aa r X i v : . [ m a t h . C O ] J a n A DEGREE SEQUENCE HAJNAL–SZEMER´EDI THEOREM
ANDREW TREGLOWN
Abstract.
We say that a graph G has a perfect H -packing if there exists a set of vertex-disjointcopies of H which cover all the vertices in G . The seminal Hajnal–Szemer´edi theorem [12] char-acterises the minimum degree that ensures a graph G contains a perfect K r -packing. Balogh,Kostochka and Treglown [4] proposed a degree sequence version of the Hajnal–Szemer´edi theoremwhich, if true, gives a strengthening of the Hajnal–Szemer´edi theorem. In this paper we prove thisconjecture asymptotically. Another fundamental result in the area is the Alon–Yuster theorem [3]which gives a minimum degree condition that ensures a graph contains a perfect H -packing foran arbitrary graph H . We give a wide-reaching generalisation of this result by answering anotherconjecture of Balogh, Kostochka and Treglown [4] on the degree sequence of a graph that forces aperfect H -packing. We also prove a degree sequence result concerning perfect transitive tournamentpackings in directed graphs. The proofs blend together the regularity and absorbing methods. Introduction
Perfect clique packings. A perfect matching in a graph G is a collection of vertex-disjointedges that together cover all the vertices of G . A theorem of Tutte [30] characterises those graphsthat contain a perfect matching. Moreover, the decision problem of whether a graph contains aperfect matching is polynomial time solvable [10].A natural generalisation of the notion of a perfect matching is a so-called perfect packing: Giventwo graphs H and G , an H -packing in G is a collection of vertex-disjoint copies of H in G . An H -packing is called perfect if it covers all the vertices of G . Perfect H -packings are also referredto as H -factors or perfect H -tilings . Hell and Kirkpatrick [13] showed that the decision problemof whether a graph G has a perfect H -packing is NP-complete precisely when H has a componentconsisting of at least 3 vertices. Therefore, for such graphs H , it is unlikely that there is a completecharacterisation of those graphs containing a perfect H -packing.The following classical result of Hajnal and Szemer´edi [12] characterises the minimum degreethat ensures a graph contains a perfect K r -packing. Theorem 1.1 (Hajnal and Szemer´edi [12]) . Every graph G whose order n is divisible by r andwhose minimum degree satisfies δ ( G ) ≥ (1 − /r ) n contains a perfect K r -packing. It is easy to see that the minimum degree condition here cannot be lowered. Earlier, Corr´adi andHajnal [7] proved Theorem 1.1 in the case when r = 3. More recently, Kierstead and Kostochka [16]gave a short proof of the Hajnal–Szemer´edi theorem.Over the last twenty years the Hajnal–Szemer´edi theorem has been generalised in a number ofdirections. For example, Kierstead and Kostochka [15] proved an Ore-type analogue of the Hajnal–Szemer´edi theorem: If G is a graph whose order n is divisible by r , then G contains a perfect K r -packing provided that d ( x ) + d ( y ) ≥ − /r ) n − x = y ∈ V ( G ). Togetherwith Balogh and Kostochka [4], the author characterised the edge density required to ensure agraph of given minimum degree contains a perfect K r -packing. In [4] the following conjecture wasalso proposed. Date : September 22, 2018. onjecture 1.2 (Balogh, Kostochka and Treglown [4]) . Let n, r ∈ N such that r divides n . Supposethat G is a graph on n vertices with degree sequence d ≤ · · · ≤ d n such that: ( α ) d i ≥ ( r − n/r + i for all i < n/r ; ( β ) d n/r +1 ≥ ( r − n/r .Then G contains a perfect K r -packing. Note that Conjecture 1.2, if true, is much stronger than the Hajnal–Szemer´edi theorem since thedegree condition allows for n/r vertices to have degree less than ( r − n/r . Moreover, the degreesequence condition in Conjecture 1.2 is best-possible. Indeed, examples in Section 4 in [4] showthat one cannot replace ( α ) with d i ≥ ( r − n/r + i − single value of i < n/r and ( β )cannot be replaced with d n/r +1 ≥ ( r − n/r −
1. Chv´atal’s theorem on Hamilton cycles [6] impliesConjecture 1.2 in the case when r = 2. In [4] it was shown that Conjecture 1.2 is true under theadditional assumption that no vertex x ∈ V ( G ) of degree less than ( r − n/r lies in a copy of K r +1 .In this paper we prove the following asymptotic version of Conjecture 1.2. Theorem 1.3.
Let γ > and r ∈ N . Then there exists an integer n = n ( γ, r ) such that thefollowing holds. Suppose that G is a graph on n ≥ n vertices where r divides n and with degreesequence d ≤ · · · ≤ d n such that: • d i ≥ ( r − n/r + i + γn for all i < n/r .Then G contains a perfect K r -packing. Keevash and Knox [14] also have announced a proof of Theorem 1.3 in the case when r = 3.Conjecture 1.2 considers a ‘P´osa-type’ degree sequence condition: P´osa’s theorem [25] states thata graph G on n ≥ d ≤ · · · ≤ d n satisfies d i ≥ i + 1 for all i < ( n − / d ⌈ n/ ⌉ ≥ ⌈ n/ ⌉ when n is odd. The aforementionedresult of Chv´atal [6] generalises P´osa’s theorem by characterising those degree sequences which forceHamiltonicity. It would be interesting to establish an analogous result for perfect K r -packings.After this paper was submitted, the author and Staden [27] have used Theorem 1.3 to prove adegree sequence version of P´osa’s conjecture on the square of a Hamilton cycle.1.2. General graph packings.
The following result of Alon and Yuster [3] initiated the topic of‘generalising’ the Hajnal–Szemer´edi theorem to perfect H -packings for an arbitrary graph H . Theorem 1.4 (Alon and Yuster [3]) . Suppose that γ > and H is a graph with χ ( H ) = r . Thenthere exists an integer n = n ( γ, H ) such that the following holds. If G is a graph whose order n ≥ n is divisible by | H | and δ ( G ) ≥ (1 − /r + γ ) n then G contains a perfect H -packing. Koml´os, S´ark¨ozy and Szemer´edi [19] proved that the term γn in Theorem 1.4 can be replacedwith a constant depending only on H . For some graphs H the minimum degree condition inTheorem 1.4 is best-possible up to the term γn . However, for other graphs H the minimum degreecondition here can be substantially reduced. Indeed, K¨uhn and Osthus [22] characterised, up toan additive constant, the minimum degree which ensures a graph G contains a perfect H -packingfor an arbitrary graph H . This characterisation involves the so-called critical chromatic number of H (see [22] for more details). K¨uhn, Osthus and Treglown [23] characterised, asymptotically, theOre-type degree condition that guarantees a graph G contains a perfect H -packing for an arbitrarygraph H .In this paper we prove the following result, thereby answering another conjecture of Balogh,Kostochka and Treglown [4]. heorem 1.5. Suppose that γ > and H is a graph with χ ( H ) = r . Then there exists an integer n = n ( γ, H ) such that the following holds. If G is a graph whose order n ≥ n is divisible by | H | ,and whose degree sequence d ≤ · · · ≤ d n satisfies • d i ≥ ( r − n/r + i + γn for all i < n/r ,then G contains a perfect H -packing. Note that Theorem 1.5 is a strong generalisation of the Alon–Yuster theorem. Further, Theo-rem 1.3 is a special case of Theorem 1.5 (so we do not prove the former directly). Previously, theauthor and Knox [17] proved Theorem 1.5 in the case when r = 2. (In fact, they proved a muchmore general result concerning embedding spanning bipartite graphs of small bandwidth.)At first sight one may ask whether the term γn in Theorem 1.5 can be replaced by a constantdependent on H . However, in Section 3 we show that for many graphs H this is not the case. (Infact, for many graphs H one cannot replace γn with o ( √ n ).)The proof of Theorem 1.5 splits into two main tasks: We use the regularity method to find an‘almost’ perfect H -packing in G and find a so-called ‘absorbing set’ that can be used to cover theremaining vertices with disjoint copies of H (see Section 2.2 for the precise definition of such a set).1.3. Perfect packings in directed graphs.
Recently, there has been a focus on generalising theHajnal–Szemer´edi theorem to the directed graph setting (see for example [5, 8, 9, 29]). Here wegive a degree sequence condition that forces a digraph to contain a perfect packing of transitivetournaments of a given size. Before we can state this result we require some notation and definitions.Given a digraph G on n vertices and x ∈ V ( G ), let d + G ( x ) and d − G ( x ) denote the out- and indegreeof x in G . We call d ∗ G ( x ) := max { d + G ( x ) , d − G ( x ) } the dominant degree of x in G . The dominantdegree sequence of G is the sequence d ∗ ≤ · · · ≤ d ∗ n of the dominant degrees of the vertices of G ordered from smallest to largest. Let T r denote the transitive tournament on r vertices. Theorem 1.6.
Let r ∈ N and γ > . Then there exists an integer n = n ( γ, r ) such that thefollowing holds. Let G be a digraph on n ≥ n vertices where r divides n . If G has dominant degreesequence d ∗ ≤ d ∗ ≤ · · · ≤ d ∗ n where d ∗ i ≥ ( r − n/r + i + γn for all i < n/r then G contains a perfect T r -packing. Notice that Theorem 1.6 implies Theorem 1.3. (In particular, since Theorem 1.3 is best-possibleup to the term γn , so is Theorem 1.6.) Further, Theorem 1.6 complements a number of existingresults on perfect T r -packings: Let n be divisible by r . Czygrinow, DeBiasio, Kierstead and Molla [8]showed that a digraph on n vertices contains a perfect T r -packing provided that G has minimumoutdegree at least ( r − n/r or minimum total degree at least 2( r − n/r −
1. Amongst otherresults, in [29] it was shown that if G has minimum dominant degree at least (1 − /r + o (1)) n then G contains a perfect T r -packing. (So Theorem 1.6 generalises this result.) It would be interestingto establish whether the term γn in Theorem 1.6 can be replaced by an additive constant.1.4. Organisation of the paper.
The paper is organised as follows. In the next section weintroduce some notation and definitions. In Section 3 we give an extremal example which showsthat, in general, one cannot replace the term γn in Theorem 1.5 with o ( √ n ). In Section 4 we statethe main auxiliary results of the paper and derive Theorems 1.5 and 1.6 from them. Szemer´edi’sRegularity lemma and related tools are introduced in Section 5. We then prove an ‘almost’ perfectpacking result for Theorem 1.6 in Section 6. In Section 7 we give an overview of our absorbingmethod and then apply it in Section 8 to prove absorbing results for Theorems 1.5 and 1.6. . Notation and preliminaries
Definitions and notation.
Let G be a (di)graph. We write V ( G ) for the vertex set of G , E ( G ) for the edge set of G and define | G | := | V ( G ) | and e ( G ) := | E ( G ) | . Given a subset X ⊆ V ( G ),we write G [ X ] for the sub(di)graph of G induced by X . We write G \ X for the sub(di)graph of G induced by V ( G ) \ X . Given a set X ⊆ V ( G ) and a (di)graph H on | X | vertices we say that X spans a copy of H in G if G [ X ] contains a copy of H . In particular, this does not necessarily meanthat X induces a copy of H in G .Suppose that G is a graph. The degree of a vertex x ∈ V ( G ) is denoted by d G ( x ) and itsneighbourhood by N G ( x ). Given a vertex x ∈ V ( G ) and a set Y ⊆ V ( G ) we write d G ( x, Y ) todenote the number of edges xy where y ∈ Y . Given disjoint vertex classes A, B ⊆ V ( G ), e G ( A, B )denotes the number of edges in G with one endpoint in A and the other in B .Given two vertices x and y of a digraph G , we write xy for the edge directed from x to y . Wedenote by N + G ( x ) and N − G ( x ) the out- and the inneighbourhood of x and by d + G ( x ) and d − G ( x ) its out-and indegree. We will write N + ( x ) for example, if this is unambiguous. Given a vertex x ∈ V ( G )and a set Y ⊆ V ( G ) we write d + G ( x, Y ) to denote the number of edges in G with startpoint x and endpoint in Y . We define d − G ( x, Y ) analogously. Recall that d ∗ G ( x ) := max { d + G ( x ) , d − G ( x ) } isthe dominant degree of x in G . If d ∗ G ( x ) = d + G ( x ) = d − G ( x ) then we implicitly consider d ∗ G ( x ) tocount the number of edges sent out from x in G . In particular, if d ∗ G ( x ) = d + G ( x ) then we define d ∗ G ( x, Y ) := d + G ( x, Y ). Otherwise, we set d ∗ G ( x, Y ) := d − G ( x, Y ).Let H be a collection of (di)graphs and G a (di)graph. An H -packing in G is a collection ofvertex-disjoint copies of elements from H in G . Given an H -packing M , we write V ( M ) for theset of vertices covered by M .Recall that T r denotes the transitive tournament of r vertices. Given 1 ≤ i ≤ r , we say a vertex x ∈ V ( T r ) is the i th vertex of T r if x has indegree i − r − i in T r .Given a (di)graph G we let G ( t ) denote the (di)graph obtained from G by replacing each vertex x ∈ V ( G ) with a set V x of t vertices so that, for all x, y ∈ V ( G ): • If xy ∈ E ( G ) then there are all possible edges in G ( t ) with startpoint in V x and endpointin V y ; • If xy E ( G ) then there are no edges in G ( t ) with startpoint in V x and endpoint in V y .We set T tr := T r ( t ) and K tr := K r ( t ).Throughout the paper, we write 0 < α ≪ β ≪ γ to mean that we can choose the constants α, β, γ from right to left. More precisely, there are increasing functions f and g such that, given γ ,whenever we choose some β ≤ f ( γ ) and α ≤ g ( β ), all calculations needed in our proof are valid.Hierarchies of other lengths are defined in the obvious way.2.2. Absorbing sets.
Let H be a (di)graph. Given a (di)graph G , a set S ⊆ V ( G ) is called an H -absorbing set for Q ⊆ V ( G ), if both G [ S ] and G [ S ∪ Q ] contain perfect H -packings. In thiscase we say that Q is H -absorbed by S . Sometimes we will simply refer to a set S ⊆ V ( G ) as an H -absorbing set if there exists a set Q ⊆ V ( G ) that is H -absorbed by S .3. An extremal example for Theorem 1.5
Let K t ,...,t r denote the complete r -partite graph with vertex classes of size t , . . . , t r . The fol-lowing result shows that for H = K t ,...,t r with t i ≥ ≤ i ≤ r ) we cannot replace the term γn in Theorem 1.5 with √ n/ r . The construction given is a generalised version of an extremalgraph from the arXiv version of [4]. Proposition 3.1.
Let r ≥ and H := K t ,...,t r with t i ≥ (for all ≤ i ≤ r ). Let n ∈ N besufficiently large so that √ n is an integer that is divisible by r | H | . Set C := √ n/ r . Then there Sfrag replacements V V V Figure 1.
An example of a graph G from Proposition 3.1 in the case when r = 3. exists a graph G on n vertices whose degree sequence d ≤ · · · ≤ d n satisfies d i ≥ ( r − n/r + i + C for all ≤ i ≤ nr but such that G does not contain a perfect H -packing. Proof.
Let G denote the graph on n vertices consisting of r vertex classes V , . . . , V r with | V | = 1, | V | = n/r + 1 + Cr , | V | = 2 n/r − − C and | V i | = n/r − C if 4 ≤ i ≤ r and which contains thefollowing edges: • All possible edges with an endpoint in V and the other endpoint in V ( G ) \ V . (In particular, G [ V ] is complete.); • All edges with an endpoint in V and the other endpoint in V ( G ) \ V ; • All edges with an endpoint in V i and the other endpoint in V ( G ) \ V i for 4 ≤ i ≤ r ; • There are √ n/ V , each of size ⌊ | V | / √ n ⌋ , ⌈ | V | / √ n ⌉ , which coverall of V (see Figure 1).In particular, note that the vertex v ∈ V sends all possible edges to V ( G ) \ V but no edges to V .Let d ≤ · · · ≤ d n denote the degree sequence of G . Notice that every vertex in V i for 3 ≤ i ≤ r − r − n/r + C . Note that ⌊ | V | / √ n ⌋ ≥ √ n/r = 6 Cr . Thus, there are √ n/ V of degree at least( r − n/r − − Cr + (6 Cr − ≥ ( r − n/r + C. The remaining n/r + 1 + Cr − √ n/ ≤ n/r − √ n/ − V have degree at least( r − n/r − Cr ≥ ( r − n/r − √ n/ C. Since d G ( v ) ≥ ( r − n/r + 1 + C for the vertex v ∈ V we have that d i ≥ ( r − n/r + i + C forall 1 ≤ i ≤ n/r .Suppose that v ∈ V lies in a copy H ′ of H in G . Then by construction of G , two of the vertexclasses U , U of H ′ must lie entirely in V . By definition of H , H ′ [ U ∪ U ] contains a path oflength 3. However, G [ V ] does not contain a path of length 3, a contradiction. Thus, v does not liein a copy of H and so G does not contain a perfect H -packing. (cid:3) For many graphs H , K¨uhn and Osthus [22] showed that a minimum degree condition substantiallysmaller than that in the Alon–Yuster theorem ensures a graph G contains a perfect H -packing.For example, if H = K t,t − ,...,t − for any t ≥ H , not χ ( H ), is the parameter that governs the minimum degree threshold that forcesa perfect H -packing. Thus, it is interesting that for such graphs H the degree sequence conditionin Theorem 1.5 is ‘best-possible’ up to the term γn . This raises the following natural question. uestion 3.2. Are there graphs H with χ ( H ) = r for which the degree sequence condition inTheorem 1.5 is ‘far’ from tight? Deriving Theorems 1.5 and 1.6
The auxiliary results for Theorem 1.6.
The proof of Theorem 1.6 splits into two maintasks. Firstly, we construct a ‘small’ T r -absorbing set M ⊆ V ( G ) with the property that both G [ M ]and G [ M ∪ Q ] contain perfect T r -packings for any ‘very small’ set Q ⊆ V ( G ) where | Q | ∈ r N . Oursecond aim is to find an ‘almost’ perfect T r -packing in G ′ := G \ M . Together this ensures that G contains a perfect T r -packing. Indeed, suppose that G ′ contains a T r -packing M covering all buta very small set of vertices Q . Then by definition of M , G [ M ∪ Q ] contains a perfect T r -packing M . Thus, M ∪ M is a perfect T r -packing in G , as desired.The following result guarantees our desired absorbing set. Theorem 4.1.
Let < /n ≪ ν ≪ η ≪ /r where n, r ∈ N and r ≥ . Suppose that G is adigraph on n vertices with dominant degree sequence d ∗ ≤ d ∗ ≤ · · · ≤ d ∗ n where d ∗ i ≥ ( r − n/r + i + ηn for all i < n/r. Then V ( G ) contains a set M so that | M | ≤ νn and M is a T r -absorbing set for any W ⊆ V ( G ) \ M such that | W | ∈ r N and | W | ≤ ν n . We prove Theorem 4.1 in Section 8.1. In order to obtain our desired absorbing set M , we firstshow that G contains many ‘connecting structures’ of a certain type. Indeed, we show that thereis a constant t ∈ N such that given any x, y ∈ V ( G ) there are ‘many’ t -sets X ⊆ V ( G ) so that both G [ X ∪ { x } ] and G [ X ∪ { y } ] contain perfect T r -packings. A result of Lo and Markstr¨om [24] thenimmediately implies the existence of the absorbing set M . A detailed exposition of our absorbingmethod is given in Section 7.The next result is used to obtain the almost perfect T r -packing in G ′ . Theorem 4.2.
Let n, r, h ∈ N where r ≥ and η > such that < /n ≪ η ≪ /r, /h . Let G bea digraph on n vertices with dominant degree sequence d ∗ ≤ d ∗ ≤ · · · ≤ d ∗ n where d ∗ i ≥ ( r − n/r + i + ηn for all i < n/r. Then G contains a T hr -packing covering all but at most ηn vertices. Note that Theorem 4.2 concerns T hr -packings rather than simply T r -packings. (Recall that T hr := T r ( h ).) In Section 6.3 we use the regularity method to prove Theorem 4.2. We give anoutline of the proof in Section 6.1.We now deduce Theorem 1.6 from Theorems 4.1 and 4.2. Proof of Theorem 1.6.
Define constants η ′ , ν, η and n ∈ N such that 0 < /n ≪ η ′ ≪ ν ≪ η ≪ γ, /r. Let G be a digraph on n ≥ n vertices as in the statement of the theorem. Apply Theorem 4.1to G with parameters ν, η to obtain a set M so that | M | ≤ νn and M is a T r -absorbing set for any W ⊆ V ( G ) \ M such that | W | ∈ r N and | W | ≤ ν n .Set G ′ := G \ M and n ′ := | G ′ | . It is easy to see that G ′ has dominant degree sequence d ∗ G ′ , ≤ d ∗ G ′ , ≤ · · · ≤ d ∗ G ′ ,n ′ where d ∗ G ′ ,i ≥ ( r − n ′ /r + i + η ′ n ′ for all i < n ′ /r. Apply Theorem 4.2 to G ′ with η ′ playing the role of η . Then G ′ contains a T r -packing M coveringall but a set W of at most η ′ n ′ ≤ ν n vertices. Since n is divisible by r , | W | is divisible by r .Hence, by definition of M , G [ M ∪ W ] contains a perfect T r -packing M . Then M ∪ M is aperfect T r -packing in G , as desired. (cid:3) .2. The auxiliary results for Theorem 1.5.
The proof of Theorem 1.5 follows the same generalstrategy as the proof of Theorem 1.6. The next result guarantees our desired absorbing set.
Theorem 4.3.
Let n, r, h ∈ N where r ≥ and ν, η > such that < /n ≪ ν ≪ η ≪ /r, /h .Suppose that H is a graph on h vertices with χ ( H ) = r . Let G be a graph on n vertices with degreesequence d ≤ d ≤ · · · ≤ d n where d i ≥ ( r − n/r + i + ηn for all i < n/r. Then V ( G ) contains a set M so that | M | ≤ νn and M is an H -absorbing set for any W ⊆ V ( G ) \ M such that | W | ∈ h N and | W | ≤ ν n . Theorem 4.3 is proved in Section 8.2. The next result is a simple consequence of Theorem 4.2.
Theorem 4.4.
Let n, r, h ∈ N where r ≥ and η > such that < /n ≪ η ≪ /r, /h . Supposethat H is a graph on h vertices with χ ( H ) = r . Let G be a graph on n vertices with degree sequence d ≤ d ≤ · · · ≤ d n where d i ≥ ( r − n/r + i + ηn for all i < n/r. Then G contains an H -packing covering all but at most ηn vertices. Proof.
Let G ′ denote the digraph obtained from G by replacing each edge xy ∈ E ( G ) with directededges xy, yx . So G ′ has dominant degree sequence d ∗ ≤ d ∗ ≤ · · · ≤ d ∗ n where d ∗ i ≥ ( r − n/r + i + ηn for all i < n/r. Thus, Theorem 4.2 implies that G ′ contains a T hr -packing M covering all but at most ηn vertices.By construction of G ′ , M corresponds to a K hr -packing in G . Notice that K hr contains a perfect H -packing. Therefore, G contains an H -packing covering all but at most ηn vertices. (cid:3) Similarly to the proof of Theorem 1.6, we now deduce Theorem 1.5 from Theorems 4.3 and 4.4.
Proof of Theorem 1.5.
Define constants η ′ , ν, η and n ∈ N such that 0 < /n ≪ η ′ ≪ ν ≪ η ≪ γ, /r, / | H | . Set h := | H | . Let G be a graph on n ≥ n vertices as in the statement of thetheorem. Apply Theorem 4.3 to G with parameters ν, η to obtain a set M so that | M | ≤ νn and M is an H -absorbing set for any W ⊆ V ( G ) \ M such that | W | ∈ h N and | W | ≤ ν n .Set G ′ := G \ M and n ′ := | G ′ | . It is easy to see that G ′ has degree sequence d G ′ , ≤ d G ′ , ≤· · · ≤ d G ′ ,n ′ where d G ′ ,i ≥ ( r − n ′ /r + i + η ′ n ′ for all i < n ′ /r. Apply Theorem 4.4 to G ′ with η ′ playing the role of η . Then G ′ contains an H -packing M coveringall but a set W of at most η ′ n ′ ≤ ν n vertices. Since n is divisible by h , | W | is divisible by h .Hence, by definition of M , G [ M ∪ W ] contains a perfect H -packing M . Then M ∪ M is aperfect H -packing in G , as desired. (cid:3) The Regularity lemma and related tools
In the proofs of Theorems 4.2 and 4.3 we will apply Szemer´edi’s Regularity lemma [28]. Beforewe state it we need some more definitions. The density of a bipartite graph G = ( A, B ) with vertexclasses A and B is defined to be d G ( A, B ) := e G ( A, B ) | A || B | . We will write d ( A, B ) if this is unambiguous. Given any ε > G is ε -regular if for all X ⊆ A and Y ⊆ B with | X | > ε | A | and | Y | > ε | B | we have that | d ( X, Y ) − d ( A, B ) | < ε .Given disjoint vertex sets A and B in a graph G , we write ( A, B ) G for the induced bipartitesubgraph of G whose vertex classes are A and B . If G is a digraph, we write ( A, B ) G for the riented bipartite subgraph of G whose vertex classes are A and B and whose edges are all theedges from A to B in G . We say ( A, B ) G is ε -regular and has density d if the underlying bipartitegraph of ( A, B ) G is ε -regular and has density d . (Note that the ordering of the pair ( A, B ) isimportant.)The Diregularity lemma is a variant of the Regularity lemma for digraphs due to Alon andShapira [1]. Its proof is similar to the undirected version. We will use the following degree form ofthe Regularity lemma which is stated in terms of both graphs and digraphs. It is derived from theDiregularity lemma. (See for example [31] for a proof of the directed version and [21] for a sketchof the undirected version.)
Lemma 5.1 (Degree form of the Regularity lemma) . For every ε ∈ (0 , and every integer M ′ there are integers M and n such that if G is a graph or digraph on n ≥ n vertices and d ∈ [0 , is any real number, then there is a partition of the vertices of G into V , V , . . . , V k and a spanningsub(di)graph G ′ of G such that the following holds: • M ′ ≤ k ≤ M ; • | V | ≤ εn ; • | V | = · · · = | V k | =: m ; • If G is a graph then d G ′ ( x ) > d G ( x ) − ( d + ε ) n for all vertices x ∈ V ( G ) ; • If G is a digraph then d + G ′ ( x ) > d + G ( x ) − ( d + ε ) n and d − G ′ ( x ) > d − G ( x ) − ( d + ε ) n for allvertices x ∈ V ( G ) ; • e ( G ′ [ V i ]) = 0 for all i ≥ ; • For all ≤ i, j ≤ k with i = j the pair ( V i , V j ) G ′ is ε -regular and has density either ordensity at least d . We call V , . . . , V k clusters , V the exceptional set and the vertices in V exceptional vertices . Werefer to G ′ as the pure (di)graph . The reduced (di)graph R of G with parameters ε , d and M ′ isthe (di)graph whose vertex set is V , . . . , V k and which V i V j is an edge precisely when ( V i , V j ) G ′ is ε -regular and has density at least d .Roughly speaking, the next result states that the reduced digraph R of a digraph G as inTheorem 4.2 ‘inherits’ the dominant degree sequence of G . Lemma 5.2.
Let M ′ , n , r ∈ N where r ≥ and let ε, d, η be positive constants such that /n ≪ /M ′ ≪ ε ≪ d ≪ η < . Let G be a digraph on n ≥ n vertices with dominant degree sequence d ∗ ≤ d ∗ ≤ · · · ≤ d ∗ n such that d ∗ i ≥ ( r − n/r + i + ηn for all i < n/r. (1) Let R be the reduced digraph of G with parameters ε, d and M ′ and set k := | R | . Then R hasdominant degree sequence d ∗ R, ≤ d ∗ R, ≤ · · · ≤ d ∗ R,k such that d ∗ R,i ≥ ( r − k/r + i + ηk/ for all i < k/r. (2) Proof.
Let G ′ denote the pure digraph of G and let V , . . . , V k denote the clusters of G and V theexceptional set. Set m := | V | = · · · = | V k | . We may assume that d ∗ R ( V ) ≤ d ∗ R ( V ) ≤ · · · ≤ d ∗ R ( V k ).Consider any i < k/r . Set S := S ≤ j ≤ i V j . So | S | = mi < mk/r ≤ n/r . Thus, by (1) there exists avertex x ∈ S such that d ∗ G ( x ) ≥ d ∗ mi ≥ ( r − n/r + mi + ηn . Suppose that x ∈ V j where 1 ≤ j ≤ i .Since km ≤ n , Lemma 5.1 implies that d ∗ R ( V j ) ≥ ( d ∗ G ′ ( x ) − | V | ) /m ≥ (( r − n/r + mi + ηn − ( d + 2 ε ) n ) /m ≥ ( r − k/r + i + ηk/ . Since d ∗ R,i = d ∗ R ( V i ) ≥ d ∗ R ( V j ) this proves that (2) is satisfied, as desired. (cid:3) n analogous proof yields the following version of Lemma 5.2 for graphs. Lemma 5.3.
Let M ′ , n , r ∈ N where r ≥ and let ε, d, η be positive constants such that /n ≪ /M ′ ≪ ε ≪ d ≪ η < . Let G be a graph on n ≥ n vertices with degree sequence d ≤ d ≤ · · · ≤ d n such that d i ≥ ( r − n/r + i + ηn for all i < n/r. Let R be the reduced graph of G with parameters ε, d and M ′ and set k := | R | . Then R has degreesequence d R, ≤ d R, ≤ · · · ≤ d R,k such that d R,i ≥ ( r − k/r + i + ηk/ for all i < k/r. The next well-known observation (see [18] for example) states that a large subgraph of a regularpair is also regular.
Lemma 5.4.
Let < ε < α and ε ′ := max { ε/α, ε } . Let ( A, B ) be an ε -regular pair of density d .Suppose A ′ ⊆ A and B ′ ⊆ B ′ where | A ′ | ≥ α | A | and | B ′ | ≥ α | B | . Then ( A ′ , B ′ ) is an ε ′ -regularpair with density d ′ where | d ′ − d | < ε . The following result will be applied in the proof of Theorem 4.2 to convert an almost perfect T r -packing in a reduced digraph into an almost perfect T hr -packing in the original digraph G . It is(for example) a special case of Corollary 2.3 in [2]. Lemma 5.5.
Let ε, d > and m, r, h ∈ N such that < /m ≪ ε ≪ d ≪ /r, /h . Let H be agraph obtained from K r by replacing every vertex of K r with m vertices and replacing each edge of K r with an ε -regular pair of density at least d . Then H contains a K hr -packing covering all but atmost εmr vertices. In the proof of Theorem 4.3 we will apply the following well-known counting lemma (often calledthe Key lemma) from [20].
Lemma 5.6 (Counting lemma [20]) . Suppose that < ε < d , that m, t ∈ N and that R is a graphwith V ( R ) = { v , . . . , v k } . Construct a graph G by replacing every vertex v i ∈ V ( R ) by a set V i of m vertices, and replacing the edges of R with ε -regular pairs of density at least d . For each v i ∈ V ( R ) let U i denote the set of t vertices in R ( t ) corresponding to v i . Let H be a subgraph of R ( t ) on h vertices and maximum degree ∆ ∈ N . Set δ := d − ε and ε := δ ∆ / (2 + ∆) . If ε ≤ ε and t − ≤ ε m then there are at least ( ε m ) h labelled copies of H in G so that if x ∈ V ( H ) lies in U i then x is embedded into V i in G . Almost perfect packings in digraphs
Outline of the proof of Theorem 4.2.
In Section 6.3 we prove Theorem 4.2. The idea ofthe proof is as follows: We apply the Regularity lemma (Lemma 5.1) to obtain a reduced digraph R of G . If R contains an almost perfect T r -packing then by applying Lemma 5.5 we conclude that G contains an almost perfect T hr -packing, as required. Otherwise, suppose that the largest T r -packingin R covers precisely d ≤ (1 − o (1)) | R | vertices. We then show that there is a { T r , T r +1 } -packingin R covering substantially more than d vertices (see Lemma 6.4). Since T r ( r ) and T r +1 ( r ) bothcontain perfect T r -packings, this implies that the blow-up R ( r ) of R contains a T r -packing coveringsubstantially more than dr vertices. Thus, crucially, the largest T r -packing in R ( r ) covers a higherproportion of vertices than the largest T r -packing in R . By repeating this argument, we obtaina blow-up R ′ of R that contains an almost perfect T r -packing. Using Lemma 5.5 we then showthat this implies that G contains an almost perfect T hr -packing, as desired. Arguments of a similarflavour were applied in [18, 11]. .2. Tools for the proof of Theorem 4.2.
In this section we give a number of tools that willbe used in the proof of Theorem 4.2. The following trivial observation will be used in the proof toconvert a { T r , T r +1 } -packing in a (reduced) digraph R into a T r -packing in the blow-up R ( r ) of R . Fact 6.1.
Suppose that r, t ∈ N such that r divides t . Then both T r ( t ) = T tr and T r +1 ( t ) = T tr +1 contain perfect T r -packings. The next result will be used to show that a blow-up of a reduced digraph R ‘inherits’ the dominantdegree sequence of R . Proposition 6.2.
Let n, r, t ∈ N and γ > where n > r . Suppose that H is a digraph on n verticeswith dominant degree sequence d ∗ H, ≤ · · · ≤ d ∗ H,n such that d ∗ H,i ≥ ( r − n/r + i + γn for all i < n/r. (3) Then H ′ := H ( t ) has dominant degree sequence d ∗ H ′ , ≤ · · · ≤ d ∗ H ′ ,nt such that d ∗ H ′ ,i ≥ ( r − nt/r + i + ( γn − t for all i < nt/r. Proof.
Given any 1 ≤ j ≤ nt , by definition of H ′ we have that d ∗ H ′ ,j = t × d ∗ H, ⌈ j/t ⌉ . (4)Suppose that j ≤ nt/r − t . Then ⌈ j/t ⌉ < n/r , so (3) and (4) imply that d ∗ H ′ ,j ≥ ( r − nt/r + ⌈ j/t ⌉ t + γnt ≥ ( r − nt/r + j + γnt. In particular, this implies that for any i < nt/r , d ∗ H ′ ,i ≥ ( r − nt/r + ( i − t ) + γnt = ( r − nt/r + i + ( γn − t, as desired. (cid:3) Proposition 6.3.
Let n, r ∈ N such that n ≥ r ≥ . Let G be a digraph on n vertices withdominant degree sequence d ∗ ≤ d ∗ ≤ · · · ≤ d ∗ n where d ∗⌈ n/r ⌉ ≥ (1 − /r ) n. (5) Then G contains a copy of T r . Proof.
Let r ′ ∈ N . Suppose that T ′ is a copy of T r ′ in G . Let V ( T ′ ) = { x , . . . , x r ′ } where x i playsthe role of the i th vertex of T r ′ . We say that T ′ is consistent if there exists 0 ≤ s ′ ≤ r ′ such that • d + G ( x i ) ≥ (1 − /r ) n for all i ≤ s ′ ; • d − G ( x i ) ≥ (1 − /r ) n for all i > s ′ .We call s ′ a turning point of T ′ . (Note that T ′ could have more than one turning point.)(5) implies that there is a copy of T in G that is consistent. Suppose that, for some 1 ≤ r ′ < r , we have found a consistent copy T ′ of T r ′ in G . As before, let V ( T ′ ) = { x , . . . , x r ′ } where x i plays the role of the i th vertex of T r ′ and let s ′ denote a turning point of T ′ . Set N ′ := T i ≤ s ′ N + G ( x i ) ∩ T i>s ′ N − G ( x i ) . Since T ′ is consistent with turning point s ′ and r ′ ≤ r − | N ′ | ≥ (cid:18) − r ′ r (cid:19) n ≥ nr . In particular, (5) implies there is a vertex x ∈ N ′ such that d ∗ G ( x ) ≥ (1 − /r ) n . Then V ( T ′ ) ∪ { x } spans a consistent copy of T r ′ +1 in G where x plays the role of the ( s ′ + 1)th vertex in T r ′ +1 . (Thisis true regardless of whether d + G ( x ) ≥ (1 − /r ) n or d − G ( x ) ≥ (1 − /r ) n .) This implies that G contains a (consistent) copy of T r , as desired. (cid:3) he next result is the main tool in the proof of Theorem 4.2. It will be used to convert a T r -packing in a reduced digraph R into a significantly larger { T r , T r +1 } -packing. Lemma 6.4.
Let η, γ > and n, r ≥ be integers such that < /n ≪ γ ≪ η ≪ /r . Let G be adigraph on n vertices with dominant degree sequence d ∗ ≤ d ∗ ≤ · · · ≤ d ∗ n where d ∗ i ≥ ( r − n/r + i + ηn for all i < n/r. (6) Further, suppose that the largest T r -packing in G covers precisely n ′ ≤ (1 − η ) n vertices. Then thereexists a { T r , T r +1 } -packing in G that covers at least n ′ + γn vertices. Proof.
By repeatedly applying Proposition 6.3, we have that n ′ ≥ ηn/
2. Define a bijection I : V ( G ) [ n ] where I ( x ) = i implies that d ∗ G ( x ) = d ∗ i . Let M denote a T r -packing in G coveringprecisely n ′ vertices so that X x ∈ V ( G ) \ V ( M ) I ( x )is maximised. Set n ′′ := n − n ′ and let G ′′ := G \ V ( M ). Claim 6.5.
There are at least γn vertices x ∈ V ( G ′′ ) such that d ∗ G ( x, V ( M )) ≥ ( r − n ′ /r + γn . Suppose for a contradiction that the claim is false. Let V ( G ′′ ) = { x , . . . , x n ′′ } where I ( x )
4. Wesay a big vertex x ∈ V ( G ′′ ) is bad if d ∗ G ( x, V ( G ′′ )) ≤ ( r − n ′′ /r + ηn/
2. At most γn big vertices x ∈ V ( G ′′ ) are bad. Indeed, otherwise there are at least γn vertices x ∈ V ( G ′′ ) such that d ∗ G ( x, V ( M )) ≥ ( r − n/r + 3 ηn/ − ( r − n ′′ /r − ηn/ ≥ ( r − n ′ /r + γn, a contradiction to our assumption.Consider any small vertex x i ∈ V ( G ′′ ). Then d ∗ G ( x i ) = d ∗ s i ≥ ( r − n/r + s i + ηn by (6) since s i < n/r . Suppose for a contradiction that d ∗ G ( x i , V ( G ′′ )) ≤ ( r − n ′′ /r + i + 2 γn . Then d ∗ G ( x i , V ( M )) ≥ ( r − n/r + s i + ηn − ( r − n ′′ /r − i − γn ≥ ( r − n ′ /r + ( s i − i ) + ηn/ . (7)Without loss of generality assume that d ∗ G ( x i ) = d + G ( x i ). Then (7) implies that there are at least( s i − i ) + ηn/ y ∈ V ( M ) with the property that if y lies in a copy T ′ r of T r in M then x i sends an edge to every vertex in V ( T ′ r ) \ { y } . In particular, ( V ( T ′ r ) ∪ { x i } ) \ { y } spans a copyof T r in G . Since I ( x i ′ ) < I ( x i ) for all i ′ < i , this implies that there is one such vertex y with s i = I ( x i ) < I ( y ). Let M ′ denote the T r -packing in G obtained from M by replacing the copy T ′ r of T r covering y with the copy of T r spanning ( V ( T ′ r ) ∪ { x i } ) \ { y } . Then M ′ is a T r -packing in G covering precisely n ′ vertices and with X x ∈ V ( G ) \ V ( M ) I ( x ) < X x ∈ V ( G ) \ V ( M ′ ) I ( x ) , a contradiction to the choice of M . So d ∗ G ( x i , V ( G ′′ )) ≥ ( r − n ′′ /r + i + 2 γn .In summary, d ∗ G ( x i , V ( G ′′ )) ≥ ( r − n ′′ /r + i + 2 γn for each small vertex x i ∈ V ( G ′′ ). Further,at most γn big vertices x ∈ V ( G ′′ ) are bad. By deleting the bad vertices from G ′′ we obtain asubdigraph G ∗ of G ′′ with dominant degree sequence d ∗ G ∗ , ≤ d ∗ G ∗ , ≤ · · · ≤ d ∗ G ∗ ,n ∗ where n ∗ := | G ∗ | so that d ∗ G ∗ ,i ≥ ( r − n ∗ /r + i + γn for all i < n ∗ /r. ence, Proposition 6.3 implies that G ∗ contains a copy T ∗ r of T r . Then M ∪ { T ∗ r } is a T r -packingin G covering more than n ′ vertices, a contradiction. This completes the proof of the claim.Given any x ∈ V ( G ′′ ) such that d ∗ G ( x, V ( M )) ≥ ( r − n ′ /r + γn there are at least γn copies T ′ r of T r in M so that x sends all possible edges to V ( T ′ r ) in G or receives all possible edges from V ( T ′ r ) in G . In particular, V ( T ′ r ) ∪ { x } spans a copy of T r +1 in G . Thus, Claim 6.5 implies thatthere exists a { T r , T r +1 } -packing in G that covers at least n ′ + γn vertices, as desired. (cid:3) Proof of Theorem 4.2.
Define additional constants ε, d, γ and M ′ ∈ N so that 0 < /n ≪ /M ′ ≪ ε ≪ d ≪ γ ≪ η ≪ /r, /h . Set z := ⌈ /γ ⌉ . Apply Lemma 5.1 with parameters ε, d and M ′ to G to obtain clusters V , . . . , V k , an exceptional set V and a pure digraph G ′ . Set m := | V | = · · · = | V k | . Let R be the reduced digraph of G with parameters ε, d and M ′ . Lemma 5.2implies that R has dominant degree sequence d ∗ R, ≤ d ∗ R, ≤ · · · ≤ d ∗ R,k such that d ∗ R,i ≥ ( r − k/r + i + ηk/ i < k/r. (8) Claim 6.6. R ′ := R ( r z ) contains a T r -packing covering at least (1 − η/ kr z = (1 − η/ | R ′ | vertices. If R contains a T r -packing covering at least (1 − η/ k vertices then Fact 6.1 implies that Claim 6.6holds. So suppose that the largest T r -packing in R covers precisely d ≤ (1 − η/ k vertices. Thenby Lemma 6.4, R contains a { T r , T r +1 } -packing that covers at least d + γk vertices. Thus, byFact 6.1, R ( r ) contains a T r -packing covering at least ( d + γk ) r vertices. (So at least a γ -proportionof the vertices in R ( r ) are covered.) Further, Proposition 6.2 and (8) imply that R ( r ) has dominantdegree sequence d ∗ R ( r ) , ≤ · · · ≤ d ∗ R ( r ) ,kr such that d ∗ R ( r ) ,i ≥ ( r − k + i + ηkr/ i < k. If R ( r ) contains a T r -packing covering at least (1 − η/ kr vertices then again Fact 6.1 implies thatthe claim holds. So suppose that the largest T r -packing in R ( r ) covers precisely d ′ ≤ (1 − η/ kr vertices. Recall that d ′ ≥ ( d + γk ) r . By Lemma 6.4, R ( r ) contains a { T r , T r +1 } -packing that coversat least d ′ + γkr ≥ ( d + 2 γk ) r vertices. Thus, by Fact 6.1, R ( r ) contains a T r -packing coveringat least ( d + 2 γk ) r vertices. (So at least a 2 γ -proportion of the vertices in R ( r ) are covered.)Repeating this argument at most z times we see that the claim holds.For each 1 ≤ i ≤ k , partition V i into classes V ∗ i , V i, , . . . , V i,r z where m ′ := | V i,j | = ⌊ m/r z ⌋ ≥ m/ (2 r z ) for all 1 ≤ j ≤ r z . Since mk ≥ (1 − ε ) n by Lemma 5.1, m ′ | R ′ | = (cid:4) m/r z (cid:5) kr z ≥ mk − kr z ≥ (1 − ε ) n. (9)Lemma 5.4 implies that if ( V i , V i ) G ′ is ε -regular with density at least d then ( V i ,j , V i ,j ) G ′ is2 εr z -regular with density at least d − ε ≥ d/ ≤ j , j ≤ r z ). In particular, we can labelthe vertex set of R ′ so that V ( R ′ ) = { V i,j : 1 ≤ i ≤ k, ≤ j ≤ r z } where V i ,j V i ,j ∈ E ( R ′ )implies that ( V i ,j , V i ,j ) G ′ is 2 εr z -regular with density at least d/ R ′ has a T r -packing M that contains at least (1 − η/ | R ′ | vertices. Considerany copy T ′ r of T r in M and let V ( T ′ r ) = { V i ,j , V i ,j , . . . , V i r ,j r } . Set V ′ to be the union of V i ,j , V i ,j , . . . , V i r ,j r . Note that 0 < /m ′ ≪ εr z ≪ d/ ≪ γ ≪ /r, /h . Thus, Lemma 5.5implies that G ′ [ V ′ ] contains a T hr -packing covering all but at most √ εr z m ′ r ≤ γm ′ r vertices. Byconsidering each copy of T r in M we conclude that G ′ ⊆ G contains a T hr -packing covering at least(1 − γ ) m ′ r × (1 − η/ | R ′ | /r ( ) ≥ (1 − γ )(1 − η/ − ε ) n ≥ (1 − η ) n vertices, as desired. (cid:3) . Overview of our absorbing approach
The ‘absorbing method’ was first used in [26] and has subsequently been applied to numerousembedding problems in extremal graph theory. In this section we give an overview of the absorbingmethod that we will apply to prove Theorems 4.1 and 4.3.Suppose that in some (di)graph (or hypergraph) G we wish to find a ‘small’ H -absorbing set M in G that absorbs any ‘very small’ set of vertices in G . To prove that such an H -absorbing set M exists it is sufficient to show that G has many ‘connecting structures’ of a certain type. Thatis, it suffices to show that there is a constant t ∈ N such that, given any x, y ∈ V ( G ), there are‘many’ t -sets X ⊆ V ( G ) so that both G [ X ∪ { x } ] and G [ X ∪ { y } ] contain perfect H -packings. Thefollowing lemma of Lo and Markstr¨om [24] makes this observation precise. Lemma 7.1 (Lo and Markstr¨om [24]) . Let h, t ∈ N and let γ > . Suppose that H is a hypergraphon h vertices. Then there exists an n ∈ N such that the following holds. Suppose that G is ahypergraph on n ≥ n vertices so that, for any x, y ∈ V ( G ) , there are at least γn th − ( th − -sets X ⊆ V ( G ) such that both G [ X ∪ { x } ] and G [ X ∪ { y } ] contain perfect H -packings. Then V ( G ) contains a set M so that • | M | ≤ ( γ/ h n/ ; • M is an H -absorbing set for any W ⊆ V ( G ) \ M such that | W | ∈ h N and | W | ≤ ( γ/ h hn/ .Moreover, the analogous result holds for digraphs H and G . (Lo and Markstr¨om [24] only proved Lemma 7.1 in the case of hypergraphs. However, the proofof the digraph case is identical.)In view of Lemma 7.1 it is natural to seek sufficient conditions that guarantee our desiredconnecting structures X for each x, y ∈ V ( G ). For this, we introduce the definition of an H -path below. In what follows we will be working in the (di)graph setting, although everything naturallyextends to hypergraphs.Suppose that H is a (di)graph on h vertices. An H -path P is a (di)graph with the followingproperties:(i) | P | = th + 1 for some t ∈ N ;(ii) V ( P ) = X ∪ · · · ∪ X t ∪ { y , . . . , y t +1 } where | X i | = h − ≤ i ≤ t ;(iii) P [ X i ∪ { y i } ] = P [ X i ∪ { y i +1 } ] = H for all 1 ≤ i ≤ t .We call y and y t +1 the endpoints of P and t the length of P . An example of a T -path of length 3is given in Figure 2. A truncated H -path Q is a (di)graph with the following properties:(i) | Q | = th − t ∈ N ;(ii) V ( Q ) = X ∪ · · · ∪ X t ∪ { y , . . . , y t } where | X i | = h − ≤ i ≤ t ;(iii) Q [ X i ∪{ y i } ] = Q [ X i ∪{ y i +1 } ] = H for all 2 ≤ i ≤ t − Q [ X ∪{ y } ] = Q [ X t ∪{ y t } ] = H .We call X and X t the endsets of Q and t the length of Q . Notice that given any H -path P oflength t and endpoints x and y , Q := P \ { x, y } is a truncated H -path of length t . In this case wesay that Q is the truncated H -path of P .The following simple observation follows immediately from the definition of an H -path. Fact 7.2.
Let H be a (di)graph. Let P be an H -path of length t with endpoints x, y and let P bean H -path of length t with endpoints y, z . Suppose that P \ { y } and P \ { y } are vertex-disjoint.Then P ∪ P is an H -path of length t + t with endpoints x, z . Suppose that P is an H -path in a (di)graph G with endpoints x and y . Set X := V ( P ) \ { x, y } .Then by condition (iii) in the definition of an H -path, both G [ X ∪ { x } ] and G [ X ∪ { y } ] containperfect H -packings. Thus, to show that the hypothesis of Lemma 7.1 holds it suffices to prove that,given any x, y ∈ V ( G ), there are at least γn th − ( th − X ⊆ V ( H ) so that X ∪ { x, y } spans Sfrag replacements y y y y X X X Figure 2.
An example of a T -path of length 3an H -path in G of length t with endpoints x and y . We will use this approach in the proofs ofTheorems 4.1 and 4.3.Let 0 < γ ≪ β ≪ /t, /h where t, h ∈ N . Suppose that H is a (di)graph on h vertices and G is a sufficiently large (di)graph on n vertices. Define an auxiliary graph G as follows. G hasvertex set V ( G ) and given distinct x, y ∈ V ( G ), xy ∈ E ( G ) precisely when there are at least βn h − H -paths in G of length one with endpoints x and y . Suppose that for each x, y ∈ V ( G ), there areat least βn t − paths of length t in G with endpoints x and y . Then Fact 7.2 together with a simplecalculation implies that there are at least γn th − H -paths in G of length t with endpoints x and y . Thus, ultimately to find the connecting structures required to apply Lemma 7.1 it suffices toprove that the auxiliary graph G is ‘well-connected’. Although G will not be formally required, itwill serve as a useful point of reference when describing how we construct our desired T r -paths inthe proof of Theorem 4.1.8. The absorbing results for Theorems 1.5 and 1.6
Proof of Theorem 4.1.
In this section we prove Theorem 4.1. For some fixed t ′ ∈ N , wewill show that, for any x, y ∈ V ( G ), there are ‘many’ T r -paths of length t ′ in G with endpoints x and y . Then by applying Lemma 7.1 this implies that G contains our desired absorbing set M .Let G be as in Theorem 4.1, H = T r and recall the definition of the auxiliary graph G given inSection 7 (where the constant β in the definition of G is chosen such that β ≪ η, /r ). Roughlyspeaking, the next result states that given any vertex x ∈ V ( G ), there are ‘many’ vertices y suchthat xy ∈ E ( G ) and y has ‘large’ dominant degree (relative to x ). This ‘expansion’ property willthen be exploited in Lemma 8.3 to conclude that G is ‘well-connected’, and thus G contains thedesired T r -paths. Lemma 8.1.
Let γ, α, η > and n, r ≥ be integers such that < /n ≪ γ ≪ α ≪ η ≪ /r . Let G be a digraph on n vertices with dominant degree sequence d ∗ ≤ d ∗ ≤ · · · ≤ d ∗ n where d ∗ i ≥ ( r − n/r + i + ηn for all i < n/r. (10) Suppose that x ∈ V ( G ) such that d ∗ G ( x ) ≥ ( r − n/r + j + ηn for some ≤ j < n/r − αn. (11) Then there exist at least γn r pairs ( y, X ) where y ∈ V ( G ) \{ x } , X ⊆ V ( G ) \{ x, y } and the followingconditions hold: (i) d ∗ G ( y ) ≥ ( r − n/r + j + ( η + γ ) n ; (ii) X ∪ { x } and X ∪ { y } both spans copies of T r in G . In particular, X ∪ { x, y } spans a copyof a T r -path of length one in G with endpoints x and y . Proof.
Without loss of generality assume that d + G ( x ) ≥ ( r − n/r + j + ηn . Suppose first that r = 2. Then (10) and (11) imply that there are at least αn vertices z ∈ N + G ( x ) such that d ∗ G ( z ) ≥ + ( η + α/ n . Fix such a vertex z and without loss of generality suppose that d ∗ G ( z ) = d + G ( z ).Then there are at least αn vertices y ∈ N + G ( z ) \ { x } such that d ∗ G ( y ) ≥ j + ( η + γ ) n . Set X := { z } .Then ( y, X ) satisfies (i) and (ii). Further, there are at least αn × αn > γn choices for ( y, X ), as desired.Next suppose that r ≥
3. Let r ′ ∈ N . Suppose that T ′ is a copy of T r ′ in G . Let V ( T ′ ) = { x , . . . , x r ′ } where x i plays the role of the i th vertex of T r ′ . We say that T ′ is consistent if thereexists 0 ≤ s ′ ≤ r ′ such that • d + G ( x i ) ≥ (1 − /r + 3 η/ n for all i ≤ s ′ ; • d − G ( x i ) ≥ (1 − /r + 3 η/ n for all i > s ′ .We call s ′ a turning point of T ′ . Claim 8.2.
There are at least η r − n r − sets X ′ ⊆ N + G ( x ) such that X ′ spans a consistent copy of T r − in G . By (10) and (11) there are at least ( r − n/r + j + ηn − n/r ≥ ηn vertices y ∈ N + G ( x ) such that d ∗ G ( y ) ≥ (1 − /r + 3 η/ n . In particular, { y } is a consistent copy of T . So if r = 3 the claimfollows. Thus, assume r ≥ ≤ r ′ ≤ r − T ′ is a consistent copy of T r ′ in G such that V ( T ′ ) ⊆ N + G ( x ).Let V ( T ′ ) = { y , . . . , y r ′ } where y i plays the role of the i th vertex of T r ′ and let s ′ denote a turningpoint of T ′ . Set N ′ := N + G ( x ) ∩ T i ≤ s ′ N + G ( y i ) ∩ T i>s ′ N − G ( y i ) . Since T ′ is consistent with turningpoint s ′ and r ′ ≤ r −
3, (11) implies that | N ′ | ≥ ( r − n/r + j + ηn − r ′ n/r ≥ n/r + ηn. In particular, (10) implies that there are at least ηn vertices w ∈ N ′ such that d ∗ G ( w ) ≥ (1 − /r +3 η/ n . Notice that V ( T ′ ) ∪ { w } ⊆ N + G ( x ) spans a consistent copy of T r ′ +1 in G where w plays therole of the ( s ′ + 1)th vertex in T r ′ +1 . (This is true regardless of whether d + G ( w ) ≥ (1 − /r + 3 η/ n or d − G ( w ) ≥ (1 − /r + 3 η/ n .)This argument shows that there are at least( ηn ) r − × r − ≥ η r − n r − sets X ′ ⊆ N + G ( x ) such that X ′ spans a consistent copy of T r − in G , thus proving the claim.Fix a set X ′ as in Claim 8.2. Let X ′ = { x , . . . , x r − } where x i plays the role of the i thvertex of a consistent copy T ′ r − of T r − in G . Let s denote a turning point of T ′ r − . Set N := N + G ( x ) ∩ T i ≤ s N + G ( x i ) ∩ T i>s N − G ( x i ) . Then | N | ( ) ≥ ( r − n/r + j + ηn − ( r − n/r = j + ηn. In particular, (10) and (11) imply that there are at least αn vertices z ∈ N such that d ∗ G ( z ) ≥ ( r − n/r + j + ( η + α/ n .Fix such a vertex z and let X := X ′ ∪ { z } . Note that X ∪ { x } spans a copy of T r in G . Withoutloss of generality suppose that d + G ( z ) ≥ ( r − n/r + j +( η + α/ n . Set N ∗ := N + G ( z ) ∩ T i ≤ s N + G ( x i ) ∩ T i>s N − G ( x i ) . Then | N ∗ | ≥ ( r − n/r + j + ( η + α/ n − ( r − n/r = j + ( η + α/ n . Thus, by (10)there are at least αn vertices y ∈ N ∗ \ { x } such that d ∗ G ( y ) ≥ ( r − n/r + j + ( η + γ ) n . Further, X ∪ { y } spans a copy of T r in G (this follows by the choice of z and y ). Fix such a vertex y . So (i)and (ii) hold. ote that there were at least η r − n r − choices for X ′ , αn choices for z and αn choices for y . Soin total there are at least η r − n r − × αn × αn × r − ≥ γn r choices for ( y, X ), as desired. (cid:3) The next result guarantees our desired T r -paths in G (implicitly implying that our auxiliarygraph G is ‘well-connected’). Lemma 8.3.
Let ξ, γ, α, η > and n, r ≥ be integers such that < /n ≪ ξ ≪ γ ≪ α ≪ η ≪ /r . Let G be a digraph on n vertices with dominant degree sequence d ∗ ≤ d ∗ ≤ · · · ≤ d ∗ n where d ∗ i ≥ ( r − n/r + i + ηn for all i < n/r. (12) Set t := (2 ⌊ /γr − α/γ ⌋ + 1) r − and t ′ := 2 ⌊ /γr − α/γ ⌋ + 1 . Given any distinct x, y ∈ V ( G ) there are at least ξn t t -sets A ⊆ V ( G ) \ { x, y } so that A ∪ { x, y } spans a T r -path of length t ′ in G with endpoints x and y . In particular, for each such set A , G [ A ∪ { x } ] and G [ A ∪ { y } ] containperfect T r -packings. Proof.
Define an additional constant β so that ξ ≪ β ≪ γ . Set t ′′ := ⌊ /γr − α/γ ⌋ . Claim 8.4.
Let ≤ i ≤ t ′′ . There are at least β i n ir pairs ( w, X ) where w ∈ V ( G ) \ { x, y } , X ⊆ V ( G ) \ { x, y, w } and | X | = ir − with the following properties: • d ∗ G ( w ) ≥ ( r − n/r + ( η + iγ ) n ; • X ∪{ x, w } spans a T r -path of length i in G with endpoints x and w . In particular, G [ X ∪{ x } ] and G [ X ∪ { w } ] both contain perfect T r -packings. We prove the claim inductively. By (12), d ∗ G ( x ) ≥ ( r − n/r + 1 + ηn . Thus, by Lemma 8.1,Claim 8.4 holds for i = 1. Suppose that for some 1 ≤ i < t ′′ , Claim 8.4 holds. Let ( w, X ) be as inClaim 8.4 with | X | = ir −
1. Set j := iγn < n/r − αn . Note that d ∗ G ( w ) ≥ ( r − n/r + ( η + iγ ) n = ( r − n/r + j + ηn. By Lemma 8.1 there exist at least γn r pairs ( w ′ , X ′ ) where w ′ ∈ V ( G ) \ { w } , X ′ ⊆ V ( G ) \ { w, w ′ } and the following conditions hold: • d ∗ G ( w ′ ) ≥ ( r − n/r + j + ( η + γ ) n = ( r − n/r + ( η + ( i + 1) γ ) n ; • X ′ ∪ { w, w ′ } spans a T r -path of length one in G with endpoints w and w ′ . That is, X ′ ∪ { w } and X ′ ∪ { w ′ } both spans copies of T r in G .Fix such a pair ( w ′ , X ′ ) so that X ′ ∪ { w ′ } is disjoint from X ∪ { x, y } ; There are at least γn r − r ( ir + 1) (cid:18) nr − (cid:19) ≥ γn r / w ′ , X ′ ).Set X ′′ := X ∪ { w } ∪ X ′ . Then by Fact 7.2, ( w ′ , X ′′ ) is such that w ′ ∈ V ( G ) \ { x, y } , X ′′ ⊆ V ( G ) \ { x, y, w ′ } and | X ′′ | = ( i + 1) r − • d ∗ G ( w ′ ) ≥ ( r − n/r + ( η + ( i + 1) γ ) n ; • X ′′ ∪ { x, w ′ } spans a T r -path of length i + 1 in G with endpoints x and w ′ . In particular, G [ X ′′ ∪ { x } ] and G [ X ′′ ∪ { w ′ } ] both contain perfect T r -packings.There are at least β i n ir choices for ( w, X ) and at least γn r / w ′ , X ′ ). Thus, in totalthere are at least β i n ir × γn r × i + 1) r )! ≥ β i +1 n ( i +1) r such pairs ( w ′ , X ′′ ). Thus, Claim 8.4 holds with respect to i + 1, as desired. ix a pair ( w, X ) as in Claim 8.4 with i := t ′′ . The proof of the next claim is analogous to thatof Claim 8.4 (so we omit it). Claim 8.5.
Let ≤ i ≤ t ′′ . There are at least β i n ir pairs ( z, Y ) where z ∈ V ( G ) , Y ⊆ V ( G ) \ { z } and | Y | = ir − with the following properties: • Y ∪ { z } is disjoint from X ∪ { w, x, y } ; • d ∗ G ( z ) ≥ ( r − n/r + ( η + iγ ) n ; • Y ∪ { y, z } spans a T r -path of length i in G with endpoints y and z . In particular, G [ Y ∪ { y } ] and G [ Y ∪ { z } ] both contain perfect T r -packings. Fix a pair ( z, Y ) as in Claim 8.5 with i := t ′′ . Note that d ∗ G ( w ) , d ∗ G ( z ) ≥ ( r − n/r + ( η + t ′′ γ ) n ≥ ( r − n/r + ( η − α − γ ) n ≥ ( r − n/r + 3 ηn/ . (13)Without loss of generality assume that d + G ( w ) = d ∗ G ( w ) and d + G ( z ) = d ∗ G ( z ) (the other cases followanalogously). Recall the definition of a consistent copy of T r ′ in G given in Lemma 8.1. By applying(13) and arguing in a similar way to the proof of Claim 8.2 we obtain the following claim. Claim 8.6.
Suppose that r ≥ . Then there are at least η r − n r − sets Z ′ ⊆ N + G ( w ) ∩ N + G ( z ) suchthat Z ′ spans a consistent copy of T r − in G . Suppose that r ≥
3. Fix a set Z ′ as in Claim 8.6 so that Z ′ is disjoint from X ∪ Y ∪ { x, y } ; Thereare at least η r − n r − − t ′′ r (cid:18) nr − (cid:19) ≥ η r − n r − / Z ′ . Let Z ′ = { z , . . . , z r − } where z i plays the role of the i th vertex of a consistent copy T ′ r − of T r − in G . Let s denote a turning point of T ′ r − . Set N := N + G ( w ) ∩ N + G ( z ) ∩ T i ≤ s N + G ( z i ) ∩ T i>s N − G ( z i ) . Then | N | ( ) ≥ ( r − n/r + 3 ηn/ − ( r − n/r = 3 ηn/ . Fix some z ′ ∈ N \ ( X ∪ Y ∪{ x, y } ). Notice that there are at least ηn choices for z ′ . Set Z := Z ′ ∪{ z ′ } .The choice of z ′ implies that Z ∪ { w } and Z ∪ { z } span copies of T r in G .If r = 2, then notice that there are at least ηn vertices z ′ ∈ N + G ( w ) ∩ N + G ( z ) that are not in X ∪ Y ∪ { x, y } . Fix such a z ′ and let Z := { z ′ } .For every r ≥
2, set A := X ∪ Y ∪ Z ∪ { w, z } . So | A | = (2 t ′′ + 1) r − t . By construction of A , A ∪ { x, y } spans a T r -path of length t ′ in G with endpoints x and y . If r ≥
3, then recall thatthere are at least β t ′′ n t ′′ r choices for ( w, X ), at least β t ′′ n t ′′ r choices for ( z, Y ), at least η r − n r − / Z ′ and at least ηn choices for z ′ . Overall, this implies that there are at least β t ′′ n t ′′ r × β t ′′ n t ′′ r × η r − n r − / × ηn × t ! ≥ ξn t choices for A . If r = 2, then a similar calculation shows there are at least ξn t choices for A , asrequired. (cid:3) We now deduce Theorem 4.1 from Lemmas 7.1 and 8.3.
Proof of Theorem 4.1.
Define additional constants ξ, γ, α such that ν ≪ ξ ≪ γ ≪ α ≪ η . Set t := (2 ⌊ /γr − α/γ ⌋ + 1) r − t ′ := 2 ⌊ /γr − α/γ ⌋ + 1 and define γ ′ > ν = ( γ ′ / r / G be as in the statement of the theorem. Lemma 8.3 implies that, for any x, y ∈ V ( G ), thereare at least ξn t t -sets X ⊆ V ( G ) \ { x, y } such that both G [ X ∪ { x } ] and G [ X ∪ { y } ] contain perfect T r -packings. Since ν ≪ ξ ≪ /r , and by definition of γ ′ , we have that ξn t ≥ γ ′ n t = γ ′ n t ′ r − . ApplyLemma 7.1 with T r , r, t ′ , γ ′ playing the roles of H, h, t, γ . Then V ( G ) contains a set M so that • | M | ≤ ( γ ′ / r n/ νn ; M is a T r -absorbing set for any W ⊆ V ( G ) \ M such that | W | ∈ r N and | W | ≤ ν n ≤ ( γ ′ / r rn/ (cid:3) Proof of Theorem 4.3.
In this section we prove Theorem 4.3. We will show that there issome fixed s ′ ∈ N such that, for any x, y ∈ V ( G ), there are ‘many’ K hr -paths P of length s ′ in G with endpoints x and y . Consider such a K hr -path P and set X := V ( P ) \ { x, y } . Since K hr containsa perfect H -packing (for any H such that | H | = h and χ ( H ) = r ), the definition of a K hr -pathimplies that G [ X ∪ { x } ] and G [ X ∪ { y } ] contain perfect H -packings. Then by applying Lemma 7.1this implies that G contains our desired H -absorbing set M .The following result guarantees our desired K hr -paths in the case when h = 1. It is an immediateconsequence of Lemma 8.3. Lemma 8.7.
Let ξ, γ, α, η > and n, r ≥ be integers such that < /n ≪ ξ ≪ γ ≪ α ≪ η ≪ /r . Let G be a graph on n vertices with degree sequence d ≤ d ≤ · · · ≤ d n where d i ≥ ( r − n/r + i + ηn for all i < n/r. Set t := (2 ⌊ /γr − α/γ ⌋ + 1) r − and t ′ := 2 ⌊ /γr − α/γ ⌋ + 1 . Given any distinct x, y ∈ V ( G ) there are at least ξn t t -sets A ⊆ V ( G ) \ { x, y } so that A ∪ { x, y } spans a K r -path of length t ′ in G with endpoints x and y . In particular, for each such set A , G [ A ∪ { x } ] and G [ A ∪ { y } ] containperfect K r -packings. We will apply Lemma 8.7 to a reduced graph R of G in order to conclude that R contains a K r -path between any two clusters. We then show that this K r -path in R corresponds to ‘many’ K hr -paths in G . For this, we first show that a special type of ‘blow-up’ of a K r -path is itself a K hr -path.Let h, t, r ∈ N such that r +1 , t ≥
3. Consider a K r -path P of length t . Let X , . . . , X t , y , . . . , y t +1 be as in the definition of P . In particular, P has endpoints y and y t +1 and | X i | = r − ≤ i ≤ t . We define P ∗ ( h ) to be the graph obtained from P as follows: • Replace each vertex set X i with a set of h ( r −
1) vertices X i ( h ) so that X i ( h ) induces acopy of K hr − in P ∗ ( h ) (for 1 ≤ i ≤ t ); • For 2 ≤ i ≤ t −
1, we replace y i with a set Y i of h vertices; y t is replaced by a set Y t of2 h − Y := { y } , Y t +1 := { y t +1 } ; • In P ∗ ( h ) there are all possible edges between Y i ∪ Y i +1 and X i ( h ) (for all 1 ≤ i ≤ t ).Given a truncated K r -path Q of length t , we define Q ∗ ( h ) analogously. In particular, if Q is thetruncated K r -path of P then Q ∗ ( h ) = P ∗ ( h ) \ { y , y t +1 } .Note that P ∗ ( h ) is a blow-up of P where the vertices in V ( P ) \ { y , y t , y t +1 } are replaced by h vertices, y t is replaced by 2 h − y and y t +1 are left ‘untouched’. In particular, thisimmediately implies the following fact. Fact 8.8.
Let h, t, r ∈ N such that r + 1 , t ≥ and let G and H be graphs. Suppose that P is a K r -path of length t and Q is a truncated K r -path of length t . If P ⊆ G then P ∗ ( h ) ⊆ G (2 h − .Further, if Q ⊆ H then Q ∗ ( h ) ⊆ H (2 h − . We now show that, crucially, P ∗ ( h ) is a K hr -path of length t . Lemma 8.9.
Let h, t, r ∈ N such that r + 1 , t ≥ . (a) Suppose that P is a K r -path of length t with endpoints y and y t +1 . Then P ∗ ( h ) is a K hr -path of length t with endpoints y and y t +1 . b) Suppose that Q is a truncated K r -path of length t with endsets X and X t . Then Q ∗ ( h ) isa truncated K hr -path of length t . Further, let Q ′ denote the graph obtained from Q ∗ ( h ) byadding a vertex x that is adjacent to every vertex in X ( h ) and a vertex y that is adjacentto every vertex in X t ( h ) . Then Q ′ is a K hr -path of length t with endpoints x and y . Proof.
We first prove (a). Let X , . . . , X t , y , . . . , y t +1 be as in the definition of P and X ( h ) , . . . ,X t ( h ), Y , . . . , Y t +1 be as in the definition of P ∗ ( h ). Firstly, note that | P ∗ ( h ) | = X ≤ i ≤ t | X i ( h ) | + 2 + X ≤ i ≤ t − | Y i | + | Y t | = h ( r − t + 2 + h ( t −
2) + (2 h −
1) = hrt + 1 . So P ∗ ( h ) satisfies condition (i) of the definition of a K hr -path of length t .For each 2 ≤ i ≤ t −
1, define Y ′ i − , y ′ i so that | Y ′ i | = h − Y i = Y ′ i − ∪ { y ′ i } . Define Y ′ t − , Y ′ t , y ′ t so that | Y ′ t − | = | Y ′ t | = h − Y t = Y ′ t − ∪ Y ′ t ∪ { y ′ t } . Set X ′ i := X i ( h ) ∪ Y ′ i for all1 ≤ i ≤ t and y ′ := y , y ′ t +1 := y t +1 . Thus, V ( P ∗ ( h )) = X ′ ∪ · · · ∪ X ′ t ∪ { y ′ , y ′ , . . . , y ′ t +1 } where | X ′ i | = h ( r −
1) + h − hr − ≤ i ≤ t . So P ∗ ( h ) satisfies condition (ii) of the definitionof a K hr -path of length t .Let 1 ≤ i ≤ t −
1. Recall that X i ( h ) induces a copy of K hr − in P ∗ ( h ). Since Y ′ i ⊆ Y i +1 andthere are all possible edges between Y i +1 and X i ( h ) in P ∗ ( h ), we have that X ′ i induces a copy ofthe complete r -partite graph K ∗ with r − h and one vertex class (namely Y ′ i ) of size h −
1. A similar argument implies that X ′ t induces a copy of K ∗ in P ∗ ( h ). Recall that y ′ i ∈ Y i for all 1 ≤ i ≤ t + 1. So by definition of P ∗ ( h ), y ′ i and y ′ i +1 send an edge to every vertex in X i ( h ) in P ∗ ( h ) (for all 1 ≤ i ≤ t ). Altogether, this implies that X ′ i ∪ { y ′ i } and X ′ i ∪ { y ′ i +1 } inducecopies of K hr in P ∗ ( h ). Thus, condition (iii) of the definition of a K hr -path of length t is satisfied.So (a) holds. It is easy to see that (b) follows from (a). (cid:3) We now apply Lemmas 8.7 and 8.9 to conclude that a graph G as in Theorem 4.3 contains many K hr -paths of a given length between any distinct x, y ∈ V ( G ). Lemma 8.10.
Let γ ′ , γ, α, η > and n, r, h ∈ N such that < /n ≪ γ ′ ≪ γ ≪ α ≪ η ≪ /r, /h where r ≥ . Let G be a graph on n vertices with degree sequence d ≤ d ≤ · · · ≤ d n where d i ≥ ( r − n/r + i + ηn for all i < n/r. (14) Set s ′ := 2 ⌊ /γr − α/γ ⌋ + 3 and s := hrs ′ − . Given any distinct x, y ∈ V ( G ) there are at least γ ′ n s s -sets A ⊆ V ( G ) \ { x, y } so that A ∪ { x, y } spans a K hr -path of length s ′ in G with endpoints x and y . In particular, for each such set A , G [ A ∪ { x } ] and G [ A ∪ { y } ] contain perfect K hr -packings. Proof.
Let ε, d, γ, α, η > M ′ ∈ N such that0 < /M ′ ≪ ε ≪ d ≪ γ ≪ α ≪ η ≪ /r, /h. Set t ′ := 2 ⌊ /γr − α/γ ⌋ + 1. Let M denote the integer obtained by applying Lemma 5.1 withparameters ε and M ′ . Let γ ′ > n ∈ N be sufficiently large such that0 < /n ≪ γ ′ ≪ /M ′ , /M. Suppose that G is a graph on n vertices as in the statement of the lemma. Apply Lemma 5.1with parameters ε, d and M ′ to G to obtain clusters V , . . . , V k , an exceptional set V and a puregraph G ′ . Set m := | V | = · · · = | V k | . So M ′ ≤ k ≤ M and (1 − ε ) n/M ≤ m ≤ n/M ′ . Let R denote the reduced graph of G with parameters ε, d and M ′ . Lemma 5.3 implies that R has degreesequence d R, ≤ d R, ≤ · · · ≤ d R,k such that d R,i ≥ ( r − k/r + i + ηk/ i < k/r. (15) onsider distinct x, y ∈ V ( G ). We may assume that x, y ∈ V . (Otherwise, we delete the atmost two clusters containing x or y , and move their vertices into V . The same properties of R still hold, just with slightly perturbed parameters k and η .) Let N R ( x ) denote the set of clusters V i ∈ V ( R ) such that d G ( x, V i ) ≥ αm . Set d R ( x ) := | N R ( x ) | . Define N R ( y ) and d R ( y ) analogously.Note that md R ( x ) + kαm + | V | ≥ d G ( x ) ( ) ≥ ( r − n/r + ηn. So as km ≤ n and | V | ≤ εn , d R ( x ) ≥ ( r − k/r + ηk/ . (16)Similarly, d R ( y ) ≥ ( r − k/r + ηk/ . Using (15) and (16), a simple greedy argument implies that there is a copy K x of K r − in R sothat: • V ( K x ) ⊆ N R ( x ); • V ( K x ) = { U , . . . , U r − } where d R ( U i ) ≥ ( r − k/r + ηk/ ≤ i ≤ r − d R ( U r − ) ≥ ( r − k/r + ηk/ V x ∈ V ( R ) such that V ( K x ) ∪ { V x } spans a copy of K r in R . Similarly, there is a copy K y of K r − in R and a cluster V y ∈ V ( R ) sothat: • V ( K y ) ⊆ N R ( y ); • V ( K y ) = { W , . . . , W r − } where d R ( W i ) ≥ ( r − k/r + ηk/ ≤ i ≤ r − d R ( W r − ) ≥ ( r − k/r + ηk/ • V ( K y ) ∪ { V y } is vertex-disjoint from V ( K x ) ∪ { V x } and spans a copy of K r in R .Remove the clusters in K x and K y from R and denote the resulting graph by R ′ . Set k ′ := | R ′ | .(15) implies that R ′ has degree sequence d R ′ , ≤ d R ′ , ≤ · · · ≤ d R ′ ,k ′ such that d R ′ ,i ≥ ( r − k ′ /r + i + ηk ′ / i < k ′ /r. (17)Apply Lemma 8.7 with R ′ , η/ G, η . So R ′ contains a K r -path P of length t ′ with endpoints V x and V y . (Here, we do not need to use the fact that there are ‘many’ such K r -paths.) By the choice of V x and V y , V := V ( P ) ∪ V ( K x ) ∪ V ( K y ) spans a truncated K r -path Q of length t ′ + 2 = s ′ in R with endsets X := V ( K x ) and X s ′ := V ( K y ).For each V ∈ V let V ′ denote a subset of V of size αm so that: • V ′ ⊆ N G ( x ) if V ∈ V ( K x ); • V ′ ⊆ N G ( y ) if V ∈ V ( K y ).(So for V ∈ V ( P ) the choice of V ′ is arbitrary.) Notice that we can choose such subsets V ′ since V ( K x ) ⊆ N R ( x ) and V ( K y ) ⊆ N R ( y ). Suppose that V, W ∈ V and
V W ∈ E ( R ). Then since( V, W ) G ′ is ε -regular with density at least d , Lemma 5.4 implies that ( V ′ , W ′ ) G ′ is ε / -regularwith density at least d/ R ′′ := R [ V ]. Since Q ⊆ R ′′ , Fact 8.8 implies that Q := Q ∗ ( h ) is a subgraph of R ′′ (2 h − Q is a truncated K hr -path of length s ′ . Let G ′′ denote thesubgraph of G whose vertex set consists of all vertices v such that v ∈ V ′ for some V ∈ V and so that G ′′ [ V ′ , W ′ ] = G ′ [ V ′ , W ′ ] for all V, W ∈ V . Apply Lemma 5.6 with G ′′ , R ′′ , Q , h − , ε / , d/ , αm playing the roles of G, R, H, t, ε, d, m . Note that | Q | = s and set ∆ := ∆( Q ) and ε := ( d/ − ε / ) ∆ / (2 + ∆). As ε ≤ ε and 2 h − ≤ ε αm , Lemma 5.6 implies that there are at least( ε αm ) s ≥ ( d ∆+1 α (1 − ε ) n/M ) s ≥ s ! γ ′ n s copies of Q in G ′′ ⊆ G .Consider any copy Q ′ of Q guaranteed by Lemma 5.6. Recall that Q has endsets X = V ( K x ) ⊆ N R ( x ) and X s ′ = V ( K y ) ⊆ N R ( y ). Lemma 5.6 guarantees that the vertices in the set X ( h ) ⊆ ( Q ′ ) corresponding to the r − X are embedded into sets V ′ ⊆ N G ( x ). Similarly,the vertices in the set X s ′ ( h ) ⊆ V ( Q ′ ) corresponding to the r − X s ′ are embeddedinto sets V ′ ⊆ N G ( y ). Thus, Lemma 8.9(b) implies that V ( Q ′ ) ∪ { x, y } spans a K hr -path in G oflength s ′ with endpoints x and y .Since there are at least s ! γ ′ n s such copies Q ′ of Q in G ′′ , there are at least γ ′ n s s -sets A ⊆ V ( G )such that A = V ( Q ′ ) for some such Q ′ . So indeed, there are at least γ ′ n s s -sets A ⊆ V ( G ) \ { x, y } so that A ∪ { x, y } spans a K hr -path of length s ′ in G with endpoints x and y . (cid:3) We now deduce Theorem 4.3 from Lemmas 7.1 and 8.10.
Proof of Theorem 4.3.
Define additional constants γ, α such that ν ≪ γ ≪ α ≪ η . Set s ′ := 2 ⌊ /γr − α/γ ⌋ + 3 and s := hrs ′ − γ ′ > ν = ( γ ′ / h /
4. In particular, γ ′ ≪ γ .Let G be as in the statement of the theorem. Lemma 8.10 implies that, for any x, y ∈ V ( G ),there are at least γ ′ n s s -sets X ⊆ V ( G ) \ { x, y } such that both G [ X ∪ { x } ] and G [ X ∪ { y } ] containperfect K hr -packings and thus, perfect H -packings. Apply Lemma 7.1 with rs ′ , γ ′ playing the rolesof t, γ . Then V ( G ) contains a set M so that • | M | ≤ ( γ ′ / h n/ νn ; • M is an H -absorbing set for any W ⊆ V ( G ) \ M such that | W | ∈ h N and | W | ≤ ν n ≤ ( γ ′ / h rn/ (cid:3) Acknowledgements
The author would like to thank Peter Keevash for a helpful conversation, in particular, forsuggesting the auxiliary graph G in Section 7 as a tool for presenting the absorbing method. Theauthor is also grateful to Daniela K¨uhn and Deryk Osthus for helpful comments, and to the refereesfor the careful reviews. References [1] N. Alon and A. Shapira, Testing subgraphs in directed graphs,
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