Finding any given 2-factor in sparse pseudorandom graphs efficiently
aa r X i v : . [ m a t h . C O ] F e b FINDING ANY GIVEN 2-FACTOR IN SPARSE PSEUDORANDOM GRAPHSEFFICIENTLY
JIE HAN, YOSHIHARU KOHAYAKAWA, PATRICK MORRIS, AND YURY PERSON
Abstract.
Given an n -vertex pseudorandom graph G and an n -vertex graph H with maximumdegree at most two, we wish to find a copy of H in G , i.e. an embedding ϕ ∶ V ( H ) → V ( G ) sothat ϕ ( u ) ϕ ( v ) ∈ E ( G ) for all uv ∈ E ( H ) . Particular instances of this problem include finding atriangle-factor and finding a Hamilton cycle in G . Here, we provide a deterministic polynomialtime algorithm that finds a given H in any suitably pseudorandom graph G . The pseudorandomgraphs we consider are ( p, λ ) -bijumbled graphs of minimum degree which is a constant proportion ofthe average degree, i.e. Ω ( pn ) . A ( p, λ ) -bijumbled graph is characterised through the discrepancyproperty: ∣ e ( A, B ) − p ∣ A ∣∣ B ∣∣ < λ √∣ A ∣∣ B ∣ for any two sets of vertices A and B . Our condition λ = O ( p n / log n ) on bijumbledness is within a log factor from being tight and provides a positiveanswer to a recent question of Nenadov.We combine novel variants of the absorption-reservoir method, a powerful tool from extremalgraph theory and random graphs. Our approach is based on that of Nenadov ( Bulletin of theLondon Mathematical Society , to appear) and on ours (arXiv:1806.01676), together with additionalideas and simplifications. Introduction
A pseudorandom graph of edge density p is a deterministic graph which shares typical propertiesof the corresponding random graph G ( n, p ) . These objects have attracted considerable attentionin computer science and mathematics. Thomason [45, 46] was the first to introduce a quantita-tive notion of a pseudorandom graph by defining so-called ( p, λ ) -jumbled graphs G which satisfy ∣ e ( U ) − p ( ∣ U ∣ )∣ ≤ λ ∣ U ∣ for every vertex subset U ⊆ V ( G ) . Ever since, there has been a great dealof investigation into the properties of pseudorandom graphs and this is still a very active area ofmodern research.The most widely studied class of jumbled graphs are the so-called ( n, d, λ ) -graphs, which wereintroduced by Alon in the 80s. These graphs have n vertices, are d -regular and their second largesteigenvalue in absolute value is at most λ . An ( n, d, λ ) -graph satisfies the expander mixing lemma [8]allowing good control of the edges between any two sets of vertices A and B : ∣ e ( A, B ) − dn ∣ A ∣∣ B ∣∣ < λ √∣ A ∣∣ B ∣ , (1) Date : 2019/02/19, 2:09am.JH was supported by FAPESP (2014/18641-5, 2013/03447-6). YK was partially supported by FAPESP (2013/03447-6) and CNPq (310974/2013-5, 311412/2018-1, 423833/2018-9). PM is supported by a Leverhulme Trust Study AbroadStudentship (SAS-2017-052 / here e ( A, B ) = e G ( A, B ) denotes the number of pairs ( a, b ) ∈ A × B so that ab is an edge of G .An illuminating survey of Krivelevich and Sudakov [34] provides a wealth of applications.There are three interesting regimes in the study of pseudorandom graphs and the class of ( n, d, λ ) -graphs is versatile enough to capture the essence of all of these regimes. In the first, one assumes λ = εn , where n is the number of vertices in a graph G and ε > d is constant and λ < d . This class then contains (non-bipartite) expanders [26]and Ramanujan graphs [38], which are prominent objects of study throughout mathematics andcomputer science. The third regime (sparse graphs) concerns λ being o ( n ) , often some power of n ,where one has better control on the distribution of edges between truly smaller sets. This case hasbeen investigated more recently and made amenable to some tools from extremal combinatorics.The focus of this paper will be on conditions under which certain spanning or almost spanningstructures are forced in sparse pseudorandom graphs. Our main motivation comes from probabilisticand extremal combinatorics, in particular the problem of universality. A graph G is called F -universal for some family F if any member F ∈ F can be embedded into G . This problem attracteda lot of attention [4–7], especially for the case where F is a class of bounded degree spanningsubgraphs. In this case we say an n -vertex graph G is ∆-universal if it contains all graphs on atmost n vertices of maximum degree ∆. A large part of the focus of the study has been on theuniversality properties of G ( n, p ) [7, 18, 20–22, 27]. It is also natural to investigate the universalityproperties of ( n, d, λ ) -graphs as was suggested by Krivelevich, Sudakov and Szab´o in [35]. Inthis setting of sparse pseudorandom graphs, a general result on universality has been proved onlyrecently in [1]. Let us comment that the case of dense graphs is well understood since the blow-up lemma of Koml´os, S´ark¨ozy and Szemer´edi [29] establishes that pseudorandom graphs of linearminimum degree contain any given bounded degree spanning structure. A little later, the secondand fourth author established jointly with Allen, B¨ottcher and H`an in [1], a variant of a blow-uplemma for regular subgraphs of pseudorandom graphs. This provides a machinery, complementingthe results of Conlon, Fox and Zhao [19] and allowing to transfer many results about dense graphsto sparse graphs in a unified way. However, these results are very general and thus do not establishtight conditions for special cases of spanning structures.Much more is known for questions about finding one particular spanning structure in a pseu-dorandom graph and the most prominent spanning structures which were considered in the lastfifteen years include perfect matchings, studied by Alon, Krivelevich and Sudakov in [34], Hamiltoncycles studied by Krivelevich and Sudakov [33], clique-factors [23, 24, 35, 41] and powers of Hamiltoncycles [3].The problem of when a triangle-factor appears in a given ( n, d, λ ) -graph has been a prominentquestion and is an instructive insight into the behaviour of pseudorandom graphs. It is easy to inferfrom the expander mixing lemma that if λ ≤ . d / n , then any ( n, d, λ ) -graph contains a triangle Note that edges in A ∩ B are counted twice. That is, disjoint copies of K covering all the vertices. in fact, every vertex lies in a triangle). An ingenious construction of Alon [9] provides an exampleof a triangle-free ( n, d, λ ) -graph with λ = Θ ( n / ) and d = Θ ( n / ) , which is essentially as denseas possible, considering the previous comments. This example can be bootstrapped, as is donein [35], to the whole possible range of d = d ( n ) , giving K -free ( n, d, λ ) -graphs with λ = Θ ( d / n ) .Further examples of (near) optimal dense pseudorandom triangle-free graphs have since been given[17, 32]. On the other hand, Krivelevich, Sudakov and Szab´o [35] proved that ( n, d, λ ) -graphswith λ = o ( d /( n log n )) contain a triangle-factor if 3 ∣ n and they made the following intriguingconjecture, which is one of the central problems in the theory of spanning structures in ( n, d, λ ) -graphs. Conjecture 1.1 (Conjecture 7.1 in [35]) . There exists an absolute constant c > such that if λ ≤ cd / n , then every ( n, d, λ ) -graph G on n ∈ N vertices has a triangle-factor. This conjecture is supported by their result that λ ≤ . d / n implies the existence of a fractionaltriangle-factor. Furthermore, a recent result of three of the authors [23, 25] states that, under thecondition λ ≤ ( / ) d / n , any ( n, d, λ ) -graph G with n sufficiently large contains a family of vertex-disjoint triangles covering all but at most n / vertices of G , thus a ‘near-perfect’ triangle-factor.A very recent, remarkable result of Nenadov [41] infers that λ ≤ cd /( n log n ) for some constant c > in a pseudorandom graph. The purpose of this work is to give a positive answerto Nenadov’s question, casting the question in terms of 2-universality and showing that we canefficiently find a given maximum degree 2 subgraph in polynomial time.In order to state our result we will switch to working with ( p, λ ) -bijumbled graphs (introducedin [28]), which give a convenient, slight variant of Thomason’s jumbledness. Bijumbled graphs G satisfy the property: ∣ e ( A, B ) − p ∣ A ∣∣ B ∣∣ < λ √∣ A ∣∣ B ∣ (2)for all A , B ⊆ V ( G ) . In particular it is easy to see by the expander mixing lemma (1) that an ( n, d, λ ) -graph is ( d / n, λ ) -(bi)jumbled. Moreover the two concepts are closely linked as a ( p, λ ) -(bi)jumbled graph is almost pn -regular, in that almost all vertices have degree close to pn .Before the current paper, the best result towards 2-universality in pseudorandom graphs is dueto Allen, B¨ottcher, H`an and two of the authors [3]. There, they proved that there exists an ε > ( p, εp / n ) -bijumbled graphs of minimum degree Ω ( pn ) contain a square of a Hamiltoncycle and hence are 2-universal. The proof is algorithmic, leading to an efficient procedure. Herewe weaken the requirement on λ to match that of Nenadov and obtain the following. Theorem 1.2.
For all δ > , there exist constants ε > and n such that, for any p ∈ ( , ] , thefollowing holds. For any n ≥ n and any given potential -factor F (that is, family of disjoint cycles A 2-factor is a 2-regular spanning subgraph. Nenadov also worked in this broader class of pseudorandom graphs. A square of a graph H is obtained by connecting its vertices at distance at most two through edges. The existence ofa square of a Hamilton cycle implies 2-universality as one can greedily find vertex-disjoint cycles of arbitrary lengths. hose lengths sum up to n ), there is a polynomial time algorithm which finds a copy of F in any ( p, λ ) -bijumbled graph G on n vertices with λ ≤ εp n / log n and minimum degree δ ( G ) ≥ δpn . In particular, Theorem 1.2 implies that such a ( p, λ ) -bijumbled graph G is 2-universal. Indeed,given a graph F ′ on at most n vertices with ∆ ( F ′ ) ≤
2, we find a supergraph F of F ′ on n vertices,so that all but at most one of the components of F are cycles. It is possible that F may haveeither one isolated vertex or a single edge but since we can easily embed a single vertex/edge into abijumbled graph G altering its minimum degree only a little, it suffices to concentrate on the casethat F is a 2-factor.1.1. Proof method.
Our proof uses the absorption-reservoir method, which has been a powerfultool in proving the existence of certain substructures and is often superior to the aforementionedblow-up lemmas. The basic idea of the method is to set up a flexible absorbing structure which cre-ates an extra ‘legroom’ when trying to embed the last o ( n ) vertices. The beginnings of this methoddate back to the early 90s, but the breakthrough in the wide applicability of these methods, how-ever, was first established by R¨odl, Ruci´nski and Szemer´edi [43, 44] in their study of Hamiltonicityin hypergraphs. There, the method was used to study dense hypergraphs but the methods havesince been adapted to other settings (see e.g. [2, 3, 36]).In his work on spanning trees in random graphs [40], Montgomery ingeniously wove sparse ‘robust’bipartite graphs (which we call sparse templates ) into the absorption-reservoir method. The firstuse of sparse templates for the absorption in the context of pseudorandom graphs was recently givenby the current authors in [24]. Our approach here builds on the ideas from our paper [24], andintroduces for the first time, an efficient version of this new type of absorption. In order to explicitlygenerate a sparse template we use bounded degree bipartite graphs with strong expansion properties.Such graphs are known as concentrators [11, 26]. We also build upon the different absorbing-typeargument due to Nenadov [41]. Our approach combines both arguments, making them constructiveby replacing certain nonalgorithmic arguments from [41] and derandomising additional argumentsat various places. 2. Proof of Theorem 1.2
The following three theorems will establish our main result. Note that the non-algorithmicversion of Theorem 2.1 was proved in [41].
Theorem 2.1 (Theorem 1.2, [41]) . For every δ > there exists a constant ε > such that, for any p ∈ ( , ] , a ( p, λ ) -bijumbled graph G on n ∈ N vertices with λ ≤ εp n / log n and minimum degree δ ( G ) ≥ δpn contains a triangle-factor, which can be found in polynomial time. Theorem 2.2.
For every δ > and L ∈ N there exist constants ε = ε ( δ, L ) > and n such thatfor any < ε < ε the following holds. Let G be a ( p, λ ) -bijumbled graph on n ≥ n vertices with p ∈ ( , / ] , λ ≤ εp n and minimum degree δ ( G ) ≥ δpn . Then in polynomial time, one can find anyfamily of vertex-disjoint cycles with lengths in the interval [ , L ] whose lengths sum up to at most n . heorem 2.3. For every δ > there exist constants L ∈ N , ε > and n such that the followingholds. For any p ∈ ( , / ] and < ε < ε , let G be a ( p, λ ) -bijumbled graph on n ≥ n vertices with λ ≤ εp n and minimum degree δ ( G ) ≥ δpn . Then in polynomial time, one can find any family ofvertex-disjoint cycles with lengths in the interval [ L + , n ] whose lengths sum up to at most n . Now we can quickly derive Theorem 1.2.
Proof of Theorem 1.2.
We consider three (not mutually exclusive) cases:( i ) at least n / F are covered by vertex-disjoint triangles,( ii ) at least n / F are covered by vertex-disjoint cycles with lengths in the interval [ , L ] where L is some absolute constant determined by Theorem 2.3 above,( iii ) at least n / F are covered by vertex-disjoint cycles with lengths in the interval [ L + , n ] .For a given 2-factor F on at most n vertices, we are in one of the three cases defined above. Let F , F and F denote the subgraphs of F , so that all triangles constitute F , all cycles with lengthsin [ , L ] constitute the subfamily F and all cycles of length at least L + F . We set n i ∶= v ( F i ) for each i ∈ [ ] .If we are in the first case ( n ≥ n /
2) we partition the vertex set V of G into three parts V ˙ ∪ V ˙ ∪ V ,so that ∣ V ∣ = n / ∣ V ∣ = ∣ V ∣ = n / G [ V i ] remains a ( p, λ ) -bijumbled graph. Moreover,every vertex v ∈ V satisfies deg ( v, V i ) ≥ δp ∣ V i ∣/ i ∈ [ ] . Clearly, one could achieve this viaa random partition and it will be possible to derandomise this approach (see Corollary 3.8). If n ≥ n , then we first apply Theorem 2.3 to embed F via some embedding ϕ into G [ V ] . ThenTheorem 2.2 asserts that F can be embedded into G ′ ∶= G [( V ˙ ∪ V ) ∖ ϕ ( V ( F ))] , since G ′ is itselfa ( p, λ ) -bijumbled graph with minimum degree δpn /
4. Finally, we apply Theorem 2.1 to embed F into the remaining graph (which is again ( p, λ ) -bijumbled graph with minimum degree at least δpn / n ≥ n then we first embed F , then F and, finally, F . The other cases n ≥ n / n ≥ n / (cid:3) Structure of the paper.
It remains to prove Theorems 2.1 – 2.3. We will only considerthe case p ≤ /
3, since the dense case can be treated fairly easily by the algorithmic version of theblow-up lemma due to Koml´os, S´ark¨ozy and Szemer´edi [30]. In Section 3 we collect some notationand useful tools and algorithms for our study. In the subsequent two sections we prove the firsttwo theorems (Theorems 2.2 and 2.3) and in Section 6 we replace one non-algorithmic argumentfrom [41] with a constructive proof.Throughout we use the shorthand ( p, λ ) -graphs to refer to ( p, λ ) -bijumbled graphs, we write logto denote the natural logarithm and we omit floor and ceiling signs in order not to clutter thearguments. The final section closes with some problems left for further study.3. Auxiliary results
Simple statements about ( p, λ ) -bijumbled graphs. Recall that we will often refer to ( p, λ ) -bijumbled graphs as ( p, λ ) -graphs. In this section, we collect some useful properties of ( p, λ ) -graphs. We will use the following notation. Given a graph G = ( V, E ) , we denote by deg ( v, U ) the umber of neighbours of v ∈ V in U ⊆ V . A u - v -path is a path P with end vertices u and v , and wecall the other vertices of P the inner vertices . For vertex subsets A, B , an A - B path is a u - v pathfor some vertices u ∈ A and v ∈ B . The length of a path is the number of its edges. Finally we willdenote by C ℓ ( , . . . , , K ) , the graph that consists of a path P of length ℓ −
2, whose end verticeshave exactly K distinct common neighbours outside of V ( P ) , for some K ∈ N . We start with thefollowing remark which follows directly from the definition (2). Remark 3.1. If ε > and A and B are subsets of a ( p, λ ) -graph with λ ≤ εp n , such that ∣ A ∣∣ B ∣ ≥ ε p n , then e ( A, B ) ≥ p ∣ A ∣∣ B ∣ . Next, we show a well-known property of bijumbled graphs; that they can not be too sparse.
Proposition 3.2.
Given ε ∈ ( , ) , there exists n ∈ N such that if G = ( V, E ) is a ( p, λ ) -graph on n ≥ n vertices with λ ≤ εp n and εp ≤ / , then p ≥ ( ε n ) − / / .Proof. Let S ⊆ V be a set of at least n / v ∈ S whose degree in G [ S ] is at most 2 p ∣ S ∣ . Indeed, we have ∑ v ∈ S deg ( v, S ) = e ( G [ S ]) ≤ p ∣ S ∣ + λ ∣ S ∣ , which implies thatthe average degree in G [ S ] is at most p ∣ S ∣ + λ ≤ p ∣ S ∣ ≤ pn .We consecutively find vertices v , . . . , v t with t = n /( + pn ) such that setting V i ∶= V ∖ { v , . . . , v i − } , we have deg ( v i , V i ) ≤ p ∣ V i ∣ ≤ pn . Thus setting U ∶= { v i ∶ i ∈ [ t ]} and W ∶= V ∖ ( U ∪ ⋃ i ∈[ t ] N ( v i )) , we have that ∣ W ∣ ≥ n − t ( + pn ) = n / e G ( U, W ) = ≥ p ∣ U ∣∣ W ∣ − λ √∣ U ∣∣ W ∣ .It follows that εp n ≥ λ ≥ p √ tn /
2. Thus, 2 ε p n ≥ t = n /( + pn ) , which implies p ≥ ( ε − / ) min { /( pn ) , / } . Rearranging we get p ≥ min {( ε n ) − / / , /( √ ε )} ≥ ( ε n ) − / / n sufficiently large. (cid:3) The following fact also concerns the edge distribution of bijumbled graphs.
Fact 3.3.
Let ε > and G be a ( p, λ ) -graph on n vertices with p ∈ ( , ] and λ ≤ εp n . ( i ) If U is a set of vertices, then there are at most ε p n /∣ U ∣ vertices w in G with ∣ N G ( w ) ∩ U ∣ < p ∣ U ∣/ . ( ii ) Given an integer t and vertex sets U , . . . , U t , W such that ∣ W ∣ > ∑ ti = ε p n /∣ U i ∣ , we canfind a vertex w ∈ W such that ∣ N G ( w ) ∩ U i ∣ ≥ p ∣ U i ∣/ for all i ∈ [ t ] in time O ( tpn ∣ W ∣) .Proof. Let U ′ be the set of vertices w such that ∣ N G ( w ) ∩ U ∣ < p ∣ U ∣/
2. From (2) we have ∣ U ′ ∣ p ∣ U ∣/ > e ( U ′ , U ) ≥ p ∣ U ∣∣ U ′ ∣ − λ √∣ U ∣∣ U ′ ∣ . The conclusion follows from rearranging.By the first part of the fact, W clearly contains a desired vertex. We find it by screening thedegree of any vertex of W into each U i , which takes time t ⋅ O ( pn ) . Since we may fail for at most ∣ W ∣ times, the conclusion follows. (cid:3) Next, given two sets A , B and C , we show how to find an A - B -path of given length such thatthe inner vertices are from C . Proposition 3.4.
Let ε > , ℓ ∈ N and G be a ( p, λ ) -graph on n vertices with p ∈ ( , ] , λ ≤ εp n and εpn ≥ . If A and B are sets of at least ℓ − εpn vertices and C is a set of at least ℓ − εn vertices, then we can find an A - B -path P of length ℓ whose inner vertices lie in C in time O ( p n ) . roof. If ℓ = e ( A, B ) > p ∣ A ∣∣ B ∣ − λ √∣ A ∣∣ B ∣ ≥ √∣ A ∣∣ B ∣ ( pεpn − λ ) ≥
0, namely, there isan edge with one end in A and the other in B . We can find such an edge by searching vertices oneby one in A . Since for each vertex it takes time O ( pn ) and by Fact 3.3 ( i ) , we may fail for at most4 εpn vertices, this can be done in time O ( p n ) . We proceed now inductively and we assume that ℓ ≥ ℓ − ( ii ) we find a vertex a ∈ A with degree at least p ∣ C ∣/ C , in time O ( p n ) (because we could focus the search on a set W of at most pn vertices in A ). Applying our inductivehypothesis to N ( a ) ∩ C , B ∖ { a } and C ∖ { a } we find an ( N ( a ) ∩ C ) - ( B ∖ { a }) -path of length ℓ − C , which together with a yields the desired path of length ℓ . (cid:3) We will use copies of C ℓ ( , . . . , , K ) in our absorbing structure. The following simple fact assertsthat we can find these copies in any large enough set of vertices. Fact 3.5.
Let ε > , K ∈ N and let G be a ( p, λ ) -graph on n vertices with p ∈ ( , ] and λ ≤ εp n .Let εp n ≥ K / , ℓ ≥ and U be a set of at least ℓ εn vertices. Then we can find a copy of C ℓ ( , . . . , , K ) , and thus also a copy of C ℓ in U , in time O ( p n ) .Proof. Let U ′ be the set of vertices v ∈ U with ∣ N ( v ) ∩ U ∣ < p ∣ U ∣/
2. Since ∣ U ∣ ≥ ℓ εn , Fact 3.3 ( i ) implies that ∣ U ′ ∣ ≤ εp n /
4. We fix a vertex u ∈ U ∖ U ′ , i.e. deg ( u , U ) ≥ p ∣ U ∣/
2. Let U ′ be the setof vertices v ∈ U with ∣ N ( v ) ∩ ( N ( u ) ∩ U )∣ < p ∣ N ( u ) ∩ U ∣/
2. Since ∣ N ( u ) ∩ U ∣ ≥ p ∣ U ∣/ ≥ εpn ,Fact 3.3 ( i ) implies that ∣ U ′ ∣ ≤ εpn /
2. Thus, we have ∣ U ′ ∪ U ′ ∣ ≤ εpn .We choose an arbitrary vertex u ∈ U ∖ ( U ′ ∪ U ′ ∪ { u }) . If ℓ = C ℓ ( , , , K ) in U , because ∣ N ( u ) ∩ N ( u ) ∩ U ∣ ≥ p ∣ U ∣/ ≥ K +
1. If ℓ ≥
5, then we first set asidea set W of K vertices from the common neighbourhood of u and u . Now due to the fact that ∣( N ( u i ) ∩ U ) ∖ ( W ∪ { u , u })∣ ≥ ℓ − εpn for i = , ∣ U ∖ ( W ∪ { u , u })∣ ≥ ℓ − εn , we find byProposition 3.4 a path of length ℓ − N ( u ) ∩ U and N ( u ) ∩ U , which together with u , u and W , forms a copy of C ℓ ( , . . . , , K ) .For the running time, by Fact 3.3 ( ii ) , we can find u and u in time O ( p n ) (because we couldfocus on a set of at most pn vertices in U ). The rest of the proof runs in time O ( p n ) , becausewe use Proposition 3.4. (cid:3) The following lemma asserts that we can (greedily) find almost spanning paths in ( p, λ ) -graphs. Lemma 3.6.
Let ε > and G be a ( p, λ ) -graph on n vertices with p ∈ ( , / ] and λ ≤ εp n . If U is a vertex subset of size greater than εn , then we can find any path of length ℓ ≤ ∣ U ∣ − εn in U intime O ( ℓ ⋅ p n ) .Proof. By Fact 3.3 there is a vertex u ∈ U of degree at least p ∣ U ∣/ U . This gives us a path oflength 0. Assume now that we found inductively a path P t = u u . . . u t of length t ≤ ⌊∣ U ∣ − εn ⌋ − ( u t , U ∖ V ( P t )) ≥ p ∣ U ∖ V ( P t )∣/
2. Then by Fact 3.3 ( i ) , as ∣ U ∖ V ( P t )∣ ≥ εn and usingthat p ≤ /
2, we have that there exists a vertex u t + ∈ N ( u t ) ∩ ( U ∖ V ( P t )) with deg ( u t + , U ∖ V ( P t )) ≥ p ∣ U ∖ V ( P t )∣/ ( i ) , Fact 3.3 ( ii ) implies that the runningtime is ℓ ⋅ O ( p n ) . (cid:3) .2. Partitioning vertex sets.
At various points in our proof, we will wish to partition our vertexset in such a way that every vertex maintains good degree to all parts of the partition. This canbe easily achieved probabilistically by choosing a random partition. However this idea can also bederandomised and achieved computationally efficiently. We use the following theorem of Alon andSpencer.
Theorem 3.7 (Theorem 16.1.2 in [13]) . Let ( a ij ) ni,j = be an n × n / -matrix. Then one can find,in polynomial time, ε , . . . , ε n ∈ { − , } such that for every ≤ i ≤ n , it holds that ∣ ∑ nj = ε j a ij ∣ ≤ √ n log ( n ) . Corollary 3.8.
Let k ∈ N ε, β, δ > and p ∈ ( , ] . Then there exists n ∈ N such that for any ( p, λ ) -graph G on n ≥ n vertices such that λ ≤ εp n , the following holds. Let U, W ⊆ V ( G ) besubsets of vertices such that ∣ U ∣ ≥ βn and for all w ∈ W , deg ( w, U ) ≥ δp ∣ U ∣ . Then in polynomialtime, we can find s ∶ = k sets U , . . . , U s ⊆ U such that U = U ˙ ∪ . . . ˙ ∪ U s , for each i we have ∣ U i ∣ = ∣ U ∣/ s and for all w ∈ W and i ∈ [ s ] , deg ( w, U i ) ≥ δp ∣ U i ∣/ .Proof. We apply Theorem 3.7 to the adjacency matrix of G , where we add an all one row and anextra column and impose that row i is all zero if i ∉ W and column j is all zero if j ∉ U . We let U ′ b = { j ∈ U ∶ ε j = ( − ) b } , for b = ,
2. The last row of the matrix guarantees that ∣∣ U ′ ∣ − ∣ U ′ ∣∣ ≤ √ ( n + ) log ( n + ) = ∶ g ( n ) . The other rows guarantee that the vertices in W have good degreeto both sets, so that after moving some vertices from one of the sets to another in order to balance ∣ U ′ ∣ and ∣ U ′ ∣ , we have that for all w ∈ W , deg ( w, U ′ i ) ≥ δp ∣ U ∣/ − g ( n ) .We can now apply the above procedure to each U ′ i , with the new minimum degrees. Repeat-ing this k times, we end up with U , . . . , U s as an equipartition of U such that for any w ∈ W ,deg ( w, U i ) ≥ δp ∣ U ∣/ s − kg ( n ) . Owing to Proposition 3.2, we are done because for sufficiently large n , 2 kg ( n ) ≤ δβpn /( s ) ≤ δp ∣ U ∣/( s ) . (cid:3) A connecting lemma.
The lemma below allows us to close many paths (whose ends are‘well-connected’ into a large set) into cycles using short paths of a fixed prescribed length. In thefollowing lemma a v - v -path refers to a cycle through v whose inner vertices are all the vertices ofthe cycle not equal to v . Lemma 3.9 (Multiple connection lemma) . For every < β, δ ′ ≤ , ℓ ≥ there exists ε > and n ∈ N such that for all ε ∈ ( , ε ) and n ≥ n the following holds. Let G be a ( p, λ ) -graph on n vertices with p ∈ ( , ] and λ ≤ εp n . Let U be a vertex subset of size at least βn and ( a i , b i ) i ∈ [ r ] asystem of pairs of vertices in G , so that every vertex occurs at most twice in ( a , . . . , a r , b , . . . , b r ) and U is disjoint from ⋃ i { a i , b i } . If r ≤ ∣ U ∣/( ℓ ) and deg ( a i , U ) , deg ( b i , U ) ≥ δ ′ p ∣ U ∣ for all i ∈ [ r ] then the following holds. In polynomial time, we can find a family Q of length ℓ a i - b i -paths Q i ,whose inner vertices are pairwise disjoint and lie in U .Proof. Fix ε ≤ δ ′ β −( ℓ + ) / ℓ . Firstly, using Corollary 3.8, in polynomial time, we can split U into U = U ˙ ∪ U such that ∣ U ∣ = ∣ U ∣ = ∣ U ∣/ ( a i , U b ) , deg ( b i , U b ) ≥ δ ′ p ∣ U ∣/ i and b = , U and then vertices of U . We initiate by letting Q ′ = ∅ , U ′ = U and U ′ = U . We will use Q ′ to denote our intermediate amily of paths and U ′ , U ′ the remaining sets of vertices that we can use. Note that throughoutwe will have ∣ V ( Q )∣ ≤ rℓ ≤ ∣ U ∣/
8, and thus ∣ U ′ ∣ , ∣ U ′ ∣ will always have size at least ∣ U ∣/ i ∈ [ r ] such that deg ( a i , U ′ ) , deg ( b i , U ′ ) ≥ δ ′ p ∣ U ∣/
8, thenusing Proposition 3.4, in time O ( p n ) we find a length ℓ − P i from a vertex in N ( a i ) ∩ U ′ to a vertex in N ( b i ) ∩ U ′ using vertices in U ′ . Add Q i ∶ = a i - P i - b i to Q and delete the vertices of P i from U ′ . At the end of this phase, let I ⊆ [ r ] be the remaining indices. Since each vertex appearsat most twice in ( a i , b i ) i ∈ [ r ] , by Fact 3.3 ( i ) , we have that ∣ I ∣ ≤ ε p n ∣ U ′ ∣ ≤ ε p n βn / ≤ ε p n ≤ δ ′ p ∣ U ∣/( ℓ ) ≤ δ ′ p ∣ U ∣/( ℓ ) , where we used ∣ U ′ ∣ ≥ ∣ U ∣/ ≥ βn /
4, and ε ≤ ε ≤ δ ′ β −( ℓ + ) / ℓ . Now we run the process again,using U in place of U . As ∣ V ( Q ′ ) ∩ U ∣ ≤ δ ′ p ∣ U ∣/ Q . Note that in each step, we need to screen thedegrees of the remaining pair a i and b i , which can be done in time O ( rpn ) . Then the applicationof Proposition 3.4 runs in time O ( p n ) . In total the algorithm runs in time O ( rp n ) . (cid:3) An explicit template. A template T with flexibility m ∈ N is a bipartite graph on 7 m verticeswith vertex parts I and J = J ˙ ∪ J , such that ∣ I ∣ = m , ∣ J ∣ = ∣ J ∣ = m , and for any ¯ J ⊆ J , with ∣ ¯ J ∣ = m , the induced graph T [ V ( T ) ∖ ¯ J ] has a perfect matching. We call J the flexible set ofvertices for the template.Sparse templates, with maximum degree smaller than some absolute constant, are very usefulin absorption arguments and can be used to design robust absorbing structures. Montgomery firstintroduced the use of such templates when applying the absorbing method in his work on spanningtrees in random graphs [40]. Ferber, Kronenberg and Luh [21] followed the same argument asMontgomery (with some small adjustments) when studying the 2-universality of the random graph.Kwan [37] also used sparse templates to study random Steiner triple systems, generalising thetemplate to a hypergraph setting and using it to define an absorbing structure for perfect matchings.Further applications were given by Ferber and Nenadov [22] in their work on universality in therandom graph and recently by the current authors in [24] which was the first use of the methodin the context of pseudorandom graphs, and by Nenadov and Pehova [42] who used the methodto study a variant of the Hajnal-Szem´eredi Theorem. The final three papers mentioned all adaptthe method to give absorbing structures which output disjoint copies of a fixed graph H (a partial H -factor), however the different absorbing structures used are interestingly all significantly distinct.It is not difficult to prove the existence of sparse templates for large enough m probabilistically;see e.g. [40, Lemma 2.8]. As we wish to give a completely algorithmic proof, in this section weshow how to build a template T efficiently. We use the following result of Lubotzky, Phillips andSarnak [38]. Theorem 3.10. [38]
For primes p, q ≡ ( mod 4 ) such that p is a quadratic residue modulo q ,one can construct an explicit ( p + ) -regular Ramanujan graph G in polynomial time (in q ) with ( q − q )/ vertices. Ramanujan graph, by definition, is a d -regular graph all of whose eigenvalues (other than d and, if bipartite, − d ) are in absolute value at most 2 √ d −
1. We will in fact use a bipartiteRamanujan graph constructed as follows. Consider the graph G provided by Theorem 3.10 – take V and V as two identical copies of V ( G ) , and join v ∈ V and v ∈ V if and only if the preimagesof v and v in V ( G ) form an edge of G . It is clear that this bipartite Ramanujan graph is still d -regular and satisfies the expander mixing lemma (1) for all A ⊆ V and B ⊆ V , where n is thenumber of vertices in each part, and λ = √ d − Proposition 3.11.
Let d ≥ / α . Let G be a bipartite d -regular Ramanujan graph on vertexset V ˙ ∪ V , with ∣ V ∣ = ∣ V ∣ = n . Suppose U ⊆ V and W ⊆ V are vertex subsets of V ( G ) such that ∣ U ∣ = ∣ W ∣ = αn and deg ( w, U ) ≥ αd / for any w ∈ W and deg ( u, W ) ≥ αd / for any u ∈ U . Then G [ U, W ] contains a perfect matching.Proof. We will verify Hall’s condition for G [ U, W ] . Note that it suffices to consider a set X ⊆ U of size ∣ X ∣ ≤ ∣ U ∣/ = αn /
2. Let Y = N ( X ) ∩ W and we aim to show that ∣ Y ∣ ≥ ∣ X ∣ . Assume to thecontrary that ∣ Y ∣ < ∣ X ∣ . We first assume that ∣ X ∣ ≤ αn /
6. By the degree condition, we obtain that e ( X, Y ) ≥ ∣ X ∣ αd /
3. On the other hand, by (1), we have e ( X, Y ) ≤ dn ∣ X ∣∣ Y ∣ + λ √∣ X ∣∣ Y ∣ < αd ∣ X ∣ + √ d ∣ X ∣∣ Y ∣ . Putting these together, we get 2 √ d ∣ X ∣∣ Y ∣ ≥ αd ∣ X ∣/
6. By d ≥ / α , this implies ∣ Y ∣ ≥ ∣ X ∣ , acontradiction. Next we assume that αn / < ∣ X ∣ ≤ αn /
2. By ∣ W ∖ Y ∣ ≥ αn / e ( X, W ∖ Y ) ≥ ( dn √∣ X ∣∣ W ∖ Y ∣ − λ ) (√∣ X ∣∣ W ∖ Y ∣) > α dn / − √ dα n / , where the second inequality follows from the fact that both factors in the product are always positive,given the restraints on ∣ X ∣ and ∣ W ∖ Y ∣ and thus the product is minimised when the factors, andhence ∣ X ∣ , ∣ W ∖ Y ∣ , are as small as possible. Since α d ≥ e ( X, W ∖ Y ) > Y . (cid:3) Lemma 3.12.
Let p ≡ ( mod 4 ) be a prime such that p ≥ . For a sufficiently large integer m , we can construct a template with flexibility m and maximum degree d ∶ = p + in polynomial time.Proof. It follows from the Siegel–Walfisz theorem [47] that we can pick a prime q ≡ ( mod 4 p ) between ( m ) / and 1 . ( m ) / for sufficiently large m . Thus 20 m ≤ q − q ≤ m . Usingquadratic reciprocity to infer that p is a quadratic residue modulo q , we have by Theorem 3.10 thatwe can construct in polynomial time a bipartite d -regular Ramanujan graph G = ( X ∪ Y, E ) with10 m ≤ ∣ X ∣ = ∣ Y ∣ ≤ m and second eigenvalue λ ≤ √ d . We first show that for any set U ⊆ X (or Y )of size at least 3 m /
2, there are at most 34000 m / d vertices v in Y (or X ) such that deg ( v, U ) < d / B the set of such vertices v . Clearly we have e ( U, B ) < d ∣ B ∣/
10. On the otherhand, by (1), we have d ∣ B ∣ > e ( U, B ) ≥ d m ∣ B ∣∣ U ∣ − λ √∣ B ∣∣ U ∣ ≥ d ∣ B ∣ − √ d ∣ B ∣ ⋅ m. This implies that 2 d ∣ B ∣/ < √ d ∣ B ∣ m , that is, ∣ B ∣ < m / d < m / d , as claimed. ow take arbitrary sets V ′′ ⊆ X , V ′′ ⊆ Y such that ∣ V ′′ ∣ = m and ∣ V ′′ ∣ = m . Next, wesequentially delete vertices from V ′′ and V ′′ as follows. ● Initiate with V ′ i ∶ = V ′′ i for i = , ● If there is a vertex v ∈ V ′ such that deg ( v, V ′ ) < d /
10, then delete v from V ′ , ● If there is a vertex v ∈ V ′ such that deg ( v, V ′ ) < d /
10, then delete v from V ′ .Note that since ∣ V ′′ i ∣ − m / d ≥ m /
2, by our claim above, at most 34000 m / d vertices will bedeleted from each set. Denote by V ′ and V ′ the resulting sets. Next, since there are at most34000 m / d vertices that have degree less than d /
10 to V ′ i , i =
1, 2, respectively, we can add verticesto V ′ and V ′ and obtain V and V such that ∣ V ∣ = m , ∣ V ∣ = m and deg ( v, V i ) ≥ d /
10 for any v ∈ V − i , i = ,
2. Finally, we pick J as a set of 2 m vertices in Y ∖ V which have degree at least d /
10 to V .We claim that T = G [ V ∪ V ∪ J ] is the desired template with flexible set J . It remains to checkthe property of T . For this, take any set J ′ of m vertices in J and consider G [ V , V ∪ J ′ ] . Sincethe assumptions of Proposition 3.11 are satisfied with α = m /∣ X ∣ ∈ [ / , / ] , G [ V , V ∪ J ′ ] hasa perfect matching and we are done. For the running time, note that in each of the steps above,it is enough to query the neighbourhood of a vertex, which can be done in constant time. So theoverall running time is polynomial in m . (cid:3) Proof of Theorem 2.2
In [24] an absorbing structure for cliques was defined. Here we generalise it for cycles as follows.Assume that T = ( I, J ∪ J , E ) is a bipartite template with flexibility m , maximum degree ∆ ( T ) ≤ K and flexible set J . It will be convenient to identify T with its edges which may be viewed as thecorresponding subset of tuples ( i, j ) ∈ [ m ] × [ m ] , hence we will also think of I as [ m ] , J as [ m ] , J as [ m + , m ] ∶ = { m + , . . . , m } and J = J ∪ J .An absorbing structure for cycles of length s + S = ( T, P , A, P , Z, Z ) which consistsof the template T with flexibility m , the two sets P and P of vertex-disjoint paths of fixed length s and three vertex sets A , Z and Z with Z ⊆ Z . Furthermore, the sets V ( P ) , V ( P ) , A and Z are pairwise disjoint and with the labelling Z = { z , . . . , z m } , Z = { z m + , . . . , z m } (so that Z ∶ = Z ∪ Z ), P ∶ = { P , P , . . . , P m } , A = { a ij ∶ ( i, j ) ∈ E ( T )} and P = { P ij ∶ ( i, j ) ∈ E ( T )} , thefollowing holds in G for ( i, j ) ∈ E ( T ) : ● a ij is adjacent to the ends of P i , i.e. closes a cycle on s + ● each a ij is adjacent to the ends of P ij , ● each z j is adjacent to the ends of P ij .In the proof of the following fact, we use a result of Micali and Vazirani [39], which constructs amaximum matching in general graphs in O (∣ E ∣∣ V ∣ / ) time. Fact 4.1.
The absorbing structure S = ( T, P , A, P , Z, Z ) has the property that, for any subset ¯ Z ⊆ Z with ∣ ¯ Z ∣ = m , the removal of ¯ Z leaves a graph with a C s + -factor, which can be found intime O ( m / ) . roof. By the property of the template T ⊆ [ m ] × [ m ] , there is a perfect matching M in [ m ] × ([ m ] ∖ ¯ J ) ∩ T with ¯ J ∶ = { j ∶ z j ∈ ¯ Z } . The above result from [39] finds M in time O ( m / ) .Then for each edge ( i, j ) ∈ M , we take the ( s + ) -cycles on { a ij } ∪ P i and { z j } ∪ P ij ; for the edges ( i, j ) ∈ E ( T ) ∖ M , we take the ( s + ) -cycle on { a ij } ∪ P ij . This gives the desired C s + -factor. (cid:3) The following lemma is a variant of Lemma 2.7 from [24].
Lemma 4.2.
Let K ∶ = . For every δ > , ℓ ≥ and α ∈ ( , α ( ℓ )] (where α ( ℓ ) ∶ = /( ℓ ( K + )) )there exists ε > such that for all ε ∈ ( , ε ) there is an n ∈ N such that the following holds forall n ≥ n . Let G be a ( p, λ ) -graph with n vertices, p ∈ ( , / ] , λ ≤ εp n , δ ( G ) ≥ δpn and suppose m = αn . Then in polynomial time we can find an absorbing structure S = ( T, P , A, P , Z, Z ) forcycles of length ℓ with flexibility m in G . Further, one can find such an S such that there is a set W ⊆ V ( G ) ∖ V ( S ) , with ∣ W ∣ = n / and deg ( v, W ) ≥ δp ∣ W ∣/ for all vertices v of G .Proof. First we choose ε = min { δ /( Kℓ ) , −( ℓ + ) , α } and let ε ∈ ( , ε ) . Then we take n largeenough. Therefore, owing to Proposition 3.2, quantities p n and pn are large as well.We consider a partition of V ( G ) = V ˙ ∪ V ˙ ∪ V ˙ ∪ V with ∣ V ∣ = ∣ V ∣ = ∣ V ∣ = ∣ V ∣ = n /
4, such thatdeg ( v, V i ) ≥ δp ∣ V i ∣/ i ∈ [ ] and v ∈ V , as given by Corollary 3.8. We fix W = V and thus the conditions on W aresatisfied. We now build our absorbing structure using vertices of V ( G ) ∖ W . Throughout the proof,we denote the intermediate partial absorbing structure by S ′ . Note that an absorbing structure forcycles of length ℓ with flexibility m which uses a template T has at most 3 ℓm ( ∆ ( T ) + ) vertices,and thus, due to the condition on α and the fact that we will have ∆ ( T ) ≤ K , we will have that ∣ V ( S ′ )∣ ≤ n /
20 throughout the proof.Let T ⊆ [ m ] × [ m ] be a bipartite template with flexibility m and flexible set J = [ m ] suchthat ∆ ( T ) ≤ K , as provided by Lemma 3.12. Pick an arbitrary collection of 3 m vertex-disjointcopies of C ℓ ( , . . . , , K ) in V (using Fact 3.5). For the i th copy of C ℓ ( , . . . , , K ) , we label thecorresponding path on ℓ − P i (so that the ends of P i have K common neighbours), andwe set P ∶ = { P , P , . . . , P m } . Then we label A = { a ij ∶ ( i, j ) ∈ E ( T )} as the vertices in the classesof K vertices in the copies of C ℓ ( , . . . , , K ) such that each a ij is connected to the ends of P i , i.e.forms a copy of C ℓ (we may then discard some extra vertices, according to the degree of x i in T ).We will pick Z = { z , . . . , z m } and P = { P ij ∶ ( i, j ) ∈ E ( T )} satisfying the definition of theabsorbing structure as follows. We choose Z in two phases, where all but at most εp n vertices for Z will be chosen in the first phase. We first use vertices in V . We recursively do the following. Wepick the smallest index j ∈ [ m ] (as long as there exists such an index) so that ∣ N G ( a ij , V ) ∖ V ( S ′ )∣ ≥ δpn /
10 for all i such that ( i, j ) ∈ T (there are at most K such i ). We pick as z j an arbitrary vertex in V ∖ ( V ( S ′ ) ∪ B j ) , where B j is the set of vertices z in G such that ∣( N G ( a ij , V ) ∖ V ( S ′ )) ∩ N G ( z )∣ < δp n /
20 for some i with ( i, j ) ∈ E ( T ) . Since ∣ N G ( a ij , V ) ∖ V ( S ′ )∣ ≥ δpn /
10 and ∆ ( T ) ≤ K ,Fact 3.3 ( i ) with U = N G ( a ij , V ) ∖ V ( S ′ ) implies that ∣ B j ∣ ≤ Kδ − ε pn ≤ n /
8, and so such achoice always exists.Having chosen z j , our next aim is to construct vertex-disjoint paths P ij of length ℓ −
2, for each ( i, j ) ∈ E ( T ) , so that the endpoints of P ij are adjacent to both a ij and z j . For this purpose, we ould like to pick two vertices y , y in U ij ∶ = ( N G ( a ij , V ) ∖ V ( S ′ )) ∩ N G ( z j ) , which are supposedto be the ends of the path P ij which we are going to construct. Since z j ∉ B j , we have ∣ U ij ∣ ≥ δp n /
20. Letting V ′ ∶ = V ∖ ( V ( S ′ ) ∪ U ij ) , we have that ∣ V ′ ∣ ≥ n /
8. From Remark 3.1, we get that e ( V ′ , U ij ) ≥ p ∣ U ij ∣∣ V ′ ∣/
2. We consider two cases. If ℓ = w from V ′ ofdegree at least p ∣ U ij ∣/ ≥ δp n / ≥ U ij (by Proposition 3.2), there is a path P ij of length 2with ends (labeled as) y and y in U ij . If ℓ ≥
5, then by Fact 3.3 ( i ) and the choice of ε , we canfind two vertices y and y ∈ U ij , whose degrees into V ′ are at least pn /
30. Proposition 3.4 thenyields the existence of a path of length ℓ − N ( y ) ∩ V ′ and N ( y ) ∩ V ′ . Togetherwith y and y this provides us with the desired path P ij .It remains still to deal with the situation (second phase), when there are no remaining appropriateindices j ∈ [ m ] . Let ˜ J ⊆ [ m ] be the set of those indices j such that for some { i, j } ∈ T we have ∣ N G ( a ij , V ) ∖ V ( S ′ )∣ < δpn /
10. Since ∣ V ∖ V ( S ′ )∣ ≥ n / ( i ) and ∆ ( T ) ≤ K that ∣ ˜ J ∣ ≤ K ( ε p n ) ≤ εp n . To finish the embedding, we will use vertices in V as well. At anypoint we will have that ∣ V ( S ′ ) ∩ V ∣ ≤ K ∣ ˜ J ∣ ℓ ≤ δpn /
40. From (3) we get deg ( v, V ∖ V ( S ′ )) ≥ δpn / v ∈ V ( G ) throughout the process and we can proceed as in the two paragraphs above,using V in place of V .Now we analyse the running time. Firstly, we pick the copies of C ℓ ( , . . . , , K ) by Fact 3.5 intime O ( p n ) . Secondly, to find a desired j ∈ [ m ] , we check ∣ N G ( a ij , V ) ∖ V ( S ′ )∣ for all vertices a ij , which takes time O ( pn ) ; with such a j , to choose z j , we search through the vertices z not in V ( S ′ ) and check ∣( N G ( a ij , V ) ∖ V ( S ′ )) ∩ N G ( z )∣ for at most K such i ’s. By Fact 3.3 ( ii ) , this takestime O ( p n ) . At last, we pick the desired path P ij of length ℓ −
2. If ℓ =
4, then we find the vertex w ∈ V ′ with degree 2 to U ij , in time O ( n ) . So we find the path P ij in time O ( pn ) . If ℓ ≥
5, we find y and y in time ∣ U ij ∣ ⋅ O ( pn ) = O ( pn ) and apply Proposition 3.4, which runs in time O ( p n ) .The overall running time is polynomial since O ( p n ) + O ( m ) ⋅ O ( pn ) = O ( pn ) and partitioningas is done by Corollary 3.8 works in polynomial time as well. (cid:3) Now we are ready to prove Theorem 2.2.
Proof of Theorem 2.2.
Let K = L ∶ = min { k ∶ k ∈ N , k > L } and fix ε ≤ ε ∶ = min { δ /( KL ( L + ) ) , ε . , ε . } , where ε . δ ′ ∶ = δ / L , α ( L ) and ε . β = /( L ( K + )) , δ ′ and L . Let n be largeenough. First, using Corollary 3.8, we find a partition of the vertex set of G into sets V ˙ ∪ V suchthat ∣ V ∣ = n / L and every vertex v ∈ V ( G ) satisfiesdeg ( v, V i ) ≥ δp ∣ V i ∣/ , (4)for i ∈ [ ] . Here V is taken to be the union of all other sets in the equipartition given by Corollary3.8, thus ∣ V ∣ = ( L − ) n / L . Let F be a collection of cycles of lengths in the interval [ , L ] ,whose lengths sum up to n . There is (at least) one length ℓ ∈ [ , L ] such that F contains at least n /(( L − ) ℓ ) cycles C ℓ . We write F = F ′ ˙ ∪ F ℓ , where F ℓ consists of cycles of length ℓ from F , while F ′ contains all other cycles. We will embed F into G in two stages. First, we greedily embed F ′ We can assume that F has n vertices as if not, we can take a supergraph by adding 4-cycles repeatedly. We canthen remove up to three vertices from G without affecting the properties of G as in the statement of Theorem 2.2. nto G [ V ] . This is possible since ∣ V ( F ′ )∣ ≤ ( L − ) nL − ≤ ( L − ) nL − = ( L − ) nL − nL ( L − ) = ∣ V ∣ − nL ( L − ) and since any set of at least 3 n /( L ( L − )) vertices in G contains a cycle of any length from theinterval [ , L ] (see Fact 3.5).In the second stage we are left with a vertex set U ⊇ V such that ∣ V ( F ℓ )∣ = ∣ U ∣ and δ ( G [ U ]) ≥ δpn /( L ) ≥ δ ′ p ∣ U ∣ , due to (4). All that remains to do is to find a C ℓ -factor in G [ U ] . We are thus ina position to apply Lemma 4.2 to G [ U ] , where one can check that the conditions there are satisfiedwith respect to ∣ U ∣ and δ ′ . Thus, in polynomial time we can construct an absorbing structure S = ( T, P , A, P , Z, Z ) for cycles of length ℓ with flexibility m = α ∣ U ∣ , where α ∶ = /( L ( K + )) ≤ α ( ℓ ) , and a vertex set W ⊆ V ( G ) ∖ V ( S ) , with ∣ W ∣ = ∣ U ∣/
4, such that for any vertex v in G , we havedeg ( v, W ) ≥ δ ′ p ∣ U ∣/
8. Let U ⊆ ( U ∖ V ( S )) be the set of vertices u such that deg ( u, Z ) ≤ p ∣ Z ∣/ ( i ) , we have that ∣ U ∣ ≤ ε p ∣ U ∣ /∣ Z ∣ = ε α − p ∣ U ∣ . We first incorporate the verticesof U into cycles of length ℓ using vertices of W ∖ U by applying Lemma 3.9 (in polynomial time)to the pairs {( u, u ) ∶ u ∈ U } . Let C be the set of disjoint cycles produced by this process.Now we greedily apply Fact 3.5 to find vertex-disjoint cycles C ℓ in G [ U ∖ ( V ( S ) ∪ V ( C ))] , untilwe are left with a set U of cardinality at most 2 ℓ εn . What remains is to find a C ℓ -factor in G [ U ˙ ∪ V ( S )] . Recall that deg ( u, Z ) ≥ p ∣ Z ∣/ u ∈ U . The assumptions of Lemma 3.9 aremet (in particular ∣ Z ∣ ≫ ∣ U ∣ ), and therefore, applying it to the pairs of vertices {( u, u ) ∶ u ∈ U } (to find paths through Z ) we find a family C of ∣ U ∣ vertex-disjoint cycles C ℓ that cover all of U (and some subset of Z ). Next, we greedily find, applying Fact 3.5, ( m − ∣ U ∣( ℓ − ))/ ℓ cycles C ℓ in Z ∖ V ( C ) , so a set Z ′ of exactly m vertices of Z remains uncovered. But then, letting Z ′′ = Z ∖ Z ′ , Fact 4.1 guarantees the existence of a C ℓ -factor on V ( S ) ∖ Z ′′ . This then gives us acopy of F in G .Note that we applied Fact 3.5 linearly many times, which took O ( p n ) running time. Moreover,we applied Corollary 3.8, Lemma 3.9, Fact 4.1 and Lemma 4.2 constantly many times. So weconclude that we can indeed find a copy of F in polynomial time. (cid:3) Proof of Theorem 2.3
Before proving Theorem 2.3, let us sketch some of the ideas that arise in the proof. Firstly wewill apply Lemma 4.2 to show the existence of an absorbing structure S = ( T, P , A, P , Z, Z ) forcycles of length 4 with flexibility m = ⌊ γn ⌋ , with γ ≤ α ( ) = /( ( K + )) , as defined in Lemma4.2. Recall that Fact 4.1 guarantees that no matter which m vertices of Z we remove, on therest of the vertices of S we can find a C -factor (which will contain exactly 3 m + ∣ E ( T )∣ copies of C ). Let us relabel the r ∶ = m + ∣ E ( T )∣ paths of length two in P ∪ P as Q = { Q , Q , . . . , Q r } ,let Q h = a h b h c h for each h ∈ [ r ] and let Y = Z ˙ ∪ A . Now the property of the absorbing structurecan be rephrased as follows. After removing exactly m vertices, Z ′ , from Z ⊆ Y , there is a perfectmatching between Q and Y ∖ Z ′ such that if Q h ∈ Q is matched with y ∈ Y , then a h yc h b h forms acopy of C . In what follows, the idea is to omit an edge (for example, a h b h ) from each of these C to get paths of length three which we will connect to longer paths. The key point is that we can do his by only omitting edges in the length two paths from Q . Thus we can simply connect verticesfrom paths in Q through short connecting paths. Eventually, this will lead to a longer path thatwill contribute to our factor and although we do not know exactly what these paths will be (as itdepends on the choice of matching to y ∈ Y ), the lengths of the paths and the vertices not in Y arefixed. More precisely, we will group the paths in Q according to the desired lengths of the cycleand connect the ones in the same group e.g. connect a h with b h − and connect b h with a h + . Atthe end of the proof, by Fact 4.1 we can match every remaining vertex y ∈ Y to one of the Q h ’s,such that a h yc h b h forms a copy of P which will contribute to some longer path which in turn ispart of a cycle in F .Our proof of Theorem 2.3 is algorithmic and we split the algorithm into three phases. Let usconcentrate here on the case where F is a full 2-factor i.e. F has n vertices. The first phase willbuild an initial segment for some of the cycles in our F , by finding short path segments which usethe vertices of Q and also, with foresight, incorporate some vertices that may be troublesome inthe last phase of our algorithm. Our second phase will incorporate the majority of the verticesinto paths. For each cycle in F we will greedily choose a path avoiding a fixed subset Z ′ ⊆ Z aswell as Z and the previously chosen vertices from the first phase. We will terminate this greedyphase with just εn candidate vertices not added to the paths. We will then use small paths through Z ′ to connect the initial path segments from phase one, the greedy paths from phase two and the εn remaining vertices. We will do this in such a way that we are left with m vertices in Z , say Z ′′ , and thus by the key property of our template T , there is a matching on T [ I ∪ J ′′ ] , where J ′′ = { j ∈ [ m ] ∶ z j ∈ Z ′′ } . This will dictate a matching between Z ′′ ∪ A and Q , which in turn tellsus how to incorporate the vertices of Z ′′ ∪ A into our cycles. This then results in disjoint cycles ofthe right size with every vertex used, that is, a copy of F in G . Proof of Theorem 2.3.
Let K = L ≥ K and fix γ ∶ = /( ( K + )) ≤ α ( ) with α ( ) defined in Lemma 4.2. Next, choose ε ≤ ε ∶ = min { δ /( K ) , ε . , ε . } , where ε . δ , α = γ , ℓ = ε . β = γ , δ ′ ∶ = δ /
16 and ℓ =
3. Let n be large enough. Let F be a graph on n vertices, whose componentsare cycles of length greater than L . We can assume that v ( F ) ≥ n − L , otherwise we can insteadconsider a supergraph by adding cycles of length L +
1. Let F consist of t cycles of lengths l ≥ ⋯ ≥ l t ,and let l = ∑ ti l i . Note that t ≤ n / L and n − L ≤ l ≤ n . We will show that F ⊆ G .Let m = γn . Apply Lemma 4.2 to get an absorbing structure S = ( T, P , A, P , Z, Z ) for cyclesof length 4 with flexibility m and a vertex set W ⊆ V ( G ) ∖ V ( S ) , with ∣ W ∣ = n /
4, such that for anyvertex v in G , we have deg ( v, W ) ≥ δp ∣ W ∣/
8. Label the vertices and paths of S as in the discussionabove. In particular, recall that r ∶ = m + ∣ E ( T )∣ . Let m ′ ∶ = εn , and let Z ′ ⊆ Z be an arbitrarysubset of size m + m ′ + t . Let V ⊆ ( V ( G ) ∖ V ( S )) ∪ ( Z ∖ Z ′ ) be the set of vertices v such thatdeg ( v, Z ′ ) ≤ p ∣ Z ′ ∣ . Write V ∶ = { v , v , . . . , v ∣ V ∣ } . By Fact 3.3 ( i ) , we have that ∣ V ∣ ≤ ε γ − p n . Wefind nonnegative integers q ij , i ∈ [ t ] , j ∈ [ ] such that the following holds: ● q i + q i + q i ≤ l i −
10, for each i ∈ [ t ] , ● ∑ ti = q i = r , ∑ ti = q i = ∣ V ∣ , and ∑ ti = q i = m ′ .Such choice can be achieved easily since r = m + ∣ E ( T )∣ and 6 r + ∣ V ∣ + m ′ ≪ l − t . e now run the first phase of our algorithm:( i ) We arbitrarily partition the set {{ a h , b h , c h } , h ∈ [ r ]} into t subsets of sizes q , q , . . . , q t and partition V into t subsets of sizes q , . . . , q t .( ii ) For i ∈ [ t ] , we fix an arbitrary linear order of the q i triples of vertices and q i vertices of V , and insert two new vertices x i , x i not in W ∪ V ∪ V ( S ) to the ordering, one to thebeginning, one to the end. Apply Lemma 3.9 to the pairs { b h − , a h } of consecutive elementsfrom each group simultaneously (we view each single vertex v in the ordering as v = a h = b h ),and get disjoint length three paths through W ∖ V joining the pairs. This is possible becausethe number of pairs we connect is at most 2 t + r + ∣ V ∣ ≤ n / L + m ( + K ) + ε γ − p n ≤ n / L + ( K + ) γn < n / δp ∣ W ∣/ − ∣ V ∣ ≥ δp ∣ W ∣/ W ∖ V , and ∣ W ∖ V ∣ ≥ n / i ∈ [ t ] , we obtain a sequence of paths on (in total) 5 q i + q i + q i + q i + exactly q i vertices from Z ). Nextwe will greedily find paths for each i ∈ [ t ] which will comprise the majority of the remainder of thecycles.( iii ) Fix U to be the vertices in ( V ( G ) ∖ ( V ( S ))) ∪ ( Z ∖ Z ′ ) which were not used in the paths chosenin the first phase. For i ∈ [ t ] , we repeatedly find a path of length exactly l i − q i − q i − q i − U using Lemma 3.6 (for this observe that there are at least ≥ εn unused vertices from U by the choice of the parameters). Denote the endpoints of the pathby x i and x i .( iv ) Arbitrarily choose m ′ vertices from U (it could happen that there are more vertices in U butonly if F has less than n vertices), partition and label them in such a way that for each i there are q i vertices u i, , . . . , u i,q i .( v ) Apply Lemma 3.9 to find paths of length 3 to connect the following set of pairs t ⋃ i = {( x i , x i ) , ( x i , u i, ) , ( u i, , u i, ) , . . . , ( u i,q i , x i )} with inner vertices from Z ′ . Note that this is possible as all the vertices of the pairs havegood degree to Z ′ and the number of pairs to connect is 2 t + ∑ i q i = t + m ′ , which is muchless than m = γn .( vi ) In the previous step we used exactly 2 m ′ + t vertices of Z ′ in length 3 paths. Thus theset Z ′′ ⊆ Z of unused vertices has size exactly m . By Fact 4.1 we can find a C -factor on V ( S ) ∖ ( Z ∖ Z ′′ ) in time O ( n / ) . Note that the paths a j y j c j b j for each C on { y j , a j , b j , c j } will complete the cycles of length exactly ( q i + q i + ) + ( l i − q i − q i − q i − ) + q i + = l i for each i ∈ [ t ] . Thus, we have found a copy of F in G .Note that we can compute the values of q ij greedily in time O ( n ) . Each of Lemma 3.9, Fact 4.1and Lemma 4.2 runs in polynomial time and we use them at most twice. Finally, we applied emma 3.6 t times. However, since the sum of the lengths of the paths we constructed is at most n , the running time is O ( p n ) . So the overall running time is polynomial. (cid:3) Let us mention here that one could also define an absorbing structure specifically for the longercycles we build in Theorem 2.3, connecting edges into paths according to the adjacencies of atemplate. Although this alternative structure would be easier to describe and would remove someof the technicalities in the above proof, we chose to instead work from the absorbing structure usedfor tiling with short cycles, for the sake of brevity.6.
A proof of Theorem 2.1
Nenadov’s proof is algorithmic, except the proof of [41, Lemma 3.5], in which he used a Hall-type result for hypergraphs due to Haxell. Here we give an alternative proof of this lemma, whichmoreover provides a polynomial time algorithm.We first need to recall some definitions from [41]. Let K − be the unique graph with 4 verticesand 5 edges. Define an ℓ -chain as a graph obtained by sequentially identifying ℓ copies of K − onvertices of degree 2. Note that an ℓ -chain contains exactly ℓ + removable .We say that a triangle in G traverses three chains D , D and D if it intersects all of them atsome removable vertices. Observe that if D , D and D are disjoint chains in G and there existsa triangle in G traversing them, then G [ V ( D ) ∪ V ( D ) ∪ V ( D )] contains a triangle-factor.Here we state [41, Lemma 3.5] and give an alternative (algorithmic) proof. Lemma 6.1 (Lemma 3.5 in [41]) . Let G be a ( p, λ ) -bijumbled graph on n vertices with λ ≤ εp n for some ε ∈ ( , / ] . Suppose we are given disjoint ℓ -chains D ′ , . . . , D ′ t ⊆ G for some t, ℓ ∈ N suchthat ℓ is even, t ≥ and λ / p ≤ t ( ℓ + ) ≤ n / . Then for any subset W ⊆ V ( G ) ∖ ⋃ i ∈ [ t ] V ( D ′ i ) of size ∣ W ∣ ≥ n / there exist disjoint ( ℓ / ) -chains D , . . . , D t ⊆ G [ W ] with the following property:for every L ⊆ [ t ] there exists L ′ ⊆ [ t ] such that G [⋃ i ∈ L V ( D i ) ∪ ⋃ i ∈ L ′ V ( D ′ i )] contains a triangle-factor, which can be found in polynomial time.Proof. We set ε ∶ = /
16. Note that a similar calculation as in the proof of Fact 3.3 ( i ) shows thatthe number of vertices which have at most εptℓ neighbours in a set of size at least t ( ℓ + )/ ≥ λ / p is at most λ t ( ℓ + )/ ( / − ε ) p t ( ℓ + ) ≤ λ ( / − ε ) < λ / . (5)Given ℓ -chains D ′ , . . . , D ′ t , we partition them arbitrarily into four groups of almost equal sizes, D , . . . , D . Note that for D and D , since each of them contains at least t ( ℓ + )/ G that have degree less than εptℓ ≤ εpn /
24 to either oftheir removable vertices is at most λ . Now we greedily pick 2 t ( ℓ / ) -chains D , . . . , D t in W butavoiding these bad vertices by [41, Lemma 3.2]. It remains to verify the ‘absorption’ property. Fixany subset L ⊆ [ t ] of ( ℓ / ) -chains D i , i ∈ L . We first greedily find triangles traversing ( ℓ / ) -chains and thus obtain triangle-factors on them) until t / t / t /
24. Because ( t / )( ℓ / + ) > t ( ℓ + )/ > λ / p , it follows from (2) (for a proof seefor example [41, Lemma 2.4]), we find a triangle with one vertex from each group. This triangletraverses the three chains containing it and thus there is a triangle-factor covering these threechains. So we can reduce the number of chains by 3.We will match the remaining t / ( ℓ / ) -chains with the ℓ -chains. We start with using ℓ -chains in D , D and recursively find triangles traversing one ( ℓ / ) -chain and one ℓ -chain in D , one ℓ -chainin D . That is, as long as there exists a vertex v in one of the ‘unmatched’ chains that sends morethan εptℓ edges to the unused removable vertices in both D and D , then we pick an edge (whoseexistence is asserted by (2)) from these neighbourhoods, namely, a triangle containing v . Note thatwhen we stop, the vertices remaining unmatched have degree at most εptℓ to the unused removablevertices of either in D or D . Note that there are still roughly half of the chains in D and D left, which contain at least ( ℓ + ) ⋅ t / D and D . Thus, by (5) thereare at most λ vertices that send low degree to either of them, namely, at most 2 λ / ℓ ( ℓ / ) -chainsare left unmatched. Now we can proceed to match the chains greedily by D and D . This ispossible because each time we match a chain, we consume ℓ + D and D , respectively, and so in total this will consume at most ( ℓ + )( λ / ℓ ) = λ ( + / ℓ ) removablevertices, which is much less than εptℓ .For the running time, note that we used [41, Lemma 3.2] in the proof, but the desired chainscan be constructed by depth-first search, which can be done in polynomial time. We also used [41,Lemma 2.4] to claim the existence of a triangle, but we can then find this triangle by brute-forcesearching the neighbourhood of a vertex, in time O ( p n ) . Finally, it takes time O ( n ) to decidewhich v to use and O ( p n ) to find the triangle containing v . Thus, the greedy process can be donein time O ( n ) . (cid:3) Concluding remarks
In this paper we answered the question of Nenadov [41] by providing a deterministic polynomialtime algorithm, which finds any given 2-factor in a ( p, εp n / log n ) -bijumbled graph on n verticesof minimum degree δpn (for any fixed δ > p > ε = ε ( δ ) >
0. This is optimal up to the O ( log n ) -factor. It also follows from the proof that the strongestcondition hinges on the fact that a triangle might be present in a 2-factor (see Theorem 2.1).Indeed, it follows from the proof of Theorem 2.2 that for a 2-factor of girth at least 4, a weakercondition suffices. The celebrated construction, due to Alon [9], of triangle-free pseudorandomgraphs has been extended by Alon and Kahale [12] to graphs without odd cycles of length 2 ℓ + ( n, Θ ( n /( ℓ + ) ) , Θ ( n /( ℓ + ) )) -graphs of odd girth at least 2 ℓ +
3. It is provedin [34, Proposition 4.12] that an ( n, d, λ ) -graph with λ ℓ − ≪ d ℓ / n contains a copy of C ℓ + . Since λ = Ω (√ d ) for, say d ≤ n /
2, we have the lower bound on d = Ω ( n /( ℓ + ) ) . As for even cycles, atheorem of Bondy and Simonovits [14], which doesn’t require any bound on λ , states that d ≫ n / ℓ already implies the existence of C ℓ . It is thus a natural avenue to further investigate the (almost) ptimal conditions of when a ( p, λ ) -bijumbled graph contains a given 2-factor of girth at least ℓ . When ℓ = n , the best condition for ( n, d, λ ) -graphs is provided by the result of Krivelevich andSudakov [33] which gives λ ≤ d ( log log n ) /( n log log log n ) , while another conjecture of theseauthors [33] states that λ ≤ cd should already be sufficient for some absolute c >
0. This conjecturewould follow from the famous toughness conjecture of Chv´atal [16], as shown by Alon [10].
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E-mail address : jie [email protected] Instituto de Matem´atica e Estat´ıstica, Universidade de S˜ao Paulo, Rua do Mat˜ao 1010, 05508-090S˜ao Paulo, Brazil
E-mail address : [email protected] Institut 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