The extremal function for partial bipartite tilings

(1)

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European Journal of Combinatorics

journal homepage:www.elsevier.com/locate/ejc

The extremal function for partial bipartite tilings

Codruţ Grosu

a

, Jan Hladký

b,c

a_{Faculty of Automatic Control and Computers, Politehnica University of Bucharest, Splaiul Independenţei 313, sector 6, Cod 060042,}

Bucharest, Romania

b_{Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Malostranské náměstí 25, 118 00,}

Prague, Czech Republic

c_{DIMAP, University of Warwick, Coventry, CV4 7AL, United Kingdom}

a r t i c l e i n f o Article history:

Available online 14 October 2011

a b s t r a c t

For a fixed bipartite graphHand givenα ∈(0,1), we determine the thresholdTH(α)which guarantees that anyn-vertex graph with

at leastTH(α)

n

2 

edges contains(1−o(1))_v(α_H₎nvertex-disjoint copies ofH. In the proof, we use a variant of a technique developed by Komlós [J. Komlós, Tiling Turán theorems, Combinatorica 20 (2) (2000) 203–218].

1. Introduction

The Turán theorem [12], one of the most important results in extremal graph theory, gives a sharp threshold, denoted by ex

(

n

,

Kr

)

, for the maximum number of edges of ann-vertex graph with no copy

ofKr. Even though the Turán theorem applies to any pair of valuesnandr, the interesting instances

are rather those whennis large compared tor. Erdős and Stone [2] extended the result by determining the asymptotic behavior of the function ex

(

n

,

H

)

for a fixed non-bipartite graphH. The same problem in the case thatHis a fixed bipartite graph is – despite considerable effort – wide open for most graphs

H. This is known as theZarankiewicz problem. Let us recall that whenHhas color classes of sizessand

t,s

≤

t, then the Kövari–Sós–Turán theorem [8] asserts that

ex

(

n

,

H

)

≤

O

(

n2−1/s

)

=

o

(

n2

).

(1)

On the other hand, a standard random graph argument gives that ex

(

n

,

Ks,t

)

≥

Ω

(

n2−(s+t−2)/(st−1)

)

.

It is natural to extend the aboveexistential questionstotiling questions. In such a setting, one asks for the maximum number of edges of ann-vertex graph which does not contain

ℓ

vertex-disjoint copies of a graphH. This quantity is denoted by ex

(

n

, ℓ

×

H

)

. Erdős and Gallai [3] gave a complete solution to the problem in the case whenH

=

K2.

(2)

Theorem 1 (Erdős–Gallai [3]).Suppose that

ℓ

≤

n

/

2. Then ex

(

n

, ℓ

×

K2

)

=

max



(ℓ

−

1

)(

n

−

ℓ

+

1

)

+



ℓ

₋

1 2



,



2

ℓ

−

1 2



.

Givenn

,

x

∈

_N,x

≤

n, we define two graphsMn,xandLn,xas follows. The graphMn,xis ann-vertex

graph whose vertex set is split into setsAandB,

|

A

| =

x

,

|

B

| =

n

−

x,Ainduces a clique,Binduces an independent set, andMn,x

[

A

,

B

] ≃

Kx,n−x. The graphLn,xis the complement ofMn,n−x, i.e., it is ann

-vertex graph whose edges induce a clique of orderx. Obviously,e

(

Mn,ℓ−1

)

=

(ℓ

−

1

)(

n

−

ℓ

+

1

)

+



_ℓ −1 2



, ande

(

Ln,2ℓ−1

)

=



2ℓ−1 2



. Moreover, it is easy to check that there are no

ℓ

vertex-disjoint edges in either of the graphsMn,ℓ−1,Ln,2ℓ−1. Therefore, when

ℓ <

2₅n

+

O

(

1

)

, the graphMn,ℓ−1is (the unique)

graph showing that ex

(

n

, ℓ

×

K2

)

≥

(ℓ

−

1

)(

n

−

ℓ

+

1

)

+



_ℓ

−1 2



. The graphLn,2ℓ−1is the unique

extremal graph for the problem otherwise.

Moon [10] started the investigation of ex

(

n

, ℓ

×

Kr

)

. Allen et al. [1] only recently determined the

behavior of ex

(

n

, ℓ

×

Kr

)

for the whole range of

ℓ

in the caser

=

3, and they made a substantial

progress for larger values ofr. Simonovits [11] determined the value ex

(

n

, ℓ

×

H

)

for a non-bipartite graphH, fixed value of

ℓ

and largen.

An equally important density parameter which can be considered in the context of tiling questions is the minimum degree of the host graph. That is, we ask what is the largest possible minimum degree of ann-vertex graph which does not contain

ℓ

vertex-disjoint copies ofH. In the caseH

=

Kr, the

precise answer is given by the Hajnal–Szemerédi theorem1_[₄_{]. An asymptotic threshold for a general}

fixed graphHwas determined by Komlós [5]. In this case, the threshold depends on a parameter which Komlós calls thecritical chromatic number. The critical chromatic number ofHis a real between

χ(

H

)

−

1 and

χ(

H

)

, defined by

χ

cr

(

H

)

:=

(χ(

H

)

−

1

)v(

H

)

v(

H

)

−

σ

,

(2)

where

σ

is the size of the smallest possible color-class in any

χ(

H

)

-coloring ofH. Let us also note that Komlós’ result [5] gives an asymptotic min-degree threshold even in the case whenHis bipartite. In this case the near-extremal graphs for the problem are complete bipartite graphs.

In the present paper we use a variation of the technique developed by Komlós to determine the asymptotic behavior of the function ex

(

n

, ℓ

×

H

)

for a fixed bipartite graphH. LetHbe an arbitrary bipartite graph. Suppose thatb:V

(

H

)

→ [

2

]

is a proper coloring ofHwhich minimizes

|

b−1

(

1

)

|

. We define quantitiess

(

H

)

:= |

b−1

₍

₁

₎

_|

_,_t

₍

_H

₎

_{:= |}

_b−1

₍

₂

₎

_|

_{. Obviously,}_s

₍

_H

₎

_≤

_t

₍

_H

₎

_{, and}_s

₍

_H

₎

₊

_t

₍

_H

₎

₌

_v(

_H

₎

_.

Furthermore, we defineV1

(

H

)

:=

b−1

(

1

)

andV2

(

H

)

:=

b−1

(

2

)

. The setsV1

(

H

)

andV2

(

H

)

are uniquely

defined provided thatHdoes not contain a balanced bipartite graph as one of its components; in this other case we fix a coloringbsatisfying the above conditions and use it to define uniquelyV1

(

H

)

and V2

(

H

)

.

Givens

,

t

∈

_N, we define a functionTs,t:

(

0

,

1

)

→

(

0

,

1

)

by setting T_s_,_t

(α)

:=

max



2s

α

s

+

t



1

−

s

α

2

(

s

+

t

)



, α

2



,

(3)

for

α

∈

(

0

,

1

)

. Note thatTs′_,_t′

=

T_s_,_twhens′

=

ksandt′

=

kt. Also, note that

Ts,s

(α)



n 2



=

ex



n

,

α

n 2

×

K2



+

o

(

n2

),

(4)

1 In its original formulation, the Hajnal–Szemerédi theorem asserts that ann-vertex graphGwith minimum-degree at least r−1

r ncontains aKr-tiling missing at mostr−1 vertices ofG, thus giving an answer only to the question of almost perfect tilings. When the minimum-degree ofGis lower, we can however add auxiliary vertices which are complete toGand obtain ann′_{-vertex graph}_G′_{such that the Hajnal–Szemerédi theorem applies to}_G′_{. The restriction of the almost perfect}_Kr_{-tiling of}_G′ toGgives aKr-tiling which is optimal in the worst case.

(3)

and, in general, fors

≤

t, the numberTs,t

(α)



n

2



is asymptotically the maximum between the number of edges ofMn,αs

s+tnandLn,αn.

Our main result is the following.

Theorem 2. Suppose that H is a bipartite graph with no isolated vertices, s

:=

s

(

H

),

t

:=

t

(

H

)

. Let

α

∈

(

0

,

1

)

and

ε >

0. Then there exists an n0

=

n0

(

s

,

t

, α, ε)

such that for any n

≥

n0, any graph G with n vertices and at least Ts,t

(α)



n 2



edges contains more than

(

1

−

ε)

α

s+tn vertex-disjoint copies of the graph H.

After some remarks below, we introduce the tools needed for our proof ofTheorem 2in Section2. The proof is then given in Section3.

(1) LetH

,

sandtbe as in the hypothesis of the theorem,

ε

′

_>

_{0 and}

_β

_∈

₍

₀

_,

₁

₎

_{. Then we may find}

an

α > β(

s

+

t

)

and an

ε < ε

′sufficiently small, such that fornlarge enough, byTheorem 2, any graph

Gwithnvertices and at leastTs,t

(α)



n 2



< (

Ts,t

(β(

s

+

t

))

+

ε

′

)



n 2



edges contains at least

β

n vertex-disjoint copies ofH. Hence ex

(

n

, β

n

×

H

)

≤

Ts,t

(β(

s

+

t

))



n 2



+

ε

′_n2_{. This asymptotically matches the}

lower bound which comes – as inTheorem 1– from graphsMn,βsn−1andLn,β(s+t)n−1. Indeed, neither of

these graphs contains

β

nvertex-disjoint copies ofH, as any such copy would require at leastsvertices in the clique subgraph ofMn,βsn−1, and at leasts

+

t

=

v(

H

)

non-isolated vertices inLn,β(s+t)n−1,

respectively. Note however that for most values ofH, the graphsMn,βsn−1andLn,β(s+t)n−1are not

extremal for the problem. For example, we can replace the independent set in the graphLn,β(s+t)n−1

by anyH-free graph. This links us to the Zarankiewicz problem, and suggests that an exact result is not within the reach of current techniques.

(2) Note thatTheorem 2could be restated using the notion of critical chromatic number. Indeed, formula Eq.(2)simplifies in the bipartite setting to

χ

cr

(

H

)

=

v(

H

)

t

(

H

)

.

In other words, the valueTs(H),t(H)

(α)

defined by Eq.(3)can be determined from

χ

cr

(

H

)

, without

knowings

(

H

)

andt

(

H

)

.

(3) The assumption onHto contain no isolated vertices inTheorem 2is made just for the sake of compactness of the statement. Indeed, letH′_{be obtained from}_H_{by removing all the isolated vertices.}

Then there is a simple relation between the sizes of optimal coverings by vertex disjoint copies ofH

andH′_{in an}_n_{-vertex graph}_G_{. Let}_x_and_x′_{be the number of vertices covered by a maximum family of}

vertex-disjoint copies ofHandH′_in_G_{, respectively. We have that}

x

=

min



v(

H

)



n

v(

H

)



,

x ′

_v(

H

)

v(

H′

)



.

(4) One can attempt to obtain an analogue ofTheorem 2for graphs with higher chromatic number. This however appears to be substantially more difficult. To indicate the difficulty, let us recall that there are two types (Mn,xandLn,x) of extremal graphs for theH-tiling problem for bipartiteH. The

graphsMn,x andLn,xhave ablock structure, i.e., their vertex set can be partitioned intoblocks(two,

in this case), such that any two vertices from the same block have almost the same neighborhoods. These two graphs appear even in the simplest case ofH

=

K2(cf.Theorem 1). However, whenHis

not balanced, if we let

α

go from 0 to 1, the transition between the two extremal structures which determine the threshold function occurs at a different time in the evolution. On the other hand, there are five types of extremal graphs for the problem of determining ex

(

n

, ℓ

×

K3

)

as shown in [1]. All the

five types have a block structure. It is plausible that whenHis a general 3-colorable graph, the same five types of extremal graphs determine the threshold function forH-tilings. However, the transitions between them occur at different times and the block sizes depend on various structural properties ofH. In particular, we have indications that the critical chromatic number alone does not determine ex

(

n

, α

n

×

H

)

in this situation.

IfF is a family of graphs, andGis a graph, anF-tilinginGis a set of vertex-disjoint subgraphs of

(4)

the vertices ofGcovered by anF-tilingF, and

|

F

| = |

V

(

F

)

|

is thesizeof the tilingF. IfFis a collection of bipartite graphs, we letV1

(

F

)

=



H∈FV1

(

H

)

andV2

(

F

)

=



H∈FV2

(

H

)

. Forn

∈

N, we write

[

n

]

to

denote the set

{

1

,

2

, . . . ,

n

}

.

2. Tools for the proof of the main result

Our main tool is Szemerédi’s Regularity Lemma (see [7,9] for surveys). To state it we need some more notation.

LetG

=

(

V

,

E

)

be ann-vertex graph. IfA

,

Bare disjoint nonempty subsets ofV

(

G

)

, thedensityof the pair

(

A

,

B

)

isd

(

A

,

B

)

=

e

(

A

,

B

)/(

|

A

∥

B

|

)

. We say that

(

A

,

B

)

is an

ε

-regular pairif

|

d

(

X

,

Y

)

−

d

(

A

,

B

)

|

<

ε

for everyX

⊂

A

,

|

X

|

> ε

|

A

|

andY

⊂

B

,

|

Y

|

> ε

|

B

|

.

The following statement asserts that large subgraphs of regular pairs are also regular.

Lemma 3. Let

(

A

,

B

)

be an

ε

-regular pair with density d, and let A′

_⊂

_A

_,

_|

_A′

_{| ≥}

_α

_|

_A

_|

_,

_B′

_⊂

_B

_,

_|

_B′

_{| ≥}

_α

_|

_B

_|

_,

α

≥

ε

. Then

(

A′

_,

_B′

₎

_{is an}

_ε

′_{-regular pair with}

_ε

′

₌

_max

_{

_ε/α,

₂

_ε

_}

_{, and for its density d}′_{we have}

_|

_d′

₋

_d

_|

< ε

.

Let

ε >

0 andd

∈ [

0

,

1

]

. An

(ε,

d

)

-regular partitionofGwithreduced graph R

=

(

V′

_,

_E′

₎

_{is a}

partitionV0

∪

˙

V1

∪ · · · ˙

˙

∪

VkofVwith

|

V0

| ≤

ε

n,

|

Vi

| = |

Vj

|

for any 1

≤

i

<

j

≤

k,V

(

R

)

= {

V1

,

V2

, . . . ,

Vk

}

, such that

(

Vi

,

Vj

)

is an

ε

-regular pair inGof density greater thandwheneverViVj

∈

E

(

R

)

, and

the subgraphG′

_⊂

_G_{induced by the}

_ε

_{-regular pairs corresponding to the edges of}_R_{has more than}

e

(

G

)

−

(

d

+

3

ε)

n2

_/

_{2 edges. In this case, we also say that}_G_{has an}

_(ε,

_d

₎

_{-reduced graph}_R_{, and call the}

setsVi

,

1

≤

i

≤

k, theclustersofG.

The following lemma is a consequence of the so-called degree version of the Regularity Lemma [7, Theorem 1.10].

Lemma 4 (Regularity Lemma).For every

ε >

0and m

∈

_Nthere is an M

=

M

(ε,

m

)

such that, if G is any graph with more than M vertices and d

∈ [

0

,

1

]

is any real number, then G has an

(ε,

d

)

-reduced graph R on k vertices, with m

≤

k

≤

M.

Given four positive numbers a

,

b

,

x

,

y we say that the pair a

,

b dominates the pair x

,

y, if max

{

x

,

y

}

/

min

{

x

,

y

} ≥

max

{

a

,

b

}

/

min

{

a

,

b

}

. The following easy lemma states thatKa,bhas an almost

perfectKs,t-tiling provided thata

,

bdominatess

,

t.

Lemma 5. For any s

,

t

∈

_Nthere exists a constant C such that the following holds. Suppose that the pair a

,

b

∈

_Ndominates s

,

t. Then the graph Ka,bcontains a Ks,t-tiling containing all but at most C vertices of Ka,b.

Proof. Ifs

=

tthen necessarilya

=

b. There obviously exists aKs,t-tiling containing all but at most C

:=

2

(

s

−

1

)

vertices ofKa,b.

With no loss of generality, we may suppose thata

≤

bands

<

t. Thenas

≤

btandbs

≤

at. A tiling with

⌊

(

bt

−

as

)/(

t2

₋

_s2

₎

_⌋

_{copies of}_K

s,twith thes-part of theKs,tplaced in thea-part of theKa,b

and

⌊

(

at

−

bs

)/(

t2

−

s2

)

⌋

copies placed the other way misses at mostC

:=

2

(

s

+

t

−

1

)

vertices of

Ka,b.

The next lemmas, versions of the Blow-up Lemma [6], assert that regular pairs have almost as good tiling properties as complete bipartite graphs.

Lemma 6. For every d

>

0

, γ

∈

(

0

,

1

)

and any two graphs R and H, there is an

ε

=

ε(

H

,

d

, γ ) >

0such that the following holds for all positive integers s. Let Rsbe the graph obtained from R by replacing every ver-tex of R by s vertices, and every edge of R by a complete bipartite graph between the corresponding s-sets. Let G be any graph obtained similarly from R by replacing every vertex of R by s vertices, and every edge of R with an

ε

-regular pair of density at least d. If Rscontains an H-tiling of size at least

γ v(

Rs

)

then so does G.

Lemma 7. For every bipartite graph H and every

γ ,

d

>

0there exists an

ε

=

ε(

H

,

d

, γ ) >

0such that the following holds. Suppose that there is an H-tiling in Ka,bof size x. Let

(

A

,

B

)

be an arbitrary

ε

-regular pair with density at least d,

|

A

| =

a,

|

B

| =

b. Then the pair

(

A

,

B

)

contains an H-tiling of size at least x

−

γ (

a

+

b

)

.

(5)

Finally, let us state a straightforward corollary of the König matching theorem.

Fact 8. Let G

=

(

A

∪

˙

B

,

E

)

be a bipartite graph with color classes A and B. If G has no matching with l

+

1

edges, then e

(

G

)

≤

lmax

{|

A

|

,

|

B

|}

. 3. The proof

In this section, we first state and prove the main technical result,Lemma 9. Then, we show how it impliesTheorem 2.

Fors

,

t

∈

_N, we setF1

:= {

Ks,t

,

Ks,t−1

,

K2

}

andF2

:= {

Kst,t2

,

Kst−1,(t−1)t

,

Kst,(t−1)t

,

K2

}

. Let us note

that whens

<

t, the sizes of the two color classes of any graph fromF∗

_:=

_F

1

∪

F2dominatesandt.

LetFbe aKs,t-tiling in a graphG,s

<

t. SupposeE0andE1are matchings inG

[

V

(

G

)

−

V

(

F

),

V1

(

F

)

]

andG

[

V2

(

F

)

]

, respectively, such that each copyKofKs,tinFhas at most one vertex matched byE0and

at most one vertex matched byE1. If anyK

∈

Fwhich has a vertex matched byE0, also has a vertex

matched byE1, then we call the pair

(

E0

,

E1

)

anF-augmentation. Note that in this caseE0andE1are

vertex disjoint, asV1

(

F

)

∩

V2

(

F

)

= ∅

.

The main step in our proof ofTheorem 2is the following lemma.

Lemma 9. Let t

>

s

≥

1

, α

∈

(

0

,

1

)

and

ε >

0. Then there exists an

ε

′

₌

_ε

′

₍

_s

_,

_t

_{, α, ε) >}

₀_{and an}

h

=

h

(

s

,

t

, α, ε) >

0such that the following holds. Suppose G is an n-vertex graph with n

≥

h and e

(

G

)

≥

Ts,t

(α)



n 2



, and F is a Ks,t-tiling in G of maximum size with

|

F

| ≤

(

1

−

ε)α

n. Then one of the following is true:

(i) there exists anF1-tiling F′in G with

|

F′

| ≥ |

F

| +

ε

′n, or

(ii) there exists an F -augmentation

(

E0

,

E1

)

such that E0contains at least

ε

′n edges.

Proof. Set

ε

′

:=

1 4min



εα

2 3t

+

1

,

ε

s

α

(

3t

+

1

)(

s

+

t

)



,

and lethbe sufficiently large.

Suppose for a contradiction that the assertions of the lemma are not true.

SetL

:=

V

(

G

)

−

V

(

F

)

andm

:= |

L

|

. LetC

:= {

V1

(

K

)

:K

∈

F

}

,

D

:= {

V2

(

K

)

:

K

∈

F

}

and C

:=



C

,

D

:=



D. We call members ofClilliputswhile members ofDaregiants. We say that giantV2

(

K

)(

K

∈

F

)

iscoupledwith lilliputV1

(

K

)

.

AsFis a maximum sizeKs,t-tiling inG, by(1)we have that

e

(

G

[

L

]

)

=

o

(

n2

).

(5)

Letrbe the number of copies ofKs,tinF. Thenr

≤

(

1

−

ε)α

n

/(

s

+

t

)

. Moreover, we have

m

=

n

−

(

s

+

t

)

r

.

(6)

Let us define an auxiliary graphH

=

(

V′

_,

_E′

₎

_{as follows. The vertex-set of}_H_is_V′

_:=

_C

_∪

_D

_∪

_L_{. For}

anyx

∈

LandK

∈

Fthe edgexV1

(

K

)

belongs toE′iff NG

(

x

)

∩

V1

(

K

)

̸= ∅

. Similarly, the edgexV2

(

K

)

belongs toE′iff NG

(

x

)

∩

V2

(

K

)

̸= ∅

. Finally, for any distinctK

,

K′

∈

Fthe edgeV2

(

K

)

V2

(

K′

)

belongs

toE′_iff_E

G

(

V2

(

K

),

V2

(

K′

))

̸= ∅

. The verticesLand the verticesCinduce two independent sets inH.

As (i) does not hold,H

[

L

,

D

]

does not contain a matching with at least

ε

′_n_{edges. It follows from}

Fact 8that

eG

(

L

,

D

)

≤

ε

′

ntmax

{

m

,

r

} ≤

t

ε

′n2

.

(7)

LetMbe a maximum matching inH

[

L

,

C

]

withledges. Obviously,l

≤

r. ByFact 8, we have that

eG

(

L

,

C

)

≤

lsmax

{

m

,

r

}

.

(8)

LetC′

⊆

Cbe the lilliputs matched byM. We writeD′

⊆

Dfor the giants coupled withC′. Set

D′

₌



D′_.

(6)

Suppose for a moment thatH

[

D′

_{] ∪}

_H

_[

_D′

_,

_D

₋

_D′

_]

_{contains a matching}_T_{with at least}

_ε

′_n_edges.

LetD′′be the giants inD′matched byT andM′the set of edges inMmatching the lilliputs coupled withD′′_{. Then}_M′_and_T_{give rise to an}_F_{-augmentation}

₍

_E

0

,

E1

)

inGwith

|

E0

| = |

M′

| ≥ |

T

| ≥

ε

′n,

contradicting our assumption that (ii) does not hold.

SoH

[

D′

_]∪

_H

_[

_D′

_,

_D

₋

_D′

_]

_{does not contain a matching with at least}

_ε

′_n_{edges. Applying}_{Theorem 1}

and passing to the graphG, we get

e

(

G

[

D′

] ∪

G

[

D′

,

D

−

D′

]

)

≤

t2ex

(

r

, ε

′n

×

K2

)

+

r



t 2



≤

2t2

ε

′nr

+

r



t 2



.

Therefore, e

(

G

[

C

∪

D

]

)

=

e

(

G

[

D′

] ∪

G

[

D′

,

D

−

D′

]

)

+

e

(

G

[

D

−

D′

]

)

+

e

(

G

[

C

]

)

+

eG

(

C

,

D

)

≤

2t2

ε

′nr

+

r



t 2



+



(

r

−

l

)

t 2



+



rs 2



+

r2st

.

(9)

Summing up the bounds(5)and(7)–(9)we get:

e

(

G

)

=

e

(

G

[

L

]

)

+

eG

(

L

,

D

)

+

eG

(

L

,

C

)

+

e

(

G

[

C

∪

D

]

)

≤

o

(

n2

)

+

t

ε

′n2

+

lsmax

{

m

,

r

} +

2

ε

′nrt2

+

r



t 2



+



(

r

−

l

)

t 2



+



rs 2



+

r2st

.

Using the convexity off

(

l

)

:=

lsmax

{

m

,

r

} +



₍ r−l)t

2



on

[

0

,

r

]

, and the fact thatrt

≤

n, we get:

e

(

G

)

≤

o

(

n2

)

+

3t

ε

′n2

+

r



t 2



+

r2st

+



rs 2



+

max



rt 2



,

rsmax

{

m

,

r

}



.

However,r2_s

_≤



rt 2



+

o

(

n2

₎

_{, and hence from}₍₆₎_{we get:} e

(

G

)

≤

o

(

n2

)

+

3t

ε

′n2

+

r



t 2



+

r2st

+



rs 2



+

max



rt 2



,

rs

(

n

−

(

s

+

t

)

r

)



<

max



(

s

+

t

)

r 2



,



rs 2



+

rs

(

n

−

rs

)



+

(

3t

+

1

)ε

′n2

,

where in the last inequality we have majorized the termr



₂t



+

o

(

n2

₎

_by

_ε

′_n2_{. But}



(

s

+

t

)

r 2



+

(

3t

+

1

)ε

′n2

≤



(

1

−

ε)α

n 2



+

εα

2_n2 4

<



1

−

ε

2



α

2n2 2

,

and



rs 2



+

rs

(

n

−

rs

)

+

(

3t

+

1

)ε

′n2

<

rsn

−

r 2_s2 2

+

ε

s

α

n2 4

(

s

+

t

)

≤

2s

α

s

+

t



1

−

α

s 2

(

s

+

t

)

+

ε(

2

−

ε)α

s 2

(

s

+

t

)

−

3

ε

4



n2 2

<

2s

α

s

+

t



1

−

α

s 2

(

s

+

t

)

−

ε

4



n2 2

.

Consequently for large enoughn,

e

(

G

) <

Ts,t

(α)



n 2



,

a contradiction.

SupposeG

=

(

V

,

E

)

is a graph andr

∈

_N. Ther-expansionofGis the graphG′

=

(

V′

,

E′

)

defined as follows. The vertex set ofG′_is_V

_{× [}

_r

_]

_{. For}_a

_,

_b

_{∈ [}

_r

_]

_{, an edge}

₍₍

_u

_,

_a

_{), (v,}

_b

₎₎

_{belongs to}_E′_iff _u

_v

(7)

belongs toE. Note that there is a natural projection

π

G′:V′

→

Vthat maps every vertex

(

u

,

a

)

from

G′to the vertexuinG. We are interested in the following property ofr-expansions. Suppose thatKis a copy of any graph fromF∗_in_G_{. Then}

_π

−1

G′

(

V

(

K

))

contains a complete bipartite graphBwith color

classes of sizess

(

K

)

randt

(

K

)

r. ByLemma 5we can tileBalmost perfectly with copies ofKs,t. IfFis

anF∗-tiling inG, we can apply the above operation on each memberK

∈

Fand obtain a new tiling

F′_{– which we call}_retiling_{– in the graph}_G′_.

We are now ready to proveTheorem 2.

Proof of Theorem 2. Note that it suffices to prove the theorem forH

≃

Ks,t.

We first deal with the particular caset

=

s. Set

α

′

_:=

₍

₁

₋

_ε/

₄

_)α

_{. Let}

_ε

1

:=

1₅

(

Ts,t

(α)

−

Ts,t

(α

′

))

,

and

ε

2be given byLemma 7for input parametersH,d

:=

ε

1and

γ

:=

αε/

8. Suppose thatk0 is

sufficiently large. LetM be the bound fromLemma 4for precision

ε

R

:=

min

{

ε

1

, ε

2

}

and minimal

number of clustersk0. LetCbe given byLemma 5for the input parameterss

,

t. Fixn0

≫

MC. Suppose

thatGis ann-vertex graph,n

≥

n0, with at leastTs,t

(α)



n

2



edges. We applyLemma 4onGto obtain an

(ε

_R

,

d

)

-reduced graphRwithkclusters,k₀

≤

k

≤

M. We have that

e

(

R

)

≥

(

Ts,t

(α)

−

d

−

3

ε

1

)



k 2



=



Ts,t

(α

′

)

+

1 5

(

Ts,t

(α)

−

Ts,t

(α

′

₎₎

 

k 2



(4)

>

ex



k

,

α

′_k 2

×

K2



.

Therefore,Rcontains at least α₂′kindependent edges. These edges correspond to regular pairs inG

which can be tiled almost perfectly with copies ofKs,t, by means ofLemmas 5and7. Elementary

calculations give that in this way we get a tiling of size at least

(

1

−

ε)α

n.

Consequently we may suppose thatt

>

s. We first define a handful of parameters. Set

α

′

_:=

6

−

4

ε

6

−

3

ε

α,

γ

:=

(

1

−

ε/

2

)α

′

_,

d

:=

2 5

(

Ts,t

(α)

−

Ts,t

(α

′

_)).

Note that

γ

=

(

1

−

2

ε/

3

)α

.

Let

ε

_Rbe given byLemma 6for input graphK_s_,_t, densityd

/

2 and approximation parameter

γ

. We may suppose that

ε

Ris sufficiently small such that

γ (

1

−

ε

R

) > (

1

−

ε)α

and

ε

R

<

d

/

2. LetCbe given

byLemma 5for inputs

,

t. Further, let

ε

′_and_h_{be given by}_{Lemma 9}_{for input parameters}

_α

′_and

_ε/

_4.

We may assume that

ε

′

< ε

. Set

p

:=

t2



4C

ε

′



,

q

:=



2t

ε

′



.

LetMbe the upper bound on the number of clusters given byLemma 4for input parametersh(for the minimal number of clusters) and

ε

Rp−q

/

2 (for the precision). Letn0

>

Mpqbe sufficiently large.

Suppose now thatGis a graph withn

>

n0vertices and at leastTs,t

(α)



n

2



edges. We first apply

Lemma 4toGwith parameters

ε

Rp−q

/

2 andh. In this way we obtain an

(ε

Rp−q

/

2

,

d

)

-reduced graph

Rwith at leasthvertices.

Let us now define a sequence of graphsR(i)_{by setting}_R(0)

₌

_R_{and letting}_R(i)_{be the}_p_-expansion

ofR(i−1)

,

i

=

1

,

2

, . . . ,

q. Note thate

(

R(i)

)

≥

Ts,t

(α

′

)



_v( R(i)) 2



for everyi

∈ {

0

,

1

, . . . ,

q

}

. LetF(i)_{be a maximum size}_K

s,t-tiling inR(i)fori

=

0

,

1

, . . . ,

q. We claim that

|

F(i)

| ≥

min



i

ε

′

v(

R(i)

)

2t

,



1

−

ε

2



α

′

_v(

R(i)

)



.

(10)

To this end it suffices to show that for anyi

≥

1, (C1) if

|

F(i−1)

_|

_{> (}

₁

₋

_ε/

₄

_)α

′

_v(

_R(i−1)

₎

_{, then} |F(i)| v(R(i)₎

≥

|F(i−1)| v(R(i−1)₎

−

εα ′ 4 , and (C2) if

|

F(i−1)

_{| ≤}

₍

₁

₋

_ε/

₄

_)α

′

_v(

_R(i−1)

₎

_{, then} |F(i)| v(R(i)₎

≥

|F(i−1)| v(R(i−1)₎

+

ε ′ 2t.

(8)

In the case (C1), according toLemma 5, the retiling ofF(i−1)_in_R(i)_{has size at least}

_|

_F(i−1)

_|

₍

_p

₋

_C

_{) >}

(

1

−

ε/

2

)α

′

v(

R(i)

)

, thus proving the statement.

Consequently we may suppose that we are in case (C2). ApplyLemma 9to the graphR(i−1)and the tilingF(i−1)_{, with parameters}

_α

′_and

_ε/

_4.

Suppose first that assertion (i) of the lemma holds. ThenR(i−1) _{contains an} _F

1-tiling F with |F|

v(R(i−1)₎

≥

|F (i−1)_|

v(R(i−1)₎

+

ε

′. By retilingF, we get aKs,t-tiling inR(i)with size at least

|

F

|

(

p

−

C

) >

i

ε

′

v(

R(i)

)/(

2t

),

thus proving the statement.

Suppose now that assertion (ii) ofLemma 9is true. ThenR(i−1)contains anF(i−1)-augmentation

(

E0

,

E1

)

with

|

E0

| ≥

ε

′

v(

R(i−1)

)

. Letr

=

p

/

t. We shall denote byTthet-expansion ofR(i−1)and byT′

ther-expansion ofT. Note thatT′_{is isomorphic to}_R(i)_.

Let us build anF2-tiling inTin the following way.

For every edgee

=

(

u

, v)

∈

E0withu

∈

V

(

F(i−1)

)

we choose an edgee′

=

(

u′

, v

′

)

inT with

π

T

(

u′

)

=

uand

π

T

(v

′

)

=

v

. We shall denote by

w

ethe vertexu′corresponding tou.

For every edgee

=

(

u

, v)

∈

E1we choose a setSeoftindependent edges in

π

T−1

(

e

)

.

For everyK

∈

F(i−1)_{we shall also choose a subgraph}_K′_of_T_{. We distinguish the following cases. If}

Khas no vertex matched byE0orE1, then we letK′

:=

T

[

π

T−1

(

K

)

]

. IfKhas a vertexumatched byE1

but no vertex matched byE0, we letK′

:=

T

[

π

T−1

(

K

−

u

)

]

. ThenK ′

_≃

_K

st,(t−1)t. Finally, ifKhas a vertex umatched by an edgee

∈

E0and a vertex

v

matched by an edge inE1, we letK′

:=

T

[

π

−1

T

(

K

−

v)

]−

w

e.

Note that in this last caseK′

_≃

_K

st−1,(t−1)t.

It is easy to see that

F

:= {

e′:e

∈

E0

} ∪ {

K′:K

∈

F(i−1)

} ∪





e∈E1 Se



is anF2-tiling inT. Moreover, we have that |F| v(T)

≥

|F(i−1)_| v(R(i−1)₎

+

ε ′ t. So the retiling ofFinT ′_{has size at}

least

|

F

|

(

r

−

C

)

≥

i

ε

′

_v(

_R(i)

_)/(

₂_t

₎

_{. This proves (C2) and also}₍₁₀₎_.

UsingLemma 3, we may subdivide every cluster corresponding to a vertex ofRintopq

equal-sized parts, by discarding some vertices if necessary. This gives us an

(ε

R

,

d

/

2

)

-reduced graphR′. By

constructionR′

≃

R(q). By(10), there is aKs,t-tilingFinR′with size at least

(

1

−

ε/

2

)α

′

v(

R′

)

. LetG′be

the subgraph ofGinduced by the clusters corresponding to the vertices ofR′_{. By applying}_{Lemma 6}_to

R′_{, we see that}_G′_{has a}_K

s,t-tiling of size at least

γ v(

G′

)

≥

γ (

1

−

ε

R

)v(

G

) > (

1

−

ε)αv(

G

)

, and so doesG.

This finishes the proof ofTheorem 2.

Acknowledgments

We would like to thank the anonymous referee for his/her very useful comments.

Most of the work was done while JH was affiliated with TU Munich. JH would like to thank the group of Anusch Taraz for a nice environment. JH was partially supported by the Grant Agency of Charles University, grant GAUK 202-10/258009, and by BAYHOST.

References

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[3] P. Erdős, T. Gallai, On maximal paths and circuits of graphs, Acta Math. Acad. Sci. Hungar. 10 (1959) 337–356. Unbound insert.

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(9)

[7] J. Komlós, M. Simonovits, Szemerédi’s regularity lemma and its applications in graph theory, in: Combinatorics, Paul Erdős is eighty, Vol. 2 (Keszthely, 1993), in: Bolyai Soc. Math. Stud., vol. 2, János Bolyai Math. Soc., Budapest, 1996, pp. 295–352. [8] T. Kövári, V. Sós, P. Turán, On a problem of K. Zarankiewicz, Colloq. Math. 3 (1954) 50–57.

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