An Ore-type theorem for perfect packings in graphs
aa r X i v : . [ m a t h . C O ] J un AN ORE-TYPE THEOREM FOR PERFECT PACKINGS IN GRAPHS
DANIELA K ¨UHN, DERYK OSTHUS AND ANDREW TREGLOWN
Abstract.
We say that a graph G has a perfect H -packing (also called an H -factor)if there exists a set of disjoint copies of H in G which together cover all the verticesof G . Given a graph H , we determine, asymptotically, the Ore-type degree conditionwhich ensures that a graph G has a perfect H -packing. More precisely, let δ Ore ( H, n )be the smallest number k such that every graph G whose order n is divisible by | H | andwith d ( x ) + d ( y ) ≥ k for all non-adjacent x = y ∈ V ( G ) contains a perfect H -packing.We determine lim n →∞ δ Ore ( H, n ) /n . Introduction
Perfect packings in graphs of large minimum degree.
Given two 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 . In this case one also saysthat G contains an H -factor . H -packings are generalisations of graph matchings (whichcorrespond to the case when H is a single edge).In the case when H is an edge, Tutte’s theorem characterises those graphs whichhave a perfect H -packing. However, for other connected graphs H no characterisationis known. Furthermore, Hell and Kirkpatrick [4] showed that the decision problemwhether a graph G has a perfect H -packing is NP-complete precisely when H has acomponent consisting of at least 3 vertices. It is natural therefore to ask for simplesufficient conditions which ensure the existence of a perfect H -packing. One such resultis a theorem of Hajnal and Szemer´edi [3] which states that a graph G whose order n is divisible by r has a perfect K r -packing provided that δ ( G ) ≥ (1 − /r ) n . It is easyto see that the minimum degree condition here is best possible. So for H = K r , theparameter which governs the existence of a perfect H -packing in a graph G of largeminimum degree is χ ( H ) = r .The first two authors [13, 14] showed that for any graph H either the so-called criticalchromatic number or the chromatic number of H is the relevant parameter. Here the critical chromatic number χ cr ( H ) of a graph H is defined as χ cr ( H ) := ( χ ( H ) − | H || H | − σ ( H ) , where σ ( H ) denotes the size of the smallest possible colour class in any χ ( H )-colouringof H . Throughout the paper, we only consider graphs H which contain at least one edge(without mentioning this explicitly), so χ cr ( H ) is well defined. Note that χ ( H ) − < Date : June 2, 2009.The authors were supported by the EPSRC, grant no. EP/F008406/1. χ cr ( H ) ≤ χ ( H ) for all graphs H , and χ cr ( H ) = χ ( H ) precisely when every χ ( H )-colouring of H has colour classes of equal size. The characterisation of when χ ( H ) or χ cr ( H ) is the relevant parameter depends on the so-called highest common factor of H ,which is defined as follows.We say that a colouring of H is optimal if it uses exactly χ ( H ) =: r colours. Givenan optimal colouring c of H , let x ≤ x ≤ · · · ≤ x r denote the sizes of the colour classesof c . We write D ( c ) := { x i +1 − x i | i = 1 , . . . , r − } , and let D ( H ) denote the union of allthe sets D ( c ) taken over all optimal colourings c of H . We denote by hcf χ ( H ) the highestcommon factor of all integers in D ( H ). If D ( H ) = { } then we define hcf χ ( H ) := ∞ .We write hcf c ( H ) for the highest common factor of all the orders of components of H .For non-bipartite graphs H we say that hcf( H ) = 1 if hcf χ ( H ) = 1. If χ ( H ) = 2 thenwe say hcf( H ) = 1 if hcf c ( H ) = 1 and hcf χ ( H ) ≤
2. (See [14] for some examples.) Put χ ∗ ( H ) := ( χ cr ( H ) if hcf( H ) = 1; χ ( H ) otherwise.Also let δ ( H, n ) denote the smallest integer k such that every graph G whose order n isdivisible by | H | and with δ ( G ) ≥ k contains a perfect H -packing. Theorem 1. [14]
For every graph H there exists a constant C = C ( H ) such that (cid:18) − χ ∗ ( H ) (cid:19) n − ≤ δ ( H, n ) ≤ (cid:18) − χ ∗ ( H ) (cid:19) n + C. Theorem 1 improved previous bounds by Alon and Yuster [1], who showed that δ ( H, n ) ≤ (1 − /χ ( H )) n + o ( n ), and by Koml´os, S´ark¨ozy and Szemer´edi [11], whoreplaced the o ( n )-term by a constant depending only on H . Further related results arediscussed in the surveys [7, 8, 12, 15, 21].1.2. Ore-type degree conditions for perfect packings.
Of course, one can alsoconsider other types of degree conditions that ensure a perfect H -packing in a graph G .One natural such condition is an Ore-type degree condition requiring a lower boundon the sum of the degrees of non-adjacent vertices of G . (The name comes from Ore’stheorem [16], which states that a graph G of order n ≥ d ( x ) + d ( y ) ≥ n for all non-adjacent x = y ∈ V ( G ).)A result of Kierstead and Kostochka [6] on equitable colourings implies that a graph G whose order n is divisible by r and with d ( x )+ d ( y ) ≥ − /r ) n − x = y ∈ V ( G ) contains a perfect K r -packing. Note that this is a strengthening of theHajnal-Szemer´edi theorem. Kawarabayashi [5] asked for the Ore-type condition whichguarantees a K − -packing in a graph G covering a given number of vertices of G . (Here K − denotes the graph obtained from K by removing an edge.) Similarly it is natural toseek an Ore-type analogue of Theorem 1. This will be the main result of this paper (butwith an o ( n )-error term). Perhaps surprisingly, the Ore-type condition needed is not‘twice the minimum degree condition’. For some graphs H it depends on the so-calledcolour extension number of H , which we will define now. Roughly speaking, this is ameasure of how many extra colours we need to properly colour H if we try to build thiscolouring by extending an ( r − H . N ORE-TYPE THEOREM FOR PERFECT PACKINGS IN GRAPHS 3
More precisely, suppose that H is a graph with χ ( H ) =: r which contains a vertex x for which the subgraph H [ N ( x )] induced by the neighbourhood of x is ( r − x ∈ V ( H ), let m x denote the smallest integer for which there existsan ( r − H [ N ( x )] that can be extended to an ( r + m x )-colouring of H .The colour extension number CE ( H ) of H is defined as CE ( H ) := min { m x | x ∈ V ( H ) with χ ( H [ N ( x )]) ≤ r − } . If χ ( H [ N ( x )]) = r − x ∈ V ( H ) we define CE ( H ) := ∞ . So every bipartitegraph H without isolated vertices has CE ( H ) = ∞ . All other bipartite graphs H have CE ( H ) = 0. In general, 1 ≤ CE ( H ) < ∞ if for any optimal colouring of H and any v ∈ V ( H ), N ( v ) lies in exactly r − H , but there exists a vertex x ∈ V ( H ) such that χ ( H [ N ( x )]) ≤ r −
2. Note that in this case CE ( H ) ≤ r − H − N ( x ) with r different colours to obtain a (2 r − H .)In order to help the readers to familiarize themselves with the notion of the colourextension number we now give a number of examples. χ ( K − ) = 3 and χ ( K − [ N ( x )]) = 2for every vertex x of K − . Thus CE ( K − ) = ∞ . Next consider the graph F ⋄ obtainedfrom the complete 3-partite graph K , , by removing an edge xy of K , , and addinga new vertex z which is adjacent to x and y only. Then χ ( F ⋄ ) = 3, χ ( F ⋄ [ N ( w )]) = 2for every vertex w = z in F ⋄ and χ ( F ⋄ [ N ( z )]) = 1. Note that in any 3-colouring of F ⋄ , x and y are coloured differently. So if we 1-colour N ( z ) = { x, y } , this colouring can beextended to a 4-colouring of F ⋄ but not a 3-colouring. Thus CE ( F ⋄ ) = 1.For each k ≥ r ≥ k + 2 we now give an example of a family of graphs H ⋄ with CE ( H ⋄ ) = k and χ ( H ⋄ ) = r . Consider a complete r -partite graph whose vertexclasses V , . . . , V r have size > k . Let H ⋄ be obtained from this graph by deleting theedges of k vertex-disjoint copies K , . . . , K k of K k +1 which lie in V ∪ · · · ∪ V k +1 , andby adding a new vertex x which is adjacent to the k ( k + 1) vertices lying in thesecopies of K k +1 as well as to all the vertices in V k +2 , . . . , V r − (see Figure 1). Note that PSfrag replacements x V V V V V Figure 1.
The graph H ⋄ in the case when k = 2, r = 5 and wheneach V i has size 3. The dashed lines indicate the deleted edges. χ ( H ⋄ ) = r . Furthermore, any vertex y ∈ V ∪ · · · ∪ V r lies in a copy of K r in H ⋄ . So χ ( H ⋄ [ N ( y )]) = r −
1. However, the subgraph D := H ⋄ [ N ( x ) ∩ V ∩ · · · ∩ V k +1 ] has a k -colouring c ′ x with colour classes V ( K ) , . . . , V ( K k ) and it is easy to check that this isthe only k -colouring of D (and so in particular χ ( D ) = k ). Thus χ ( H ⋄ [ N ( x )]) = r − DANIELA K ¨UHN, DERYK OSTHUS AND ANDREW TREGLOWN and the only ( r − H ⋄ [ N ( x )] is the one which agrees with c ′ x on D andcolours each of V k +2 , . . . , V r − with a new colour. Let c x denote this colouring. Whenextending c x to a proper colouring of H ⋄ we cannot reuse the r − c x since every y ∈ V ( H ⋄ ) \ N ( x ) is adjacent to a vertex in each colour class of c x . As χ ( H ⋄ − N ( x )) = r − ( r − k −
2) = k + 2 this means that we require r + k colours in totalto extend c x to a proper colouring of H ⋄ . Thus CE ( H ⋄ ) = k .Let χ Ore ( H ) := ( χ ( H ) if hcf( H ) = 1 or CE ( H ) = ∞ ;max n χ cr ( H ) , χ ( H ) − CE ( H )+2 o otherwise.Recall that CE ( K − ) = ∞ and CE ( F ⋄ ) = 1, where F ⋄ was defined above. So χ Ore ( K − ) = χ ( K − ) = 3. Any 3-colouring of F ⋄ has one colour class of size 3 and two colour classes ofsize 2. So hcf( F ⋄ ) = 1 and thus χ Ore ( F ⋄ ) = max { χ cr ( F ⋄ ) , − / } = max { / , / } =14 / . Note that if hcf( H ) = 1 and CE ( H ) = 0 then χ Ore ( H ) = χ cr ( H ) (an odd cycle oflength at least 5 provides an example of such a graph H ). On the other hand, one canchoose the sizes of the vertex classes V i in the preceding example H ⋄ so that χ Ore ( H ⋄ )lies strictly between χ cr ( H ⋄ ) and χ ( H ⋄ ). (For instance, take k large, | V | = k + 1, | V | = 2 k and | V i | = 2 k + 1 for all i ≥
3. Then χ cr ( H ⋄ ) is close to χ ( H ⋄ ) − / H ⋄ ) = 1 and so χ Ore ( H ⋄ ) = χ ( H ⋄ ) − / ( k + 2).)Given a graph H , let δ Ore ( H, n ) be the smallest integer k such that every graph G whose order n is divisible by | H | and with d ( x ) + d ( y ) ≥ k for all non-adjacent x = y ∈ V ( G ) contains a perfect H -packing. Roughly speaking, our next result statesthat when considering an Ore-type degree condition, for any graph H , χ Ore ( H ) is therelevant parameter which governs the existence of a perfect H -packing. In particular,it implies that we do not have a ‘dichotomy’ involving only χ ( H ) and χ cr ( H ) as inTheorem 1. Theorem 2.
For every graph H and each η > there exists a constant C = C ( H ) andan integer n = n ( H, η ) such that if n ≥ n then (cid:18) − χ Ore ( H ) (cid:19) n − C ≤ δ Ore ( H, n ) ≤ (cid:18) − χ Ore ( H ) + η (cid:19) n. So for example, Theorem 2 implies that lim n →∞ δ Ore ( K − , n ) /n = 4 / n →∞ δ Ore ( F ⋄ , n ) /n = 9 / H there are infinitely many values of n for which we can take C = 2 inTheorem 2. In fact, if hcf( H ) = 1 or CE ( H ) = ∞ then C = 2 suffices for all n divisibleby | H | . In general C ≤ | H | (see Section 2). It would be interesting to know whetherone can replace the error term ηn by a constant depending only on H .1.3. Almost perfect packings.
The critical chromatic number was first introducedby Koml´os [9], who showed that it is the relevant parameter when considering ‘almost’perfect H -packings. N ORE-TYPE THEOREM FOR PERFECT PACKINGS IN GRAPHS 5
Theorem 3. [9]
For every graph H and each γ > there exists an integer n = n ( γ, H ) such that every graph G of order n ≥ n and minimum degree at least (1 − /χ cr ( H )) n contains an H -packing which covers all but at most γn vertices of G . It is easy to see that the bound on the minimum degree in Theorem 3 is best possible.In the proof of Theorem 2 we will use the following result which provides an Ore-type analogue of Theorem 3. Again, the critical chromatic number is the relevantparameter for any graph H . In particular, this means that Theorem 4 is a generalizationof Theorem 3. The proof of Theorem 4 is almost identical to that of Theorem 3. Asketch of the proof is given in Section 5. Full details can be found in [19]. Theorem 4.
For every graph H and each η > there exists an integer n = n ( H, η ) such that if G is a graph on n ≥ n vertices and d ( x ) + d ( y ) ≥ (cid:18) − χ cr ( H ) (cid:19) n for all non-adjacent x = y ∈ V ( G ) then G has an H -packing covering all but at most ηn vertices. Shokoufandeh and Zhao [17] showed that in Theorem 3 the bound on the number ofuncovered vertices can be reduced to a constant depending only on H . We conjecturethat this should also be the case for Theorem 4.1.4. Copies of H covering a given vertex. In the proof of Theorem 2 it will be usefulto determine the Ore-type degree condition which guarantees a copy of H covering agiven vertex of G . Let δ ′ Ore ( H, n ) denote the smallest integer k such that whenever w isa vertex of a graph G of order n with d ( x ) + d ( y ) ≥ k for all non-adjacent x = y ∈ V ( G )then G contains a copy of H covering w . Define χ ′ Ore ( H ) := ( χ ( H ) if CE ( H ) = ∞ ; χ ( H ) − CE ( H )+2 otherwise. Theorem 5.
For every graph H and every η > there exists an integer n = n ( H, η ) and a constant C = C ( H ) such that if n ≥ n then (cid:18) − χ ′ Ore ( H ) (cid:19) n − C ≤ δ ′ Ore ( H, n ) ≤ (cid:18) − χ ′ Ore ( H ) + η (cid:19) n. Theorem 5 is proved in Section 3. As in the case of perfect H -packings, the Ore-type degree condition in Theorem 5 does not quite match the bound needed for thecorresponding minimum degree version. Indeed, let δ ′ ( H, n ) denote the smallest integer k such that whenever w is a vertex of a graph G of order n with δ ( G ) ≥ k then G contains a copy of H covering w . Together with the Erd˝os-Stone theorem the nextresult implies that asymptotically δ ′ ( H, n ) is the same as the minimum degree neededto force any copy of H in a graph of order n . Proposition 6.
For every graph H and every η > there exists an integer n = n ( H, η ) such that if n ≥ n then (cid:18) − χ ( H ) − (cid:19) n − ≤ δ ′ ( H, n ) ≤ (cid:18) − χ ( H ) − η (cid:19) n. DANIELA K ¨UHN, DERYK OSTHUS AND ANDREW TREGLOWN
The lower bound on δ ′ ( H, n ) in Proposition 6 follows by considering a complete( χ ( H ) − G whose vertex classes are as equal as possible. The proof ofthe upper bound is similar to Case 1 of the proof of Lemma 13 (see Section 6). Detailscan be found in [20].We will not use Proposition 6 in the proof of Theorem 2, but we have included itas it helps to explain the difference between Theorems 1 and 2. Indeed, Theorem 3and Proposition 6 show that the minimum degree which ensures an almost perfect H -packing is larger than the minimum degree which guarantees a copy of H covering anygiven vertex. In contrast, Theorems 4 and 5 imply that for some H this is not truein the Ore-type case. So it is natural that δ Ore ( H, n ) involves this property explicitly(since the property that every vertex is contained in a copy of H is clearly necessary toensure a perfect H -packing). In fact, this is the only real difference to the expressionfor δ ( H, n ) in Theorem 1: note that we have χ Ore ( H ) = max { χ ∗ ( H ) , χ ′ Ore ( H ) } and thusTheorems 1, 2 and 5 imply that δ Ore ( H, n ) = max (cid:8) δ ( H, n ) , δ ′ Ore ( H, n ) (cid:9) + o ( n ) . Forcing a single copy of H . In view of Theorem 5, one might also wonder whatOre-type degree condition ensures at least one copy of H (i.e. we do not require everyvertex to lie in a copy of H ). It is easy to see that if G is of order n then the conditionis similar to the condition on the minimum degree. Proposition 7.
For every graph H and every η > there exists an integer n = n ( H, η ) such that if n ≥ n and G is a graph on n vertices which satisfies d ( x ) + d ( y ) ≥ (cid:18) − χ ( H ) − η (cid:19) n for all non-adjacent x = y ∈ V ( G ) , then G contains a copy of H . Proposition 7 immediately follows from the Erd˝os-Stone theorem and the followingobservation (which we expect to be known, but we were unable to find a reference):
Proposition 8.
Let G be a graph with d ( x ) + d ( y ) ≥ k for all non-adjacent x = y ∈ V ( G ) . Then G has average degree at least k . To prove Proposition 8, let A be the set of vertices in G whose degree is less than k and let B be the set of remaining vertices. Let G denote the complement of G and let F denote the bipartite subgraph of G induced by A and B . Hall’s theorem implies that F has a matching covering all of A (Hall’s condition can be verified by noting that forall X ⊆ A the number of edges in F between X and the neighbourhood of X is at least | X | ( n − k −
1) and at most | N ( X ) | ( n − k − G which are contained in this matching.1.6. Other structures.
As mentioned earlier, packing and embedding results in graphsof large minimum degree have also been studied for other structures. It would beinteresting to obtain Ore-type analogues for some of these: e.g. for the P´osa-Seymourconjecture which states that every graph G on n vertices with δ ( G ) ≥ rr +1 n contains the r th power of a Hamilton cycle. [7] contains a discussion of other Ore-type results. N ORE-TYPE THEOREM FOR PERFECT PACKINGS IN GRAPHS 7 Notation and extremal examples
Throughout this paper we omit floors and ceilings whenever this does not affect theargument. We write e ( G ) to denote the number of edges of a graph G , | G | for its order, δ ( G ) and ∆( G ) for its minimum and maximum degrees respectively and χ ( G ) for itschromatic number.Given disjoint A, B ⊆ V ( G ), an A - B edge is an edge of G with one endvertex in A and the other in B . The number of these edges is denoted by e G ( A, B ) or e ( A, B ) if thisis unambiguous. We write (
A, B ) G for the bipartite subgraph of G with vertex classes A and B whose edges are precisely the A - B edges in G .Let us now prove the lower bound in Theorem 2. The next proposition deals withthe case when CE ( H ) = ∞ . Proposition 9.
Let H be a graph with CE ( H ) = ∞ . Let n ≥ | H | . Then there exists agraph G of order n with d ( x ) + d ( y ) ≥ (cid:18) − χ ( H ) (cid:19) n − for all non-adjacent x = y ∈ V ( G ) containing a vertex that does not belong to a copyof H . (In particular, G has no perfect H -packing.) Proof.
Let r := χ ( H ). Consider the complete r -partite graph of order n whose vertexclasses V ′ , V ′ , V , . . . , V r have sizes as equal as possible, where | V ′ | ≤ | V ′ | ≤ | V | ≤ · · · ≤| V r | . Note that n − | V ′ | − | V ′ | ≥ n − n/r .Let G be obtained from this graph by moving all but one vertex, w say, from V ′ to V ′ ,by making the set V ⊇ V ′ thus obtained from V ′ into a clique and by deleting all theedges between w and the vertices in V .Any vertex y ∈ V ∪ · · · ∪ V r satisfies d ( y ) ≥ n − ⌈ nr ⌉ ≥ (1 − /χ ( H )) n −
1. Thus d ( y ) + d ( y ) ≥ − /χ ( H )) n − y = y ∈ V ( G ) \ ( { w } ∪ V ).Moreover, d ( w ) = n − | V ′ | − | V ′ | ≥ n − n/r and for any z ∈ V we have d ( z ) = n − d ( w ) + d ( z ) ≥ − /χ ( H )) n −
2. Hence G satisfies our Ore-type degree condition.The neighbourhood of w in G induces an ( r − G . Therefore,since χ ( H [ N ( x )]) = r − x ∈ V ( H ), w cannot play the role of any vertex in H .So G does not contain a copy of H covering w . (cid:3) The following proposition will be used for the case when H is non-bipartite and CE ( H ) < ∞ . Proposition 10.
Let H be a graph with r := χ ( H ) ≥ for which m := CE ( H ) < ∞ .Then there are infinitely many graphs G whose order n is divisible by | H | and such that d ( x ) + d ( y ) ≥ − r − m +2 ! n − for all non-adjacent x = y ∈ V ( G ) containing a vertex that does not belong to a copyof H . (In particular, G has no perfect H -packing.) Proof.
Let t ∈ N be such that (( m + 2) r − r −
2) divides t . Define s := 2 | H | / (( m +2) r − G ′ be the complete ( r + m − V of size DANIELA K ¨UHN, DERYK OSTHUS AND ANDREW TREGLOWN st − m vertex classes V , . . . , V m +1 of size st and r − V m +2 , . . . , V r + m − of size | H | t − ( m +1) str − . Let G be obtained from G ′ by adding a vertex w to G ′ such that w is adjacent to precisely those vertices in V m +2 ∪ · · · ∪ V r + m − . So | G | = | H | t .Any y ∈ V ∪ · · · ∪ V m +1 satisfies d ( y ) + d ( w ) ≥ | H | t − ( m + 2) st − (cid:18) − m + 2( m + 2) r − (cid:19) | G | − . Furthermore, given any y = y ∈ V i for some m + 2 ≤ i ≤ r + m −
1, we have d ( y ) + d ( y ) = 2 | H | t − (cid:18) | H | t − ( m + 1) str − (cid:19) = 2 | G | − r − (cid:18) − m + 1)( m + 2) r − (cid:19) | G | = 2 | G | − r − m + 2)( r − m + 2) r − | G | = 2 (cid:18) − m + 2( m + 2) r − (cid:19) | G | . Since d ( y ) + d ( y ′ ) ≥ d ( y ) + d ( w ) for any y = y ′ ∈ V i with 1 ≤ i ≤ m + 1 this impliesthat G satisfies our Ore-type degree condition.Suppose that w belongs to some copy H w of H in G . Since χ ( G ) = m + r −
1, anoptimal colouring of G induces an ( m + r − H w and an ( r − G [ N ( w )]. But then w must be playing the role of a vertex x ∈ V ( H ) such that χ ( H [ N ( x )]) ≤ r −
2, contradicting the definition of m = CE ( H ). (cid:3) We will now use Propositions 9 and 10 to prove the lower bound of Theorem 2.
Proof of Theorem 2 (lower bound).
In the case when hcf( H ) = 1 the lowerbound follows from the lower bound in Theorem 1. Proposition 9 settles the case when CE ( H ) = ∞ . So we may assume that hcf( H ) = 1 and CE ( H ) < ∞ . In this case, thelower bound in Theorem 1 also implies that(1) δ Ore ( H, n ) ≥ − /χ cr ( H )) n − H ). Suppose first that H is bipartite. Since CE ( H ) < ∞ this meansthat H must have an isolated vertex and so CE ( H ) = 0. Thus χ Ore ( H ) = χ cr ( H ) andso we are done by (1).So suppose next that χ ( H ) ≥
3. In this case the proof of Proposition 10 implies thelower bound whenever n is divisible by (( m + 2) r − r − | H | . To deduce the lowerbound for any n ≥ (( m + 2) r − r − | H | which is divisible by | H | we proceed asfollows. Let n ′ be the largest integer such that n ′ ≤ n and n ′ is divisible by (( m + 2) r − r − | H | . Construct a graph G of order n ′ as in the proof of Proposition 10. Thenadd n − n ′ < (( m + 2) r − r − | H | new vertices to V so that these vertices havethe same neighbourhoods as the original vertices in V . Then | G | = n and by the sameargument as in Proposition 10, G does not contain a perfect H -packing. Moreover, it iseasy to check that d ( x ) + d ( y ) ≥ − / ( r − / ( m + 2))) n − | H | for all non-adjacent x = y ∈ V ( G ). (cid:3) N ORE-TYPE THEOREM FOR PERFECT PACKINGS IN GRAPHS 9 Some useful results
In Section 2 we proved the lower bound on δ Ore ( H, n ) in Theorem 2. The followingtwo results together imply the upper bound.
Lemma 11.
Let H be a graph and let η > . There exists an integer n = n ( H, η ) such that if G is a graph whose order n ≥ n is divisible by | H | and d ( x ) + d ( y ) ≥ (cid:18) − χ ( H ) + η (cid:19) n for all non-adjacent x = y ∈ V ( G ) then G contains a perfect H -packing. Lemma 12.
Let η > and suppose that H is a graph such that hcf( H ) = 1 and CE ( H ) < ∞ . There exists an integer n = n ( H, η ) such that if G is a graph whoseorder n ≥ n is divisible by | H | and d ( x ) + d ( y ) ≥ max ( − χ ( H ) − CE ( H )+2 + η ! n, (cid:18) − χ cr ( H ) + η (cid:19) n ) (2) for all non-adjacent x = y ∈ V ( G ) then G contains a perfect H -packing. Note that Lemma 11 implies the upper bound on δ ( H, n ) by Alon and Yuster (whichwe mentioned in Section 1). We now deduce Lemma 11 from Lemma 12.
Proof of Lemma 11.
Let h := | H | and r := χ ( H ). Given any k ≥
2, define H ∗ tobe the complete ( r + 1)-partite graph with one vertex class of size 1, one vertex classof size hk − r − hk . Let H ′ be obtained from H ∗ byremoving an edge between some vertex y in a vertex class of size hk and the vertex inthe singleton vertex class. So χ ( H ′ ) = r + 1, | H ′ | = hkr and χ ( H ′ [ N ( y )]) = r − CE ( H ′ ) = 0 since N ( y ) lies in r − H ′ . It is easy to seethat H ′ contains a perfect H -packing and that hcf( H ′ ) = 1. So χ Ore ( H ′ ) = χ cr ( H ′ ) =( χ ( H ′ ) − | H ′ || H ′ |− σ ( H ′ ) = r | H ′ || H ′ |− . In particular, we can choose k sufficiently large toguarantee that 1 /χ cr ( H ′ ) ≥ /χ ( H ) − η/ G as in Lemma 11. Choose a ≤ kr such that n − ah is divisibleby | H ′ | = hkr . Apply Proposition 7 to obtain a disjoint copies of H in G . Remove these a copies of H from G to obtain a graph G ′ whose order is divisible by | H ′ | and whichsatisfies d G ′ ( x ) + d G ′ ( x ) ≥ (cid:18) − χ ( H ) + η (cid:19) | G ′ | ≥ (cid:18) − χ cr ( H ′ ) + η (cid:19) | G ′ | for all non-adjacent x = x ∈ V ( G ′ ). Apply Lemma 12 to find a perfect H ′ -packingin G ′ . In particular, this induces a perfect H -packing in G ′ . Thus, together with allthose copies of H in G − G ′ we have chosen before, we obtain a perfect H -packing in G . (cid:3) Thus to prove Theorem 2 it remains to prove Lemma 12, which we will do in Section 7.In order to deal with the ‘exceptional’ vertices in the proof of Lemma 12 we use thefollowing result which implies that every vertex w of a graph G as in Lemma 12 iscontained in a copy of H . We prove Lemma 13 in Section 6. Lemma 13.
Let H be a graph such that m := CE ( H ) < ∞ . Let r := χ ( H ) and η > .There exists an integer n = n ( η, H ) such that whenever G is a graph on n ≥ n vertices with d ( x ) + d ( y ) ≥ − r − m +2 + η ! n (3) for all non-adjacent x = y ∈ V ( G ) then every vertex of G lies in a copy of H in G . The above results also imply Theorem 5:
Proof of Theorem 5.
The lower bound in the case when CE ( H ) = ∞ follows fromProposition 9. If CE ( H ) < ∞ and χ ( H ) ≥ n and as in the proof of the lower bound in Theorem 2 itcan be used to derive the lower bound for any n . If CE ( H ) < ∞ and χ ( H ) = 2 then CE ( H ) = 0 and so the lower bound is trivial. The upper bound follows from Lemmas 11and 13. (cid:3) In our proof of Lemma 12 we will also use the following result, Lemma 12 from [14]. Itgives a sufficient condition on the sizes of the vertex classes of a complete χ ( H )-partitegraph G which ensures that G has a perfect H -packing. Lemma 14 is the point wherethe assumption that hcf( H ) = 1 is crucial – it is false for graphs with hcf( H ) = 1. Lemma 14.
Let H be a graph with hcf( H ) = 1 . Put r := χ ( H ) and γ := ( r − σ ( H ) / ( | H | − σ ( H )) . Let < β ≪ λ ≪ γ, − γ, / | H | be positive constants. Supposethat G is a complete r -partite graph with vertex classes U , . . . , U r such that | G | ≫ | H | is divisible by | H | , (1 − λ / ) | U r | ≤ γ | U i | ≤ (1 − λ ) | U r | for all i < r and such that | | U i | − | U j | | ≤ β | G | whenever ≤ i < j < r . Then G contains a perfect H -packing. Here (and later on) we write 0 < a ≪ a ≪ a ≤ a , a , a from right to left. More precisely, there are increasing functions f and g such that, given a , whenever we choose some a ≤ f ( a ) and a ≤ g ( a ), allcalculations needed in the proof of Lemma 14 are valid.4. The Regularity lemma and the Blow-up lemma
In the proof of Lemma 12 we will use Szemer´edi’s Regularity lemma [18] and the Blow-up lemma of Koml´os, S´ark¨ozy and Szemer´edi [10]. In this section we will introduce allthe information we require about these two results. To do this, we firstly introducesome more notation. The density of a bipartite graph G with vertex classes A and B isdefined to be d ( A, B ) := e ( A, B ) | A || B | . Given any ε >
0, we say that G is ε -regular if for all sets X ⊆ A and Y ⊆ B with | X | ≥ ε | A | and | Y | ≥ ε | B | we have | d ( A, B ) − d ( X, Y ) | < ε . In this case we also say N ORE-TYPE THEOREM FOR PERFECT PACKINGS IN GRAPHS 11 that (
A, B ) is an ε -regular pair . Given d ∈ [0 ,
1) we say that G is ( ε, d ) -super-regular ifall sets X ⊆ A and Y ⊆ B with | X | ≥ ε | A | and | Y | ≥ ε | B | satisfy d ( X, Y ) > d and,furthermore, if d G ( a ) > d | B | for all a ∈ A and d G ( b ) > d | A | for all b ∈ B .We will use the following degree form of Szemer´edi’s Regularity lemma which can beeasily derived from the classical version. Lemma 15 (Regularity lemma) . For every ε > and each integer ℓ there is an M = M ( ε, ℓ ) such that if G is any graph on at least M vertices and d ∈ [0 , , then thereexists a partition of V ( G ) into ℓ +1 classes V , V , ..., V ℓ , and a spanning subgraph G ′ ⊆ G with the following properties: • ℓ ≤ ℓ ≤ M, | V | ≤ ε | G | , | V | = · · · = | V ℓ | =: L , • d G ′ ( v ) > d G ( v ) − ( d + ε ) | G | for all v ∈ V ( G ) , • e ( G ′ [ V i ]) = 0 for all i ≥ , • for all ≤ i < j ≤ ℓ the graph ( V i , V j ) G ′ is ε -regular and has density either orgreater than d . The sets V , . . . , V ℓ are called clusters , V is called the exceptional set and the verticesin V exceptional vertices . We refer to G ′ as the pure graph of G . Clearly, we may assumethat ( V i , V j ) G is not ε -regular or has density at most d whenever ( V i , V j ) G ′ contains noedges (for all 1 ≤ i < j ≤ ℓ ). The reduced graph R of G is the graph whose verticesare V , . . . , V ℓ and in which V i is adjacent to V j whenever ( V i , V j ) G ′ is ε -regular and hasdensity greater than d .A well-known fact is that the minimum degree of a graph G is almost inherited byits reduced graph. We now prove an analogue of this for an Ore-type degree condition. Lemma 16.
Given a constant c , let G be a graph such that d G ( x ) + d G ( y ) ≥ c | G | for allnon-adjacent x = y ∈ V ( G ) . Suppose we have applied Lemma 15 with parameters ε and d to G . Let R be the corresponding reduced graph. Then d R ( V i ) + d R ( V j ) > ( c − d − ε ) | R | for all non-adjacent V i = V j ∈ V ( R ) . Proof.
Let V , . . . , V ℓ denote the clusters obtained from Lemma 15. Let L := | V | = · · · = | V ℓ | , let V denote the exceptional set and let G ′ be the pure graph. Set G ′′ := G ′ − V . Consider any pair V i V j of clusters which does not form an edge in R . Pick x ∈ V i and y ∈ V j such that xy E ( G ). So d G ( x ) + d G ( y ) ≥ c | G | and thus d G ′′ ( x ) + d G ′′ ( y ) > ( c − d − ε ) | G | . However, by definition of G ′′ , each cluster containing a neighbour of x in G ′′ must be a neighbour of V i in R and the analogue holds for the clusters containingthe neighbours of y . Thus d R ( V i ) + d R ( V j ) ≥ ( d G ′′ ( x ) + d G ′′ ( y )) /L ≥ ( c − d − ε ) | R | ,as required. (cid:3) We will also use the following Embedding lemma. The proof is based on a simplegreedy argument, see e.g. Lemma 7.5.2 in [2] or Theorem 2.1 in [12] for details.
Lemma 17 (Embedding lemma) . Let H be an r -partite graph with vertex classes X , . . . , X r and let ε, d, n be constants such that < /n ≪ ε ≪ d, / | H | . Let G be an r -partite graph with vertex classes V , . . . , V r of size at least n such that ( V i , V j ) G is ε -regular and has density at least d whenever H contains an edge between X i and X j (for all ≤ i < j ≤ r ). Then G contains a copy of H such that X i ⊆ V i . The Blow-up lemma of Koml´os, S´ark¨ozy and Szemer´edi [10] states that one can evenfind a spanning subgraph H in G provided that H has bounded maximum degree andthe bipartite pairs forming G are super-regular. Lemma 18 (Blow-up lemma) . Given a graph R with V ( R ) = { , . . . , r } and d, ∆ > ,there is a constant ε = ε ( d, ∆ , r ) > such that the following holds. Given L , . . . , L r ∈ N and < ε ≤ ε , let R ∗ be the graph obtained from R by replacing each vertex i ∈ V ( R ) with a set V i of L i new vertices and joining all vertices in V i to all vertices in V j preciselywhen ij ∈ E ( R ) . Let G be a spanning subgraph of R ∗ such that for every ij ∈ E ( R ) the bipartite graph ( V i , V j ) G is ( ε, d ) -super-regular. Then G contains a copy of everysubgraph H of R ∗ with ∆( H ) ≤ ∆ . Sketch proof of Theorem 4
Let G be a sufficiently large graph on n vertices so that d ( x ) + d ( y ) ≥ (cid:18) − χ cr ( H ) (cid:19) n for all non-adjacent x = y ∈ V ( G ). Let r := χ ( H ). Denote by B the complete r -partite graph with one vertex class of size ( r − σ ( H ) and r − | H | − σ ( H ). Note that χ cr ( B ) = χ cr ( H ) and B has a perfect H -packing. Thus byconsidering B instead of H if necessary, it is sufficient to prove the theorem under theadded assumption that H is a complete r -partite graph with one vertex class of size σ ∈ N and r − ω ∈ N . It suffices to consider the case when σ < ω .(It is easy to deduce the case σ = ω from this using the same trick as in the proof ofLemma 11.) Let H ′ denote the complete r -partite graph with one vertex class of size σ and r − ω − G . Claim 19.
Let < τ, /ℓ ≪ d ′ ≪ γ, / | H | . Let R ′ be a graph on ℓ ′ ≥ ℓ vertices suchthat d ( x ) + d ( y ) ≥ − /χ cr ( H ) − d ′ ) ℓ ′ for all non-adjacent x = y ∈ V ( R ′ ) . Supposethat the maximum number of vertices in R ′ covered by an H -packing is N ≤ (1 − γ ) ℓ ′ .Then R ′ contains a collection of vertex-disjoint copies of H, H ′ and K r which togethercover at least N + τ ℓ ′ vertices. The proof of Claim 19 is almost identical to that of Lemma 15 from [9]. Full detailscan be found in [19]. Here we briefly outline the proof. Let L denote the set of verticesnot covered by the largest H -packing in R ′ . Since the subgraph of R ′ induced by L must contain a small number of edges (else it will contain a copy of H ), most verticesin L have ‘small’ degree in this subgraph. Furthermore, all but at most | H | − x have degree at least (1 − /χ cr ( H ) − d ′ ) ℓ ′ in R ′ (otherwise we have a copy of K | H | and thus of H in L ). Now we proceed exactly as in Lemma 15 from [9]. Indeed,for many such vertices x we can combine x with a suitable copy of H in the packing andreplace it with a copy of H ′ and a copy of K r containing x . Thus we obtain our desiredcollection of vertex-disjoint copies of H , H ′ and K r .Consider a graph F and t ∈ N . Let F ( t ) denote the graph obtained from F byreplacing every vertex x ∈ V ( F ) by a set U x of t independent vertices, and joining each N ORE-TYPE THEOREM FOR PERFECT PACKINGS IN GRAPHS 13 u ∈ U x to each v ∈ U y precisely when xy is an edge in F . In other words we replace theedges of F by copies of K t,t . We will refer to F ( t ) as a blown-up copy of F .Define constants 0 ≪ /ℓ ≪ ε ≪ ε ′ ≪ d ≪ d ′ ≪ γ ≪ η. Apply the Regularity lemma with parameters ε , d and ℓ to G to obtain clusters V , . . . , V ℓ of size L , an exceptional set V and a reduced graph R . By Lemma 16we have that d R ( V i ) + d R ( V j ) ≥ (cid:18) − χ cr ( H ) − d (cid:19) | R | for all V i = V j with V i V j E ( R ).We wish to find an H -packing either in R or in a blown-up copy of R , which covers atleast a (1 − γ )-proportion of the clusters. Let N denote the maximum number of clustersin R covered by an H -packing. If N ≥ (1 − γ ) ℓ we are done. If not, then by Claim 19 R contains a collection of vertex-disjoint copies of H , H ′ and K r which together coverat least N + τ ℓ vertices. Let t := ( ω − σ ) | H | . It is not hard to see that H ( t ), H ′ ( t ) and K r ( t ) all have perfect H -packings. Thus R := R ( t ) contains an H -packing covering at least ( N + τ ℓ ) t vertices. Since | R | = ℓt ,a larger proportion of the vertices in R are covered by this H -packing compared to the H -packing in R . If ( N + τ ℓ ) t ≥ (1 − γ ) ℓt we are done. Otherwise we can continue thisprocess: By applying Claim 19 to R we see that R ( t ) = R ( t ) contains an H -packingcovering a substantially larger proportion of the vertices than the previous H -packing.Eventually we obtain a graph R ′ , where R ′ = R ( t k ) for some constant k = k ( H, γ, d ′ ),such that there is an H -packing H covering at least (1 − γ ) | R ′ | clusters in R ′ .Suppose that ( A, B ) is an ε -regular pair of density at least d . By removing at most t k vertices from A and B , we can partition both A and B into t k equal subclusters A , . . . , A t k and B , . . . , B t k respectively. We can do this in such a way that each ( A i , B j )is an ε ′ -regular pair with density at least d − ε . So since each edge of R corresponds toan ε -regular pair of density at least d , the edges of R ′ can be viewed as correspondingto ε ′ -regular pairs with density at least d − ε . (Thus for each V i ∈ V ( R ) there are t k vertices in R ′ which correspond to subclusters of V i .)Suppose that H ∗ is a copy of H in H . Consider the subgraph H ∗ G of G whose vertexset consists of all those vertices lying in the clusters of H ∗ , and whose edge set consistsof all those edges which lie in an ε ′ -regular pair corresponding to an edge in H ∗ . Itis easy to see (for example by repeated use of Lemma 17) that there is an H -packingcovering almost all the vertices in H ∗ G . We find such an H -packing for each copy of H in H . Since H covers at least (1 − γ ) | R ′ | of the clusters in R ′ , and since γ ≪ η , the unionof all these H -packings covers at least (1 − η ) n vertices of G , as desired.6. Proof of Lemma 13
Let H be as in the statement of the lemma and let G be a graph of sufficiently largeorder n which satisfies (3). Recall that r = χ ( H ) and m = CE ( H ). Let x be anyvertex of G . We have to find a copy of H in G which contains x . Suppose first that r = 2. Then H must have an isolated vertex v (since CE ( H ) < ∞ ). So we can apply Proposition 7 to find a copy of H − v in G − x and thus a copy of H in G (where x playsthe role of v ).So suppose that r ≥
3. Choose additional constants ε, d and α such that0 < ε ≪ d ≪ α ≪ η and let ℓ := 1 /ε . Apply the Regularity lemma with parameters ε, d, ℓ to G to obtainclusters V , . . . , V ℓ of size L , an exceptional set V , a pure graph G ′ and a reducedgraph R . Let k := ( m + 2) r − . Lemma 16 implies that(4) d R ( V i ) + d R ( V j ) ≥ − r − m +2 + η ! | R | = 2 (cid:18) − m + 2 k + η (cid:19) | R | for all V i = V j ∈ V ( R ) with V i V j E ( R ). By adding the vertices of one cluster to V if necessary (and deleting this cluster from R ) we may assume that x ∈ V . (So now | V | ≤ εn .) We say that x is adjacent to a cluster V i ∈ V ( R ) if x is adjacent to at least αL vertices of V i in G . We denote by S the set of clusters V i ∈ V ( R ) that x is adjacentto, and define s := | S | / | R | . Also, we write S := V ( R ) \ S . Note that(5) d G ( x ) ≤ | S | L + | S | αL + | V | ≤ ( s + α + 2 ε ) n ≤ ( s + 2 α ) n and so(6) s ≥ δ ( G ) n − α ( ) ≥ − r − m +2 + 2 η ! − α ≥ − m + 2) k + η. In particular s > r ≥
3. Our aim now is to find either a copy K ′ r of K r in R containing r − x (i.e. | V ( K ′ r ) ∩ S | ≥ r − K ′ r + m of K r + m in R containing r − x . In both cases we could apply theEmbedding lemma (Lemma 17) to find the desired copy H x of H in G . Indeed, in thecase where we find K ′ r + m we could use x to play the role of a vertex y ∈ V ( H ) for whichthere exists an ( r − H [ N ( y )] that can be extended to an ( r + m )-colouringof H . The neighbourhood N H ( y ) of y would be embedded into the clusters belongingto V ( K ′ r + m ) ∩ S and H − N H ( y ) would be embedded into the clusters belonging to V ( K ′ r + m ) (so here we use the fact that CE ( H ) = m ). In the case where we find K ′ r , x can play the role of any vertex of H . Given some optimal colouring of H , the verticesof H which have a different colour than x are embedded into the clusters in V ( K ′ r ) ∩ S (so we only use that χ ( H ) = r in this case).Let C be the set of clusters U ∈ S with d R ( U ) < (1 − ( m + 2) /k + η/ | R | . By (4), C induces a clique. So we may assume that | C | < r , since otherwise we have our copy K ′ r of K r . Suppose now that for some 1 ≤ i ≤ r − i clusters U , . . . , U i ∈ S \ C such that U , . . . , U i form a copy K ′ i of K i in R . Then(7) | \ ≤ j ≤ i N R ( U j ) | ≥ − ( i − | R | + i X j =1 d R ( U j ) ≥ (cid:18) − i ( m + 2) k + η/ (cid:19) | R | . Case 1. − s ≤ (2 m + 2) /k N ORE-TYPE THEOREM FOR PERFECT PACKINGS IN GRAPHS 15
In this case, we will find a copy of K r which contains at least r − S . Supposethat i ≤ r − U , . . . , U i as above. Then 1 − i ( m + 2) /k ≥ (2 m + 2) /k and so (7) implies that the common neighbourhood N R ( K ′ i ) of K ′ i satisfies | N R ( K ′ i ) | ≥ (1 − s + η/ | R | . So we can choose U i +1 ∈ S \ C to extend K ′ i into a copy of K i +1 in R [ S \ C ] (we can avoid C when choosing U i +1 since | C | < r ≪ η | R | ). If i = r −
1, then1 − i ( m +2) k = mk ≥
0. So | N R ( K ′ i ) | ≥ η | R | / K ′ i = K ′ r − into thedesired copy of K r using an arbitrary vertex of R . Case 2. − s ≥ (2 m + 2) /k In this case, we will either find a copy of K r which contains at least r − S orfind a copy of K r + m which contains at least r − S . Suppose that i ≤ r − U , . . . , U i as described before Case 1 which form a copy K ′ i of K i in R [ S \ C ]. Note that1 − i ( m + 2) k ≥ k − ( r − m + 2) k = 3( m + 2) − k ≥ m + 2) k ( ) ≥ − s. Thus (7) implies that we can choose a cluster U i +1 ∈ S \ C which forms a K i +1 togetherwith K ′ i . This shows that we can find a copy K ′ r − of K r − which lies in R [ S \ C ]. Notethat (7) also implies that the common neighbourhood N R ( K ′ r − ) of K ′ r − satisfies(8) | N R ( K ′ r − ) | ≥ (cid:18) − ( r − m + 2) k + η (cid:19) | R | = (cid:18) m + 1) k + η (cid:19) | R | . Now we aim to extend K ′ r − into a copy K ′ r + m of K r + m . We will aim to find theadditional vertices in S . Suppose for some 0 ≤ i ≤ m + 1 we have found i clusters W , . . . , W i ∈ S which together with K ′ r − form a copy K ′ r − i of K r − i in R . Wewill need a lower bound on d R ( W j ) for all j = 1 , . . . , i . To derive this, note that thedefinition of S implies that W j contains a vertex y which is not adjacent to x in G .So (3) and (5) and the inequality in Case 2 imply that d G ( y ) ≥ (cid:18) (cid:18) − m + 2 k + η (cid:19) − s − α (cid:19) n ≥ (cid:18) − k + η (cid:19) n and so d G ′ ( y ) ≥ (1 − /k + η/ n . But each cluster containing a neighbour of y in G ′ must be a neighbour of W j in R . Hence(9) d R ( W j ) ≥ d G ′ ( y ) − | V | L ≥ (cid:18) − k (cid:19) | R | . So the common neighbourhood N R ( K ′ r − i ) of K ′ r − i satisfies(10) | N R ( K ′ r − i ) | ≥ | N R ( K ′ r − ) |− i | R | + i X j =1 d R ( W j ) ( ) , ( ) ≥ (cid:18) m + 1) k − i k + η (cid:19) | R | ≥ η | R | . So we can choose a vertex W i +1 ∈ V ( R ) \ C that is a common neighbour of the clustersin K ′ r − i . Suppose that W i +1 ∈ S . Then together with K ′ r − this forms a copy K ′ r − of K r − in R [ S \ C ]. Now (7) implies that | N R ( K ′ r − ) | ≥ ( m/k + η/ | R | and so we canextend K ′ r − to a copy of K r with at least r − S . So we may assume that W i +1 ∈ S . Continuing in this way, we obtain a copy of K r + m having r − S ,as required. 7. Proof of Lemma 12
Preliminaries and an outline of the proof.
Let H , G and η > r := χ ( H ). Choose t ∈ N such that t | H | ( r − ≥ r/η . Let z := t ( r − σ ( H ) and z := t ( | H | − σ ( H )). Put γ := z /z . Note that 0 < γ < H ) = 1. Define B ∗ to be the complete r -partite graph with one vertex class of size z and r − z . Then B ∗ has a perfect H -packing and η | B ∗ | / ≥ r .Moreover, χ cr ( B ∗ ) = χ cr ( H ) = ( r − | H || H | − σ ( H ) = r − r − σ ( H ) | H | − σ ( H ) = r − γ. (11)Choose s ∈ N and a new constant λ such that 0 < λ ≪ η, γ, − γ as well as s := γ (1 + λ ) s ∈ N and s ≤ s . Let B ′ denote the complete r -partite graph with one vertexclass of size s and r − s . Thus, χ cr ( B ′ ) = ( r − | B ′ || B ′ | − s = r − γ (1 + λ ) . (12)Note that the proportion γ (1+ λ ) of the size of the smallest vertex class of B ′ compared tothe size of one of the larger classes is slightly larger than the corresponding proportion γ associated with B ∗ . We can therefore choose s and λ in such a way that B ′ has aperfect B ∗ -packing, and thus a perfect H -packing. (Indeed, the perfect B ∗ -packingwould consist of ‘most’ but not all of the copies of B ∗ having their smallest vertex classlying in the smallest vertex class of B ′ .)We now give an outline for the proof of Lemma 12. We first apply the Regularitylemma to G to obtain a reduced graph R . Since R almost inherits the Ore-type conditionon G we may apply Theorem 4 to find an almost perfect B ′ -packing of R . We thenremove all clusters from R that are not covered by this B ′ -packing and add the verticesin these clusters to the exceptional set V .For each exceptional vertex x ∈ V , we apply Lemma 13 to find a copy of H in G containing x , and remove the vertices in this copy from G . Thus some vertices in clustersin R will be removed from G . The copies of H will be chosen to be disjoint for differentexceptional vertices.Our aim is to apply the Blow-up lemma to each copy B ′ i of B ′ in the B ′ -packingof R in order to find an H -packing in G which covers all the vertices belonging to (themodified) clusters in B ′ i . Then all these H -packings together with all those copies of H chosen for the exceptional vertices would form a perfect H -packing in G . However, to dothis, we need that the complete r -partite graph F ∗ i whose j th vertex class is the unionof all the clusters in the j th vertex class of B ′ i has a perfect H -packing. Lemma 14 givesa condition which guarantees this.To apply Lemma 14 we need that | F ∗ i | is divisible by | H | . We will remove a boundednumber of further copies of H from G to ensure this (see Section 7.4). Furthermore, werequire that F ∗ i has r − u say, and that its othervertex class is a little larger than γu . But this condition will be satisfied automatically N ORE-TYPE THEOREM FOR PERFECT PACKINGS IN GRAPHS 17 by the choice of the sizes of the vertex classes in B ′ . In fact, this is the reason why wechose a B ′ -packing in R rather than a B ∗ -packing. The above strategy is based on thatin [13]. However, there are additional difficulties.7.2. Applying the Regularity lemma and modifying the reduced graph.
Wedefine further constants satisfying0 < ε ≪ d ≪ η ≪ β ≪ α ≪ λ ≪ η, γ, − γ. We also choose η so that η ≪ | B ′ | . Throughout the proof we assume that the order n of our graph G is sufficiently largefor our calculations to hold. Apply the Regularity lemma with parameters ε , d and ℓ := 1 /ε to obtain clusters V , . . . , V ℓ of size L , an exceptional set V , a pure graph G ′ and a reduced graph R . Let m := CE ( H ). By Lemma 16 we have that d R ( V j ) + d R ( V j ) ≥ max ( − r − m +2 + η ! | R | , (cid:18) − χ cr ( H ) + η (cid:19) | R | ) for all V j = V j ∈ V ( R ) with V j V j E ( R ). Together with (11) and (12) this impliesthat d R ( V j ) + d R ( V j ) ≥ (cid:18) − χ cr ( B ′ ) (cid:19) | R | for all V j = V j ∈ V ( R ) with V j V j E ( R ). So we can apply Theorem 4 to R to obtaina B ′ -packing covering all but at most η | R | vertices. We denote the copies of B ′ in thispacking by B ′ , . . . , B ′ ℓ ′ . We delete all the clusters not contained in some B ′ i from R andadd all vertices lying in these clusters to V . So | V | ≤ εn + η n ≤ η n . We now referto R as this modified reduced graph. We still have that d R ( V j ) + d R ( V j ) ≥ max ( − r − m +2 + η ! | R | , (cid:18) − χ cr ( H ) + η (cid:19) | R | ) (13)for all V j = V j ∈ V ( R ) with V j V j E ( R ). Recall that by definition of B ′ , each B ′ i contains a perfect B ∗ -packing. Fix such a B ∗ -packing for each i = 1 , . . . , ℓ ′ . The unionof all these B ∗ -packings gives us a perfect B ∗ -packing B ∗ in R .Given any B ′ i , it is easy to check that we can replace each cluster V j ∈ V ( B ′ i ) witha subcluster of size L ′ := (1 − ε | B ′ | ) L such that for each edge V j V j of B ′ i the chosensubclusters of V j and V j form a (2 ε, d/ G ′ . We do this for each i = 1 , . . . , ℓ ′ and add all the vertices not belonging to our chosen subclusters to V . Wenow refer to these subclusters as the clusters of R . Then for every edge V j V j of R thepair ( V j , V j ) G ′ is still 2 ε -regular and has density more than d/
2. Moreover,(14) | V | ≤ η n + ε | B ′ | n ≤ η n. We now partition each cluster V j into a red part V redj and a blue part V bluej where | | V redj | − | V bluej | | ≤ εL ′ and | | N G ( x ) ∩ V redj | − | N G ( x ) ∩ V bluej | | ≤ εL ′ for all x ∈ V ( G ).(Consider a random partition to see that there are V redj and V bluej with these properties.) Together all these partitions of the clusters yield a partition of V ( G ) − V into a set V red of red vertices and a set V blue of blue vertices. In Section 7.3 we will choose certaincopies of H in G to cover the exceptional vertices in V , but each of these copies willavoid the red vertices. All the vertices contained in these copies of H will be removedfrom the clusters they belong to. However, for every edge V j V j of B ′ i the modifiedbipartite subgraph of G ′ whose vertex classes are the remainders of V j and V j will stillbe (5 ε, d/ V redj ∪ V redj . Furthermore,all edges in R will still correspond to 5 ε -regular pairs of density more than d/
5. AfterSection 7.3 we will only remove a bounded number of further vertices from the clusters,which will not affect the super-regularity significantly.7.3.
Incorporating the exceptional vertices.
In this section we cover all the excep-tional vertices with vertex-disjoint copies of H . Let G blue denote the induced subgraphof G with vertex set V blue ∪ V . The definition of V blue , (2) and (14) together imply that d G blue ( x )+ d G blue ( y ) ≥ max ( − r − m +2 + η ! | G blue | , (cid:18) − χ cr ( H ) + η (cid:19) | G blue | ) for all non-adjacent x = y ∈ V ( G blue ). Let v , . . . , v | V | be an enumeration of theexceptional vertices. Lemma 13 gives us a copy H v of H in G blue covering v . Deletethe vertices of H v from G blue and apply the lemma again to find a copy H v of H covering v . We would like to continue this way. However, for later purposes it isconvenient to be able to assume that from each cluster we only delete a small proportionof vertices during this process. So before choosing the copy H v j for v j (say), we call B ′ i bad if it contains a cluster meeting the copies H v , . . . , H v j − that we have chosenbefore in at least βL ′ vertices. So at most | V || H | / ( βL ′ ) ≤ η | H | n/ ( βL ′ ) ≤ ηℓ ′ /
10 ofthe B ′ i are bad. We delete all the vertices belonging to clusters in bad B ′ i from G blue .Since there are at most ηn/ ≤ η | G blue | / H v j . Thus we can cover all the exceptional vertices. We remove all the verticeslying in the copies H v , . . . , H v | V | of H from the clusters they belong to (and from G ).7.4. Making the blow-up of each B ∈ B ∗ divisible by | H | . Given a subgraph S ⊆ R we write V G ( S ) for the set of all those vertices of G that belong to a clusterin S . Our aim now is to find, for each B ′ i in our B ′ -packing in R , an H -packing in G covering all the vertices in V G ( B ′ i ). Thus, taking the union of these H -packings and thecopies of H containing the vertices in V , we will obtain a perfect H -packing in G . Ifwe can ensure that the complete r -partite graph whose j th vertex class is the union ofall clusters in the j th vertex class of B ′ i has a perfect H -packing, then by the Blow-up lemma the subgraph of G ′ corresponding to B ′ i will have a perfect H -packing. ByLemma 14 the former will turn out to be the case provided that | H | divides | V G ( B ′ i ) | .So our next aim is to remove a bounded number of copies of H from G to ensure that | V G ( B ′ i ) | is divisible by | H | for all i = 1 , . . . , ℓ ′ . This in turn will be achieved by ensuringthat | H | divides | V G ( B ) | for all B ∈ B ∗ .Consider the auxiliary graph F whose vertices are the elements of B ∗ where B , B ∈B ∗ are adjacent in F if R contains a copy of K r with one vertex in B and r − B or vice versa. N ORE-TYPE THEOREM FOR PERFECT PACKINGS IN GRAPHS 19
Suppose first that F is connected. Consider a spanning tree T of F with root B ∈ B ∗ ,say. If B , B ∈ B ∗ are adjacent in F then by the Embedding lemma G contains a copyof H with one vertex in V G ( B ) and all the other vertices in V G ( B ), or vice versa. (Tosee this, let K ′ r be a copy of K r in R with one vertex V ∈ V R ( B ) and all other verticesin V R ( B ). Choose any V ′ ∈ V R ( B ) which is adjacent to all of V ( K ′ r ) \ { V } . Then ourcopy of H will have one vertex, v say, in V . All other vertices of H lying in the samecolour class as v will be embedded into V ′ and all the remaining vertices of H will beembedded into V ( K ′ r ) \ { V } .) In fact, we can choose | H | − H .So by removing at most | H | − H we can ensure | V G ( B ) | is divisibleby | H | .We can use this observation to ‘shift the remainders mod | H | ’ along T to achievethat | H | divides | V G ( B ) | for all B ∈ B ∗ as follows. Let j max be the largest distance ofsome B ∈ B ∗ from B in T . Then for all B ∈ B ∗ of distance j max from B we can removecopies of H as indicated above to ensure that | H | divides | V G ( B ) | . We can repeat thisfor all those B ∈ B ∗ of distance j max − B etc. until | V G ( B ) | is divisible by | H | for all B ∈ B ∗ . (This follows as P B ∈B ∗ | V G ( B ) | is divisible by | H | since | G | is divisibleby | H | .)So we may assume that F is not connected. Let C denote the set of all componentsof F . Given C ∈ C , we denote by V R ( C ) ⊆ V ( R ) the set of all those clusters whichbelong to some B ∈ B ∗ with B ∈ C . We write V G ( C ) ⊆ V ( G ) for the union of all theclusters in V R ( C ). We will show that we can remove a bounded number of copies of H from G to achieve that | V G ( C ) | is divisible by | H | for all C ∈ C . As in the case when F is connected, we can then ‘shift the remainders mod | H | ’ along a spanning tree of eachcomponent to make | V G ( B ) | divisible by | H | for all B ∈ B ∗ .In the case when r = 2 this is straightforward. Indeed, in this case H contains anisolated vertex (since CE ( H ) < ∞ ). So given any C ∈ C we can apply the Embeddinglemma to find | H | − H in G such that one vertex (playing therole of the isolated vertex) lies in V G ( C ) and the other vertices lie in V G ( C ′ ) for some C ′ ∈ C \ { C } . By removing a suitable number of such copies we can ensure that | H | divides | V G ( C ) | . Since in the above argument we can choose any C ′ ∈ C \ { C } to containthe remaining vertices of our copy of H (and since | G | is divisible by | H | ) we can applythis argument repeatedly to make | V G ( C ′′ ) | divisible by | H | for all C ′′ ∈ C .So now we consider the case when r ≥
3. We need the following claim.
Claim 20.
Let C , C ∈ C and let V ∈ V R ( C ) . Then | N R ( V ) ∩ V R ( C ) | < (cid:18) − r − γ (cid:19) | V R ( C ) | . Proof.
Suppose not. Then there exists some B ∈ B ∗ such that B ∈ C and | N R ( V ) ∩ B | ≥ (cid:18) − r − γ (cid:19) | B | = | B | − ( r − z + z r − z /z = | B | − z. Hence V has a neighbour in at least r − B . So R contains a copy of K r with one vertex, namely V , in a copy B ∈ B ∗ and r − B . So B and B areadjacent in F . But they lie in different components of F , a contradiction. (cid:3) We now show that we can remove a bounded number of copies of H from G tomake | V G ( C ) | divisible by | H | for some C ∈ C . (In particular, if F consists of exactlytwo components C and C ′ this also ensures that | V G ( C ′ ) | is divisible by | H | .) Claim 21.
There exists a component C ∈ C with | V R ( C ) | ≤ | R | / for which we canensure that | H | divides | V G ( C ) | by removing at most | H | − copies of H from G . Proof.
To prove the claim we will distinguish two cases.
Case 1.
There exists a component C ∈ C with | V R ( C ) | ≤ | R | / and such that there isa cluster V ∈ V R ( C ) with d R ( V ) ≥ (1 − /χ cr ( H ) + η/ | R | . Recall that K − r +1 is a K r +1 with one edge removed. We call the two non-adjacent verticesof K − r +1 small . We say that a copy K ′ of K − r +1 in R is good if either (i) V ( K ′ ) ∩ V R ( C )consists of a small vertex of K ′ or (ii) V ( K ′ ) \ V R ( C ) consists of a small vertex of K ′ .Once we have found a good K ′ , we can use the Embedding lemma to find at most | H |− H in G such that their removal from G ensures that | V G ( C ) | is divisible by | H | , as desired. (In case (i) precisely one vertex in each of these copiesof H lies in V G ( C ) while in case (ii) precisely | H | − H lies in V G ( C ).) So it suffices to find a good copy of K − r +1 .Let S denote the set of neighbours of V outside V R ( C ) in R . Let K be the set ofvertices V ∈ S with d R ( V ) < (1 − /χ cr ( H ) + η/ | R | . By (13), K induces a clique in R . If | K | ≥ r , then we have a found a good copy of K − r +1 (consisting of V and r verticesof K ). So we may assume that | K | < r .Since r ≥ d R ( V ) ≥ (1 / η/ | R | . So | S \ K | ≥ η | R | / − r >
0. Thuswe can choose V ∈ S \ K . By (11) the number of common neighbours of V and V in R is at least(15) (cid:18) − r − γ + η (cid:19) | R | . We first consider the case when at least (1 − r − γ + η ) | V ( R ) \ V R ( C ) | common neigh-bours of V and V lie outside V R ( C ). We claim that we can find V , . . . , V r ∈ S \ K whichform a K r with V and V . Suppose that we have found V , . . . , V i where 2 ≤ i ≤ r − S imply that for j ≥ V j outside V R ( C ) is at least (1 − / ( r − γ )) | V ( R ) \ V R ( C ) | . Togetherwith (15), this implies that the common neighbourhood of V , . . . , V i outside V R ( C ) hassize at least(16) (cid:18) − ir − γ + η (cid:19) | V ( R ) \ V R ( C ) | ≥ η | V ( R ) \ V R ( C ) | > r > | K | . This shows that we can find V i +1 and more generally V , . . . , V r as required. A similarcalculation as in (16), shows that the common neighbourhood of V , . . . , V r outside V R ( C ) is non-empty and so contains some vertex V r +1 say. Together with V , . . . , V r , V r +1 forms a good copy of K − r +1 .Now consider the case when at least (1 − r − γ + η ) | V R ( C ) | common neighboursof V and V lie inside V R ( C ). Since η | V R ( C ) | / ≥ η | B ∗ | / ≥ r we can argue as in theprevious case. Indeed, this time we choose V , . . . , V r inside V R ( C ) to obtain a copyof K r in R with one vertex, namely V , outside V R ( C ). We also choose a vertex V r +1N ORE-TYPE THEOREM FOR PERFECT PACKINGS IN GRAPHS 21 inside V R ( C ) that is adjacent to V , V , . . . , V r . Again, V , . . . , V r +1 form a good copyof K − r +1 . Case 2.
Every component C ∈ C with | V R ( C ) | ≤ | R | / is such that d R ( V ) < (1 − /χ cr ( H ) + η/ | R | for all V ∈ V R ( C ) . Together with (13) this implies that V V ∈ E ( R ) for all V ∈ V R ( C ), V ∈ V R ( C )where C , C ∈ C are such that | V R ( C ) | , | V R ( C ) | ≤ | R | /
2. But this means that thereis only one component C ′ ∈ C with | V R ( C ′ ) | ≤ | R | /
2. So F consists of precisely twocomponents C ′ and C ′′ where V R ( C ′ ) forms a clique in R and | V R ( C ′′ ) | > | R | / . We first consider the case when r = 3. Note that R contains an edge between V R ( C ′ )and V R ( C ′′ ). Indeed, if not then for any V ′ ∈ V R ( C ′ ) and V ′′ ∈ V R ( C ′′ ) by (13) we havethat d R ( V ′ ) + d R ( V ′′ ) ≥ − /χ cr ( H ) + η/ | R | > | R | and so there must be an edgefrom V ′ to V R ( C ′′ ) or from V ′′ to V R ( C ′ ), a contradiction.So since | V R ( C ′ ) | ≥ | B ∗ | ≥ r + m we have a copy K ′ r + m of K r + m in V R ( C ′ ) such thatthere is a cluster V ′′ ∈ V R ( C ′′ ) adjacent to one of the clusters, V ′ say, of K ′ r + m . Usingthe definition of m and the Embedding lemma we can find at most | H | − H in G each containing precisely one vertex in V G ( C ′′ ) such that their removal ensuresthat | H | divides | V R ( C ′ ) | and thus also | V R ( C ′′ ) | . (Indeed, by definition of m thereexists a vertex y of H such that χ ( H [ N ( y )]) = r − N ( y ) can be extended to an ( r + m )-colouring of H . So in our copies of H the vertex y will lie in V ′′ , N ( y ) will lie in V ′ and the remaining vertices of H will lie in V ( K ′ r + m ).)Now suppose that r ≥
4. We claim that there exists V ′′ ∈ V R ( C ′′ ) which sendsat least r edges to V R ( C ′ ) in R . Suppose not. Then no V ∈ V R ( C ′′ ) is joined toall of V R ( C ′ ). Together with the definition of C ′ and (13) this implies that d R ( V ) ≥ (1 − /χ cr ( H ) + η/ | R | . But then | V R ( C ′ ) | < | R | /χ cr ( H ) since otherwise V is joined to η | R | / ≥ r vertices in V R ( C ′ ). By assumption there are less than r | V R ( C ′′ ) | < r | R | edgesbetween V R ( C ′ ) and V R ( C ′′ ) in R . Moreover, by (13) and since | V R ( C ′ ) | < | R | /χ cr ( H )every cluster in V R ( C ′ ) sends at least (1 − /χ cr ( H )+ η/ | R | > η | R | / V R ( C ′′ ).So η | R || V R ( C ′ ) | / < r | R | . But | V R ( C ′ ) | ≥ | B ∗ | ≥ r/η by definition of B ∗ and so η | R || V R ( C ′ ) | / ≥ r | R | , a contradiction. So indeed there exists a vertex V ′′ ∈ V R ( C ′′ )sending at least r edges to V R ( C ′ ). As before, we can remove at most | H | − H from G to ensure that | H | divides both | V R ( C ′ ) | and | V R ( C ′′ ) | . (cid:3) Claim 22.
We can make | V G ( B ) | divisible by | H | for all B ∈ B ∗ by removing at most |B ∗ || H | copies of H from G . Proof.
Our first aim is to take out some copies of H in G to achieve that | V G ( C ) | isdivisible by | H | for each C ∈ C . We apply Claim 21 to remove at most | H | − H from G to ensure that | V G ( C ) | is divisible by | H | for some component C ∈ C with | V R ( C ) | ≤ | R | /
2. Next we consider the graphs F := F − V ( C ) and R := R − V R ( C )instead of F and R . Claim 20 and (13) together imply that d R ( V j ) + d R ( V j ) ≥ (cid:18) − r − γ + η (cid:19) | R | for all V j = V j ∈ V ( R ) with V j V j E ( R ). Now suppose that |C| ≥
3. Thensimilarly as in the proof of Claim 21 we can find a component C ∈ C with | V R ( C ) | ≤ | R | / | H | − H from G we ensure that | H | divides | V G ( C ) | . As | G | was divisible by | H | we can continue in this fashion to achievethat | V G ( C ) | is divisible by | H | for each C ∈ C .During this process we have to take out at most ( |C| − | H | −
1) copies of H in G .Now consider each C ∈ C separately. By proceeding as in the connected case for each C and taking out at most ( | C | − | H | −
1) further copies of H in each case, we can make | V G ( B ) | divisible by | H | for all B ∈ B ∗ . Hence, in total we have taken out at most( |C| − | H | −
1) + ( |B ∗ | − |C| )( | H | − ≤ |B ∗ || H | copies of H . (Note that |B ∗ || H | isalso an upper bound on the number of copies of H removed from G in the case when r = 2.) (cid:3) Applying the Blow-up lemma.
We now consider all the copies B ′ , . . . , B ′ ℓ ′ of B ′ in the B ′ -packing of R , where the vertices of R are the modified clusters (i.e. they donot contain the vertices contained in the copies of H removed in Sections 7.3 and 7.4).For each i ≤ ℓ ′ let G ′ i denote the r -partite subgraph of G ′ whose j th vertex class isthe union of all the clusters lying in the j th vertex class of B ′ i (for j = 1 , . . . , r ). InSection 7.4 we made | G ′ i | = | V G ( B ′ i ) | divisible by | H | for each i . Moreover, in Section 7.3we removed at most βL ′ vertices from each cluster. In Section 7.4 we removed only abounded number of further vertices. So altogether we removed at most 2 βL ′ verticesfrom each cluster. Since β ≪ λ ≪ γ, − γ we may apply Lemma 14 to conclude that thecomplete r -partite graph whose vertex classes are the same as the vertex classes of G ′ i has a perfect H -packing.We observed at the end of Section 7.2 that the choice of those copies of H removed inSection 7.3 ensures that all the bipartite subgraphs corresponding to edges of B ′ i are still(5 ε, d/ G ′ i are still(6 ε, d/ i = 1 , . . . , ℓ ′ , we may apply the Blow-up lemmato find a perfect H -packing in G ′ i . All these H -packings together with the copies of H chosen previously form a perfect H -packing in G , as desired. References [1] N. Alon and R. Yuster, H -factors in dense graphs, J. Combin. Theory B (1996), 269–282.[2] R. Diestel, Graph Theory , Graduate texts in Mathematics 173 (3rd Edition), Springer-Verlag 2005.[3] A. Hajnal and E. Szemer´edi, Proof of a conjecture of Erd˝os,
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Computer Science Review (2007), 12–26.Daniela K¨uhn, Deryk Osthus & Andrew TreglownSchool of MathematicsUniversity of BirminghamEdgbastonBirminghamB15 2TTUK E-mail addresses: { kuehn,osthus,treglowa }}