoshi } @ime.usp.br nstitut f¨ur Mathematik, Freie Universit¨at Berlin, Arnimallee 3, 14195 Berlin, Germany andBerlin Mathematical School, Germany E-mail address : [email protected] Institut f¨ur Mathematik, Technische Universit¨at Ilmenau, 98684 Ilmenau, Germany
E-mail address : [email protected]@tu-ilmenau.de