Large planar subgraphs in dense graphs

Journal of Combinatorial Theory, Series B 95 (2005) 263–282www.elsevier.com/locate/jctb

Large planar subgraphs in dense graphs

Daniela Kühna, Deryk Osthusa, Anusch TarazbaSchool of Mathematics, Birmingham University, Edgbaston, Birmingham B15 2TT, UK

bZentrum Mathematik, Technische Universität München, Boltzmannstr. 3, 85747 Garching bei München,Germany

Received 11 December 2002Available online 15 June 2005

Abstract

Weprove sufficient and essentially necessary conditions in terms of theminimumdegree for a graphto contain planar subgraphs with many edges. For example, for all positive� every sufficiently largegraphGwith minimum degree at least(2/3+ �)|G| contains a triangulation as a spanning subgraph,whereas this need not be the case when the minimum degree is less than 2|G|/3.© 2005 Elsevier Inc. All rights reserved.

Keywords:Extremal graph theory; Regularity lemma; Planar subgraphs

1. Introduction

1.1. Results

In this paper, we study the following extremal question: Given a functionm = m(n),how large does the minimum degree of a graphGof ordernhave to be in order to guaranteea planar subgraph with at leastm(n) edges?If m�n, the answer is easy. Indeed, suppose that the minimum degree ofG is at least

one. Then every componentC of G has a spanning tree with|C| − 1� |C|/2 edges. SoGhas a (planar) spanning forest with at leastn/2 edges, which is best possible ifG consistsof independent edges. Similarly, it is easy to see that ifG has minimum degree at least two,thenG contains a planar subgraph withn edges, which is best possible ifG is a cycle.

E-mail addresses:kuehn@for.maths.bham.ac.uk(D. Kühn), osthus@maths.bham.ac.uk(D. Osthus),taraz@ma.tum.de(A. Taraz).

0095-8956/$ - see front matter © 2005 Elsevier Inc. All rights reserved.doi:10.1016/j.jctb.2005.04.004

264 D. Kühn et al. / Journal of Combinatorial Theory, Series B 95 (2005) 263–282

On the other hand, ifG is bipartite, then the facial cycles of any planar subgraph havelength at least four and so Euler’s formula implies that no planar subgraph ofG has morethan 2n− 4 edges. So as long as the minimum degree is at mostn/2, we cannot hope for aplanar subgraph with more than 2n−4 edges. Our first theorem shows that a much smallerminimum degree already guarantees a planar subgraph with roughly 2n edges.

Theorem 1. For every0 < ε < 1 there existsn0 = n0(ε) such that every graph G ofordern�n0 andminimum degree��1500√n/ε2 contains a planar subgraph with at least(2− ε)n edges.

The planar subgraph we find consists of copies ofK2,s together with a few stars (heres is quite large and its value is not the same for all copies). The result is essentially bestpossible in two ways. Firstly, there are graphs with minimum degree

√n/2 and girth at

least 6[6], (see also [3]). Hence Euler’s formula shows that any planar subgraph of sucha graph can have at most32(n− 2) edges (as all of its facial cycles have length at least 6).Secondly, for��n/2 consider the graph consisting ofn/2� disjoint copies of the completebipartite graphK�,�. It obviously has minimum degree�, but again by Euler’s formula itcannot contain a planar subgraph with more than(2 · 2� − 4)n/2� = 2n − 2n/� edges.This shows that as long as the minimum degree� of G is o(n), we cannot ask for a planarsubgraph ofG with 2n − C edges, whereC does not depend onn. So if we want at least2n−C edges in a planar subgraph, then a necessary condition is that��2n/C, i.e.�mustbe linear inn. Our second theorem shows that the linearity of� is also sufficient.

Theorem 2. For every� > 0 there existsC = C(�) such that every graph G of order n andminimum degree at least�n contains a planar subgraph with at least2n− C edges.

The planar subgraph we construct in the proof consists of a bounded number of disjointquadrangulations. As we have already seen, the above result is best possible up to the valueof the constantC as long as the minimum degree is at mostn/2. If however the minimumdegree is a little larger than this, we can already guarantee a planar subgraph which is atriangulation apart from a constant number of missing edges:

Theorem 3. For every� > 0 there existsC = C(�) such that every graph G of order n andminimum degree at least

(12 + �

)n contains a planar subgraph with at least3n−C edges.

This time, the planar subgraph we construct in the proof consists of a bounded numberof disjoint triangulations. Again, the result is best possible in the sense that the constantChas to depend on� and the additional term�n in the bound on the minimum degree cannotbe replaced by a sublinear one (see Proposition14).Finally, we seek a spanning triangulation, i.e. a planar subgraph with 3n − 6 edges. As

pointed out to us by Bollobás, the following 3-partite graphGshows that aminimumdegreeof 2n/3 is necessary for this.G is obtained from two disjoint cliquesC1 andC2 of ordern/3 by adding an independent setX of n/3 new vertices and joining each of them to all thevertices in the two cliques. SoGhas minimum degree 2n/3−1. Observe that any spanningtriangulation inG would have two facial trianglesT1 andT2 which share an edge and are

D. Kühn et al. / Journal of Combinatorial Theory, Series B 95 (2005) 263–282 265

such thatTi contains a vertex ofCi (i = 1,2). But this is impossible since every triangle ofG containing a vertex ofCi can have at most one vertex outsideCi , namely inX. However,to guarantee a triangulation, it suffices to increase the minimum degree by a small amount.

Theorem 4. For every� > 0 there exists an integern0 = n0(�) such that every graph G oforder n�n0 and minimum degree at least(23 + �)n contains a triangulation as a spanningsubgraph.

In [19] the first two authors show that for sufficiently large graphs a minimum degree of2n/3 suffices. However, the proof of this is rather more involved than that of Theorem 4.We also obtain an analogue of Theorem 4 for quadrangulations, i.e. plane subgraphs with

2n− 4 edges in which every face is bounded by a 4-cycle.

Theorem 5. For every� > 0 there exists an integern0 = n0(�) such that every graph Gof order n�n0 and minimum degree at least(12 + �)n contains a quadrangulation as aspanning subgraph.

The disjoint union of two cliques of ordern/2 shows that apart from the error term�n,the minimum degree in Theorem5 cannot be reduced.

1.2. Open questions and related results

There is a conjecture of Bollobás and Komlós [12] which would immediately implyTheorems 4 and 5. It asserts that for every� > 0 and allr,� ∈ N there are� > 0 andn0 ∈ N such that every graphG of ordern�n0 and minimum degree at least(1− 1

r+ �)n

contains a copy of every graphH of ordern whose chromatic number is at mostr, whosemaximum degree is at most� and whose band-width is at most�n. (Theband-widthof agraphH is the smallest integerk for which there exists an enumerationv1, . . . , v|H | of thevertices ofH such that every edgevivj ∈ H satisfies|i − j |�k.) Indeed, to derive, e.g.Theorem 4 from this conjecture it suffices to find for alln ∈ N a 3-partite triangulation ofordern which has both bounded maximum degree and bounded band-width. It is easy tosee that such triangulations exist (e.g. modify the graphH1 in Fig. 3 below).Theorems 1–5 give a fairly accurate picture of the maximum size of a planar subgraph

when we consider graphs whose minimum degree� is much larger than√n. However, we

are not aware of any nontrivial lower bounds when� lies between 2 and√n. An easy upper

bound is obtained as follows. For��3 let �2� = �2�(n) be the largest integer such thatthere are graphsG of ordern, minimum degree at least�2� and girth at least 2�. (The orderof magnitude of�2� is only known for� = 3,4 and 6, see e.g. [2,6].) So all facial cycles ina planar subgraph of such a graphG have length at least 2� and thus Euler’s formula givesus an upper bound on the size of a planar subgraph ofG. We believe that in general thisupper bound is close to the truth (except maybe when the minimum degree is only a littlelarger than�2�+2).The problem of finding a large planar subgraph in a random graph was investigated by

Schlatter [20], the case of triangulations was already considered earlier by Bollobás andFrieze [5].

266 D. Kühn et al. / Journal of Combinatorial Theory, Series B 95 (2005) 263–282

1.3. Algorithmic aspects

Our proofs immediately show that the planar subgraphs guaranteed by Theorems1–5 canbe found in polynomial time. For graphs with high minimum degree we therefore obtainimproved approximation algorithms for the maximum planar subgraph problem which in agiven graphG asks for a planar subgraph with the maximum number of edges. Cˇalinescuet al. [7] showed that this problem is Max SNP-hard: there is a constantε such that therecannot exist a polynomial time approximation algorithm with approximation ratio betterthan 1− ε, unlessP = NP . Recently, Faria et al. [11] proved that it is Max SNP-hardeven for cubic graphs. The best known approximation algorithm for arbitrary input graphshas an approximation ratio of 4/9 [7]. (Note that a ratio of 1/3 is already achieved byproducing spanning trees for all connected components.) On the other hand, our proof ofTheorem 4 implies that for any� > 0 the maximum planar subgraph problem can be solvedin polynomial time for graphswithminimumdegree at least(23+�)n. Our remaining resultsgive improved approximation algorithms for graphs whose minimum degree is sufficientlylarge for the respective results to apply.The paper is organized as follows. In Section 2, we give a brief sketch of the proofs of

Theorems 1–5. In Section 3, we collect some notation and all the information about theRegularity lemma and the Blow-up lemma we need for the proofs of Theorems 2–5. Theproofs themselves are then given in the final section.

2. Sketch of proofs

The proof of Theorem 1 is rather different from those of the other results. (Since theminimum degree iso(n), one cannot apply the Regularity lemma or the Blow-up lemma.)Instead, the strategy is to repeatedly find a suitable greedy covering of part of the verticesof the original graphGwith disjoint complete bipartite graphsK2,s , wheres is large. (Notethat if s is large then the planar graphH := K2,s has roughly 2|H | edges.) These partialcoverings (which will overlap a little) are then combined into a single planar graph of therequired size.We now give a sketch of the proofs of Theorems 2–5. The structure of these proofs is

similar: we first apply the Regularity lemma (Lemma 7) to obtain a partition of the verticesof G into a large but constant number of clusters. SinceG has large minimum degree, thisis also true for the ‘reduced graph’R (whose vertices are the clusters and whose edgescorrespond to the pairs of clusters which are regular and have sufficient density). We willuse this to cover almost all vertices ofRby suitable disjoint graphsH of bounded size. Thenwe apply the Blow-up lemma (Lemma 10) to find spanning planar graphsP of the requireddensity within the subgraphsH ′ of G corresponding to these graphsH. However, we alsohave to ensure that the exceptional vertices ofG (i.e. the small proportion of those verticesof Gwhich do not belong to some suchH ′) can be incorporated into these planar graphsPwithout reducing their density. This also follows from the Blow-up lemma provided that wecan assign each exceptional vertexv to someH which contains enough clusters with manyneighbours ofv in such a way that to eachH we assign only a small number of exceptionalvertices.

D. Kühn et al. / Journal of Combinatorial Theory, Series B 95 (2005) 263–282 267

In the proof of Theorem2 the graphsH will be stars of bounded size and the planargraphsPwe seek within the graphsH ′ will be quadrangulations. For Theorem 3 we wantthe planar graphsP to be triangulations, which means that the graphsH can no longer bebipartite. Thus an obvious choice forH would be a triangle, but we cannot hope to coveralmost all vertices of the reduced graphR by disjoint triangles since its minimum degreemay be only a little larger than|R|/2. However, a recent result of Komlós (Theorem 13)implies that we can takeH to be the complete 3-partite graphKa,a,1 (wherea is large) asit is in some sense close to being bipartite.In the proof of Theorem 4 the minimum degree of the reduced graphRexceeds 2|R|/3

and hence the theorem of Corrádi and Hajnal [8] implies thatRcan be covered by disjointtriangles. However, this is not sufficient for our purposes as this time we seek a singletriangulation containing all vertices ofG (instead of a disjoint union of boundedly manytriangulations as in the proof of Theorem 3). So we have to ‘glue together’ the differenttriangulations corresponding to the triangles coveringR. For this we use suitable edges ofR joining these triangles (as well as some additional vertices ofG). Thus instead of merelycoveringRby disjoint triangles, we will start with the second power of a Hamilton path ofR. The latter is guaranteed by a result of Fan and Kierstead [10].The proof of Theorem5 is similar to that of Theorem4 but the gluing process is somewhat

simpler. Insteadof the secondpower, this time it suffices toworkwith an ‘ordinary’Hamiltonpath.

3. Notation and tools

Throughout this paper we omit floors and ceilings whenever this does not affect theargument. We writee(G) for the number of edges of a graphG, |G| for its order,�(G) forits minimum degree,�(G) for its maximum degree and�(G) for its chromatic number. Ifthis is not ambiguous, we also writen for the order of a graphG. We denote the degree of avertexx ∈ GbydG(x)and the set of its neighbours byNG(x).GivendisjointA,B ⊆ V (G),anA–B edgeis an edge ofG with one endvertex inA and the other inB, the number ofthese edges is denoted byeG(A,B) or e(A,B) if this is unambiguous. We write(A,B)Gfor the bipartite subgraph ofG whose vertex classes areA andB and whose edges are allA–B edges inG. More generally, we write(A,B) for a bipartite graph with vertex classesAandB. Given a plane graphG, afacial cycleofG is a cycle inGwhich is the boundary ofa face.G is a triangulation if all its faces are bounded by triangles and aquadrangulationif all faces are bounded by four-cycles. So by Euler’s formula a triangulation has 3n − 6edges whereas a quadrangulation has 2n− 4 edges.In the remainder of this sectionwecollect all the informationweneedabout theRegularity

lemma and the Blow-up lemma. See [18,12] for surveys about these. Let us start with somemore notation. Thedensityof a bipartite graphG = (A,B) is defined to be

d(A,B) := e(A,B)

|A||B| .

Givenε > 0, we say thatG is ε-regular if for all setsX ⊆ A andY ⊆ B with |X|�ε|A|and |Y |�ε|B| we have|d(A,B) − d(X, Y )| < ε. Given d ∈ [0,1], we say thatG is

268 D. Kühn et al. / Journal of Combinatorial Theory, Series B 95 (2005) 263–282

(ε, d)-super-regularif all setsX ⊆ A andY ⊆ B with |X|�ε|A| and|Y |�ε|B| satisfyd(X, Y ) > d and, furthermore, ifdG(a) > d|B| for all a ∈ A anddG(b) > d|A| for allb ∈ B.We will often use the following simple fact, which follows from the definition ofε-

regularity.

Proposition 6. Given anε-regular bipartite graph(A,B) of density> d and a setX ⊆ A

with |X|�ε|A|, there are less thanε|B| vertices in B which have at most(d − ε)|X|neighbours in X.

We will use the following degree form of Szemerédi’s regularity lemma which can beeasily derived from the classical version. Proofs of the latter are for example included in[4,9].

Lemma 7 (Regularity lemma). For all ε > 0 and all integersk0 there is anN = N(ε, k0)

such that for every numberd ∈ [0,1] and for every graph G on at least N vertices thereexist a partition ofV (G) into V0, V1, . . . , Vk and a spanning subgraphG′ of G such thatthe following holds:

• k0�k�N ,• |V0|�ε|G|,• |V1| = · · · = |Vk| =: L,• dG′(x) > dG(x)− (d + ε)|G| for all verticesx ∈ G,• for all i�1 the graphG′[Vi] is empty,• for all 1� i < j�k the graph(Vi, Vj )G′ is ε-regular and has density either0 or > d.

The setsVi (i�1) are calledclusters, V0 is called theexceptional set. Given clusters andG′ as in Lemma7, thereduced graph Ris the graph whose vertices areV1, . . . , Vk and inwhichVi is joined toVj whenever(Vi, Vj )G′ is ε-regular and has density> d. ThusViVjis an edge ofR if and only ifG′ has an edge betweenVi andVj .

Proposition 8. Let H be a subgraph of the reduced graph R with�(H)��. Then eachvertexVi of H contains a subsetV ′

i of size(1− ε�)L such that for every edgeViVj of Hthe graph(V ′

i , V′j )G′ is (ε/(1− ε�), d − (1+ �)ε)-super-regular.

Proof. Consider an edgeViVj of H. By Proposition6, there are less thanεL vertices inVi which have at most(d − ε)L neighbours inVj (in the graphG′). So for every vertexVi of H we can choose a setV ′

i ⊆ Vi of size(1− ε�)L such that for each neighbourVjof Vi in H all verticesx ∈ V ′

i have more than(d − ε)L neighbours inVj . It can be easilychecked that for every edgeViVj of H the graph(V ′

i , V′j )G′ is (ε/(1− ε�), d − (1+ �)ε)-

super-regular. �

We will often use the following well known and simple fact. Its proof is the only placein this paper where the degree form of the Regularity lemma is more convenient than theclassical form.

D. Kühn et al. / Journal of Combinatorial Theory, Series B 95 (2005) 263–282 269

Proposition 9. For every� > 0 there existε0 = ε0(�) and d0 = d0(�) such that for allε�ε0, d�d0 and everyc�0 an application of Lemma7 to a graph G of minimum degreeat least(c + �)|G| yields a reduced graph R of minimum degree at least(c + �/2)|R|.

Proof. Suppose that there is a vertexVi ∈ R whose degree inR is less than(c+ �/2)k. LetWdenote the union of all those clustersVj (j �= i) for which (Vi, Vj )G′ has density 0. Letu be any vertex inVi . Then

dG′(u) � |NG′(u) ∩W | + dR(Vi) · L+ |NG′(u) ∩ V0| < 0+ (c + �/2)kL+ εn

� (c + �/2+ ε)n.

On the other hand, Lemma 7 states thatdG′(u) > dG(u)− (d + ε)n�(c + � − d − ε)n, acontradiction, provided that��2d + 4ε. �

We will also use the Blow-up lemma of Komlós et al. [14]. It implies that dense regularpairs behave like complete bipartite graphs with respect to containing bounded degreegraphs as subgraphs.

Lemma 10(Blow-up lemma). Given a graph R on{1, . . . , r} and numbersd, c,� > 0,there are positive numbersε0 = ε0(d,�, r, c) and � = �(d,�, r, c)�1/2 such that thefollowing holds. GivenL ∈ N and ε�ε0, let R(L) be the graph obtained from R byreplacing each vertexi ∈ R with a setVi of L new vertices and joining all vertices inVi toall vertices inVj whenever ij is an edge of R. Let G be a spanning subgraph ofR(L) suchthat for every edgeij ∈ R the graph(Vi, Vj )G is (ε, d)-super-regular. Then G containsa copy of every subgraph H ofR(L) with �(H)��. Furthermore, we can additionallyrequire that for verticesx ∈ H ⊆ R(L) lying in Vi their images in the copy of H in G arecontained in(arbitrary) given setsCx ⊆ Vi provided that|Cx |�cL for each such x andprovided that in eachVi there are at most�L such vertices x.

We say that the verticesx in Lemma10 areimage restricted toCx .

4. Proofs

4.1. Planar subgraphs of size2n− εn

In our proof of Theorem 1 we will use the following well-known upper bound on thenumber of edges ofK2,s-free graphs (see e.g. [3, Chapter VI, Theorems 2.2 and 2.3]).

Theorem 11. Let s�2 be an integer. Then every graph G withe(G)�√sn3/2 contains a

copy ofK2,s .Moreover, every bipartite graphG = (A,B)with e(G)�√s|A||B|1/2+|B|

contains a copy ofK2,s with 2 vertices in A and s vertices in B.

Proof of Theorem 1. Throughout the proof we assume thatn is sufficiently large for ourestimates to hold. For allk�1 set sk := 2k

2+2/εk. We first greedily choose as manydisjoint copies ofK2,s1 in G as possible. LetP1 be the union of all theseK2,s1’s, X1 :=

270 D. Kühn et al. / Journal of Combinatorial Theory, Series B 95 (2005) 263–282

V (P1) and letY1 := V (G) \X1. ThusG[Y1] isK2,s1-free and so Theorem11 implies thate(G[Y1])�√

s1|Y1|3/2. LetY ′1 be the set of all those vertices inY1 which have at most�/2

neighbours inX1. Then

�|Y ′1|/2�2e(G[Y1])�2√s1n3/2,

and thus

|Y ′1|�

4√s1n

3/2

�. (1)

Let Y ∗1 := Y1 \ Y ′

1. Next we greedily choose (as many as possible) disjoint copies ofK2,s2in (X1, Y ∗

1 )G having 2 vertices inX1 ands2 vertices inY∗1 . LetP2 be the union of all these

K2,s2’s, X2 := V (P2) ∩ X1 andY2 := V (P2) ∩ Y ∗1 . Let Y

′2 be the set of all those vertices

in Y ∗1 \ Y2 which have at most�/22 neighbours inX2. Thus each vertex inY ′

2 has at least�/22 neighbours inX1 \X2 and so

e(X1 \X2, Y ′2)��|Y ′

2|/22.On the other hand,(X1 \X2, Y ′

2)G does not contain aK2,s2 with 2 vertices inX1 \X2 ands2 vertices inY ′

2. Thus Theorem11 implies

e(X1 \X2, Y ′2)�

√s2|X1 \X2| |Y ′

2|1/2 + |Y ′2|�

√s2n

3/2 + |Y ′2|

and therefore

|Y ′2|�

√s2n

3/2

�/22 − 1� 5

√s2n

3/2

�. (2)

Let Y ∗2 := Y ∗

1 \ (Y2 ∪ Y ′2) and greedily choose (again as many as possible) disjoint copies

of K2,s3 in (X2, Y∗2 )G having 2 vertices inX2 ands3 vertices inY

∗2 . LetP3 be the union of

all theseK2,s3’s, X3 := V (P3) ∩ X2 andY3 := V (P3) ∩ Y ∗2 . Let Y

′3 be the set of all those

vertices inY ∗2 \ Y3 which have at most�/23 neighbours inX3. Let Y ∗

3 := Y ∗2 \ (Y3 ∪ Y ′

3)

and continue in this fashion untilPi = ∅ (and thusXi = Yi = ∅ andY ′i = Y ∗

i−1). Let i bethe smallest index such thatPi = ∅. Thusi�√

log n sinces√log n > n.Using that|Xk−1 \ Xk|� |Xk−1|�2n/sk−1 for all 3�k� i, a calculation similar to the

casek = 2 shows that

|Y ′k|�

5 · 2k−1√skn3/2�sk−1

. (3)

Moreover, sinceY ′i = Y ∗

i−1,

|X1| +i−1∑k=2

|Yk| +i∑

k=1|Y ′k| = n. (4)

Set

P := (P1 −X2) ∪ (P2 −X3) ∪ · · · ∪ (Pi−2 −Xi−1) ∪ Pi−1.

D. Kühn et al. / Journal of Combinatorial Theory, Series B 95 (2005) 263–282 271

ClearlyP is a planar subgraph ofG. Notice that when removingXk fromPk−1, we destroyat mostsk−1|Xk| of its edges, but this is negligible compared toe(Pk) = 2|Yk|, assk growsrather rapidly withk. Also, recall that|Xk|�2n/sk for k�2. Hence

e(P ) �i−1∑k=1

e(Pk)−i−1∑k=2

sk−1|Xk|

� 2s1|X1|s1 + 2

+i−1∑k=2

2|Yk| −i−1∑k=2

2sk−1nsk

(4)= 2

(n−

i∑k=1

|Y ′k|)

− 4|X1|s1 + 2

−i−1∑k=2

εn

22k−2

(1,2,3)� 2n− 32n3/2

�√ε

− 80n3/2

�ε−

i∑k=3

5 · 2k√skn3/2�sk−1

− 4n

s1− εn

3

� 2n− εn

9−

i∑k=3

5εk/2−1n3/2

2k2/2−3k+2�− εn

2− εn

3

� 2n− 80n3/2

�− 17εn

18� (2− ε)n

as required. �

We remark that the proof of Theorem1 shows that we can letε be any function ofnwithε(n)�1. Note that it does not make sense to takeε(n)�n−1/4.

4.2. Planar subgraphs of size2n− C

For the proof of Theorem 2 we need the following simple proposition.

Proposition 12. Given0 < ��1/2 and a graph G of minimum degree at least�n, thereexists a setS of disjoint substars of G such that every vertex of G lies in someS ∈ S andsuch that each such S satisfies1��(S)�1/�.

Proof. Construct the stars inS greedily as follows. Suppose that we have already covereda setX ⊆ V (G) with a setS ′ of disjoint substars ofG such that 1��(S)�1/� for everyS ∈ S ′. Choosex ∈ V (G) \ X. If x has a neighboury outsideX, we may add the starconsisting of the edgexy to S ′. So suppose that all neighbours ofx lie in X. If x is joinedto a leafy of some starS ∈ S ′ then, if |S|�3, we can replaceSby S − y and add the newstarxy to S ′ or, if |S| = 2, we can replaceSby S ∪ xy. If x is only joined to midpoints ofstars inS ′, then one such star must have at most 1/� − 1 leaves and so we can addx to thisstar. �

Proof of Theorem 2. Clearly, we may assume that��1/2. Let ε0(�) and d0(�) =: dbe as given in Proposition 9. Letε0(d/2,8/�,1+ 2/�, �/4) =: ε∗ and�(d/2,8/�,1+

272 D. Kühn et al. / Journal of Combinatorial Theory, Series B 95 (2005) 263–282

2/�, �/4) =: � be as defined in the Blow-up lemma (Lemma10). Put

ε := min

{ε0(�),

ε∗

2,�3�

72,�d

6

}.

Clearly, it suffices to show that every graphGwhose ordern is sufficiently large comparedwith � contains a planar subgraph with at least 2n − 4N(ε,2) vertices, whereN(ε,2) isgiven by the Regularity lemma (Lemma7). So throughout the proof we assume thatn issufficiently large.We first apply the Regularity lemma toG to obtain an exceptional setV0 and clusters

V1, . . . , Vk where 2�k�N(ε,2). Let L andG′ be as defined in the Regularity lemmaand letR denote the reduced graph. Thus Proposition 9 implies that�(R)��k/2. So byProposition 12 there exists a setS of disjoint substars ofRsuch that every vertex ofR liesin some star fromS and such that 1��(S)�2/� for eachS ∈ S.NextweapplyProposition 8 to obtain setsV ′

i ⊆ Vi of size(1−2ε/�)L =: L′ such that forall the edgesViVj ofRlying in somestar fromS the graph(V ′

i , V′j )G′ is(2ε, d−(1+2/�)ε)-

super-regular. Henceforth, we will think ofRand of the stars inS as graphs whose verticesare the new setsV ′

i . Add all vertices ofGwhich do not lie in someV′i to the exceptional set

V0. By adding further vertices toV0 if necessary, we may assume thatL′ is even. We stilldenote the enlarged exceptional set byV0. Thus|V0|�εn+ 2εkL/� + k�3εn/�.Given a vertexv ∈ V0 and a starS ∈ S, we say thatS is v-friendly if there is a vertex

V ′i ∈ S such thatv has at least�L′/4 neighbours inV ′

i . Let Nv denote the number ofv-friendly starsS ∈ S. Then

�n/2< (� − 3ε/�)n�dG(v)− |V0|�Nv(1+ 2/�)L′ +∑S∈S

|S|�L′/4

and therefore, since∑

S∈S |S| = k,

Nv >�

3L′

(�n

2− �kL′

4

)� �2n

12L′ .

So

2|V0|�L′ � 6εn

��L′ < Nv

for every vertexv ∈ V0. But this implies that we can greedily assign each vertexv ∈ V0 toav-friendly starS ∈ S in such a way that to everyS ∈ S we assign at most�L′/2 verticesfrom V0.Consider a fixedS ∈ S and letX ⊆ V0 be the set of all vertices assigned toS. LetU1

be the centre ofSand letU2, . . . , U|S| be its other vertices. So eachU� is a set of the formV ′i . Fix any bipartite quadrangulationPS of maximum degree 4�(S)�8/� whose vertexclasses areU1 andU2 ∪ · · · ∪ U|S| such that for each�� |S| there is a setC� of at leastL′/4� |X| facial 4-cycles ofPS with the property that, firstly, eachC ∈ C� has two of itsvertices inU�, secondly, these vertices are distinct for differentC ∈ C� and thirdly, eachfacial 4-cycle ofPS lies in at most one suchC�. Recalling thatL′ is even, it is not difficultto see that such quadrangulations exist (see Fig.1).

D. Kühn et al. / Journal of Combinatorial Theory, Series B 95 (2005) 263–282 273

Fig. 1. A quadrangulationPS which corresponds to a starS with three leaves. The black vertices belong toU1.The shaded faces indicate a possible choice forC1.

As each edge ofS corresponds to a(2ε, d/2)-super-regular subgraph ofG′, the Blow-uplemma (Lemma10) implies that the subgraph ofG′ corresponding toS (that isG′[U1 ∪· · · ∪ U|S|]) contains a spanning copy ofPS such that every vertexv ∈ X is joined to twoopposite vertices on some facial 4-cycle ofPS and such that these 4-cycles differ for distinctverticesv ∈ X. Indeed, this can be achieved as follows. By definition, eachv ∈ X has atleast�L′/4 neighbours in someU� (1��� |S|). Assignv to a cycleCv ∈ C� such that thesecyclesCv differ for distinct suchv. When applying the Blow-up lemma, for eachv ∈ X thetwo vertices inV (Cv)∩U� are image restricted to the neighbourhood ofv in U�. (This canbe done since the vertices inV (Cv) ∩ U� are distinct for differentv.)The graph obtained fromPS by inserting all the verticesv ∈ X in their facial 4-cycles

Cv is still a quadrangulation. HenceG contains a planar subgraph which is a disjoint unionof |S| quadrangulations and thus has 2n− 4|S|�2n− 4N(ε,2) edges. �

4.3. Planar subgraphs of size3n− C

Thecritical chromatic number�cr(H)of agraphH is definedas(�(H)−1)|H |/(|H |−�),where� denotes the minimum size of the smallest colour class in a colouring ofH with�(H) colours. For the proof of Theorem 3 we need the following result of Komlós [13,Theorem 8].

Theorem 13. For everyε > 0 and every graphH there exists an integerk0 = k0(H, ε)

such that all but at mostεk vertices of every graphR of orderk�k0 and minimum degree�(R)�(1− 1/�cr(H))k can be covered by disjoint copies ofH .

Note that Theorem13 immediately implies that for allε, � > 0 there exists an integern0 = n0(ε, �) such that every graphR of ordern�n0 and minimum degree at least�ncontains a planar graph with at least 2n − εn edges. Indeed, letH := K2,s in Theorem13, wheres is sufficiently large compared toε and�. Then the critical chromatic numberof H is close to one and the disjoint union of all copies ofH given by Theorem 13 isa planar subgraph ofR of the required size. Similarly, as there exist large triangulations

274 D. Kühn et al. / Journal of Combinatorial Theory, Series B 95 (2005) 263–282

Fig. 2. A triangulation apart from the shaded faces (into which the exceptional vertices will be inserted).

whose critical chromatic number is close to 2 (e.g. modify the graph in Fig.2), Theorem 13implies that Theorem 3 is true for largen if we only ask for a planar subgraph with 3n− εn

edges.

Proof of Theorem 3. Bymaking� smaller, we may assume that 1/� is an integer divisibleby 4. Letε0(�) andd0(�) =: d be as given in Proposition 9. Seta := 2/� andH := Ka,a,1,the complete 3-partite graph with vertex classes of sizea, a and 1. Letε0(d/2,8a,2a +1, �/4) =: ε∗ and�(d/2,8a,2a+1, �/4) =: �beas defined in theBlow-up lemma (Lemma10). Put

ε := min

{ε0(�),

ε∗

2,

�3�

640,�d

12

}

and letk0 := k0(H, ε) be defined as in Theorem13. Clearly, it suffices to show that everygraphG whose ordern is sufficiently large compared with� contains a planar subgraphwith at least 3n − 6N(ε, k0) vertices, whereN(ε, k0) is given by the Regularity lemma(Lemma 7).We first apply the Regularity lemma toG to obtain an exceptional setV0 and clusters

V1, . . . , Vk wherek0�k�N(ε, k0). Let L andG′ be as defined in the Regularity lemmaand letR denote the reduced graph. Thus Proposition 9 implies that�(R)�(1/2+ �/2)k.As �cr(H) = 2(2a + 1)/2a = 2 + 1/a and therefore�(R)�(1− 1/�cr(H))k, we can

D. Kühn et al. / Journal of Combinatorial Theory, Series B 95 (2005) 263–282 275

apply Theorem13 to obtain a setH of disjoint copies ofH in R such that all but at mostεk vertices ofR lie in the unionH ′ of all these copies. As�(H ′) = 2a, we may applyProposition 8 to find for everyVi ∈ V (H ′) a setV ′

i ⊆ Vi of size(1− 2aε)L =: L′ suchthat for every edgeViVj ∈ H ′ the graph(V ′

i , V′j )G′ is (2ε, d − (1+ 2a)ε)-super-regular.

We add all vertices ofG which do not lie in someV ′i to the exceptional setV0 and still

denote this enlarged set byV0. Thus

|V0|�εn+ εkL+ 2aεkL�4aεn.

PutR′ := R[V (H ′)]. We will think of R′ and of the graphs inH as graphs whose verticesare the new setsV ′

i .Given a vertexv ∈ V0 andS ∈ H, we say thatS is v-friendly if there are verticesV ′

i andV ′j lying in different classes of theKa,a ⊆ S such thatv has at least�L′/4 neighbours inbothV ′

i andV′j . LetNv denote the number ofv-friendly S ∈ H. Then

(1/2+ �/2)n < (1/2+ � − 4aε)n�dG(v)− |V0|� Nv(2a + 1)L′ + |H|(a + 1+ �a/4)L′

and therefore

Nv >(1/2+ �/2)n

(2a + 1)L′ − k(a + 1+ �a/4)L′

(2a + 1)2L′

� n

(2a + 1)L′

(1

2+ �

2− a(1+ 1/a + �/4)

2a

)� n�

5L′�

8= �2n

40L′ .

So

2|V0|�L′ � 8aεn

�L′ < Nv

for every vertexv ∈ V0. But this implies that we can successively assign each vertexv ∈ V0to av-friendly S ∈ H in such a way that to everyS ∈ H we assign at most�L′/2 verticesfrom V0.Consider a fixedS ∈ H and the setX ⊆ V0 of all vertices assigned toS. Let PS be

any 3-partite plane graph which satisfies the following three properties. Firstly, the classesof PS have sizesaL′, aL′ andL′, respectively. Secondly,�(PS)�8a and, thirdly,PS is atriangulation apart from|X| disjoint facial 4-cycles and the vertices of each of these 4-cycleslie in the two larger vertex classes ofPS . Such plane graphs exist, see e.g. Fig.2.Since each edge ofScorresponds to a(2ε, d/2)-super-regular subgraph ofG′, the Blow-

up lemma (Lemma 10) implies that the subgraph ofG′ corresponding toS contains aspanning copy ofPS where every vertexv ∈ X is joined to all vertices on one of the facial4-cycles inPS and these 4-cycles differ for distinct vertices fromX. (The latter can beachieved in a similar way as in the proof of Theorem 2.) Thus by inserting the vertices fromX into these facial 4-cycles ofPS we obtain a triangulation. Proceeding similarly for everyelement ofH, we obtain a spanning planar subgraph ofGwhich is the disjoint union of|H|triangulations and thus has 3n− 6|H|�3n− 6N(ε, k0) edges. �

276 D. Kühn et al. / Journal of Combinatorial Theory, Series B 95 (2005) 263–282

As a special case, the following proposition implies that the constantC in Theorem3must depend on� and that the extra�n in the condition on the minimum degree cannot bereplaced by a sublinear term.

Proposition 14. For all positive integers k and n which satisfyn/2+ k = r(2k + 1) forsome integerr�2 there is a graph G of order n and minimum degreen/2+ k which doesnot contain a planar subgraph with more than3n− 6− n/12k edges.

Proof. LetG be the graph obtained from a disjoint union ofr cliquesG1, . . . ,Gr of order2k+1 by adding a setYof n/2−k new vertices and joining every vertex inY to every vertexin V (G1) ∪ · · · ∪ V (Gr) =: X. SoG has ordern and minimum degreen/2+ k. Considera planar subgraphP of Gwith a maximum number of edges. PutC := 3n− 6− e(P ). Wewill show thatC�n/12k. LetE be a set ofC edges such thatP + E is a triangulation,Tsay. ThusE ∩ E(G) = ∅. Call an edgee ∈ E useful forGi if either

• ehas an endvertex inGi (and thus both endvertices ofe lie in X) or• ehas both endvertices inYand is an edge of a facial triangle ofTwhich contains a vertexof Gi .

We claim that for everyi there is an edge inE which is useful forGi . Since a given edgefrom E lies in two faces ofT and hence is useful for at most two cliquesGi , this wouldimply that

C = |E|� r

2= n/2+ k

4k + 2� n

8k + 4� n

12k

as desired. So fixi�r and let us now show that there is an edge inEwhich is useful forGi .Suppose not. Then every vertex ofGi lies in a facial triangle ofTwhich is contained inG.So each such triangle contains at least one edge ofGi . We say that all these facial trianglesofTare oftypeI and all other facial triangles (i.e. those which do not contain an edge ofGi)are oftypeII. So no vertex ofX−V (Gi) lies in a facial triangle of type I and thus there arefacial triangles of type II. SinceT is a triangulation, there is a path in the dual graph from atriangle of type I to a triangle of type II. Hence there is a triangle of type I which shares anedge with some triangleD of type II. ButD cannot be contained inG, and so it contains anedgee fromE. It is now easy to check thate is useful forGi , a contradiction. �

4.4. Triangulations and quadrangulations

ThesquareG2 of a graphG is the graph obtained fromG by adding an edge betweenevery two vertices of distance two inG. For the proof of Theorem4wewill use the followingresult of Fan and Kierstead [10]. (It was extended to arbitrary powers of Hamilton cyclesby Komlós et al. [17], see also [16].)

Theorem 15. Every graph of minimum degree at least2|G|/3 contains the square of aHamilton path.

D. Kühn et al. / Journal of Combinatorial Theory, Series B 95 (2005) 263–282 277

Proof of Theorem 4. Clearly, we may assume that� < 1/3. Apply Proposition 9 toobtain ε0(�) and d0(�). Put d := min{�, d0(�)}. Let ε0(d/2,8,3, (d/2)4) =: ε∗ and�(d/2,8,3, (d/2)4) =: � be as given in the Blow-up lemma (Lemma 10). Set

ε := min

{ε0(�),

ε∗

3,

��

252,d3

16

}

andk0 := max{2/ε,20/�}. Throughout the proof we assume thatn is sufficiently large forour estimates to hold.Apply the Regularity lemma (Lemma7) toG to obtain an exceptional setV0 and clusters

V1, . . . , Vk wherek0�k�N(ε, k0). LetL andG′ be as defined in the Regularity lemma. Byaddingatmost 2 of theVi to theexceptional setV0 if necessary,wemayassume that 3 dividesk.Westill denote theenlargedexceptional set byV0.Thus|V0|�εn+2L�εn+2n/k0�2εn.Let R denote the reduced graph. By Proposition 9 we have�(R)�(2/3+ �/2)k − 2. SoTheorem15 implies thatRcontains the square of a Hamilton pathP.As�(P 2) = 4, wemayapply Proposition 8 to obtain adjusted clustersV ′

i ⊆ Vi (i�1) of size(1−4ε)L =: L′ suchthat every edge ofP 2 corresponds to a(2ε, d−5ε)-super-regular subgraph ofG′.We add allvertices that do not lie in someV ′

i to the exceptional setV0. Thus|V0|�2εn+4εkL�6εn.Given a vertexx ∈ R, we will write V ′(x) for the adjusted cluster corresponding tox.Since|V ′(x)|, |V ′(y)|�L/2 for every edgexy ∈ R, it follows from theε-regularity ofthe original pair that the graph(V ′(x), V ′(y))G′ corresponding toxy is 2ε-regular and hasdensity> d − ε.Partition the vertices ofP 2 into k′ := k/3 disjoint setsD1, . . . , Dk′ , each containing

3 consecutive vertices ofP. So the vertices in eachDi induce a triangle ofP 2. For all1� i < k′ letNi be the number of vertices ofRwhich are joined to at least five of the sixvertices inDi ∪Di+1. Then

6�(R)− 2e(R[Di ∪Di+1]) � eR(Di ∪Di+1, V (R) \ (Di ∪Di+1))� 6Ni + 4|R| (5)

and thus

Ni��(R)− 2|R|/3− e(R[Di ∪Di+1])/3��k/2− 2− 5> 0.

So for each 1� i < k′ we can find a vertexai ∈ R as well as verticessi, ti ∈ Di andui+1, wi+1 ∈ Di+1 with siui+1 ∈ P 2 and such that inR each ofsi, ti , ui+1, wi+1 isjoined toai . (Here the verticesai need not be distinct for differenti.) As each edge ofR corresponds to a 2ε-regular subgraph ofG′ of density> d − ε, it easily follows fromrepeated applications of Proposition6 that there are verticesxi �= yi in V ′(ai) such that inthe graphG′ their common neighbourhood in each ofV ′(si), V ′(ti), V ′(ui+1), V ′(wi+1)has size at least(d − 3ε)2L′. Moreover, all these verticesxi andyi can be chosen to bedistinct.Roughly speaking, theproof nowproceedsas follows.Weapply theBlow-up lemmato obtain for alli an (almost) triangulation which is a spanning subgraph of the subgraphof G′ corresponding toDi . (Each exceptional vertex will also be added to one of thesetriangulations.) The verticesxi andyi will be used to ‘glue together’ all these triangulations

278 D. Kühn et al. / Journal of Combinatorial Theory, Series B 95 (2005) 263–282

into a single triangulation containing all vertices ofG. In this gluing process we will alsouse two edges betweenV ′(si) andV ′(ui+1).So letSi ⊆ V ′(si) be any set consisting of(d−3ε)3L′ vertices which lie in the common

neighbourhood ofxi andyi but are not of the formxj or yj (1�j < k′). Note that thisis possible since(d − 3ε)3L′ �(d − 3ε)2L′ − 2k′. DefineTi, Ui+1 andWi+1 similarly.Since we still have|Ui+1|�2εL′, we can apply Proposition6 again to find a setS′

i ⊆Si of size (d − 3ε)4L′ � |Si | − 2εL′ such that inG′ each vertex fromS′

i has at least(d − 3ε)|Ui+1|�(d − 3ε)4L′ neighbours inUi+1.Remove allxi andyi from the adjusted clusters to which they belong (but do not add

them toV0). Then the sizes of the clusters thus obtained lie betweenL′ − 2k′ andL′. Set� := �(L′ − 2k′)/4�. By moving a constant number of vertices intoV0 if necessary, wemay assume that for all 1� i�k′ every cluster belonging toDi has size 4� =: L′′. We stilldenote byV ′(x) the (re)-adjusted cluster corresponding to a vertexx ∈ R and byV0 theenlarged exceptional set. Thus|V0|�7εn and each pair of clusters inDi still correspondsto a (3ε, d/2)-super-regular subgraph ofG′. Furthermore, we can easily ensure that eachnewly adjusted cluster of the formV ′(si), V ′(ti), V ′(ui) or V ′(wi) still containsS′

i , Ti , UiorWi respectively.LetH1,H2 andH3 be the 3-partite plane graphs of order 3L′′ given in Fig. 3. So eachHi

has maximum degree 8 and all of its vertex classes have sizeL′′ = 4�. Moreover, bothH1andH2 are triangulations apart from two disjoint facial 4-cycles. InH1 the vertices on these4-cycles lie in the same two vertex classes while inH2 one of the 4-cycles has its vertices inthe first and second vertex class and the other one in the second and third vertex class.H3 isa triangulation apart from one facial 4-cycle. Formally,H1 can be constructed as follows:begin with a set of 2� + 1 cyclesCi = a1i a

2i a3i a4i of length four, where 1� i�2� + 1.

Next, for all i�2� andj�4, connectaji to aji+1. This gives a bipartite quadrangulation of

the plane. Now for all eveni with 1 < i�2�, subdivide each edge ofCi by inserting onenew vertex (which thus has degree two at this stage). If the new vertex is adjacent toa

ji

andaj+1i say (where the superscripts are modulo 4), then add edges from the new vertex

to each ofaji±1 andaj+1i±1 . Altogether, this gives us a 3-partite triangulation except for two

facial 4-cycles. Finally, to ensure that the vertex classes have equal size, we remove the fourvertices onC2 and instead insert the edgesa

j1a

j3 (j = 1, . . . ,4). We obtainH3 fromH1 as

follows: denote the new vertex which is incident toa11 anda21 by v. Now delete the edge

a11a21 and add the two edgesva

31 andva

41. Finally, we obtainH2 fromH1 as follows: define

v as above and letw denote the new vertex which is incident toa31a41. This time, delete the

two edgesa11a21, a

31a41 and add the two edgesva

31 andwa

11.

The Blow-up lemma implies that for all 1� i�k′ the subgraph ofG′ corresponding toR[Di] contains a spanning copy of each ofH1,H2 andH3. However, before we apply theBlow-up lemma we also have to take care of the exceptional vertices. So given a vertexv ∈ V0 and 1� i�k′, we say thatDi isv-friendly if each of the three newly adjusted clustersin Di contains at least�L′′ neighbours ofv. LetNv denote the number ofv-friendlyDi ’s.Then

(2/3+ �/2)n < dG(v)− |V0|�Nv3L′′ + k′(2+ �)L′′.

D. Kühn et al. / Journal of Combinatorial Theory, Series B 95 (2005) 263–282 279

H3

H1 H2

Fig. 3. The graphsH1,H2 andH3, the only non-triangular facial cycles are indicated with thick lines.

Thus

Nv >n

3L′′

(2

3+ �

2− k(2+ �)L′′

3n

)� n

3L′′( �

2− �

3

)= �n

18L′′

and hence

2|V0|�L′′ � 14εn

�L′′ < Nv

for everyv ∈ V0. This shows that we can successively assign each exceptional vertexv ∈ V0to somev-friendlyDi in such a way that to eachDi we assign at most�L′′/2 vertices.We are now ready to construct our spanning triangulation ofG. We first apply the Blow-

up lemma to find a spanning copyP1 of H3 in the subgraph ofG′ corresponding toR[D1]so that the vertices of the unique facial 4-cycle inP1 lie alternately inS′

1 andT1 and so thatevery exceptional vertexv assigned toD1 is joined to all vertices on some facial triangleof P1 where these facial triangles are disjoint for distinct such verticesv ∈ V0. (This canbe done in a similar way as in the proof of Theorem2 sinceH3 contains at least�L′′/2disjoint facial triangles which are also disjoint from the unique facial 4-cycle ofH3.) Letx1S, y

1S ∈ S′

1 andx1T , y

1T ∈ T1 be the vertices of the facial 4-cycle ofP1 and call this cycle

C1ST .

280 D. Kühn et al. / Journal of Combinatorial Theory, Series B 95 (2005) 263–282

yT2

yS2

xT2

xS2

P2

x

yT1 y1S

xT1

P1

yU2

y2

y1

xU x2W

x1

W

2

S1

Fig. 4. Gluing two almost-triangulationsP1 andP2.

For 1< i < k′, we now say thatDi is of typeI if the unordered pairssi, ti andui, wicoincide and oftype II if they differ. The pair si, ti will be used to ‘glue’ the (almost)triangulationPi corresponding toDi to that corresponding toDi+1, whereas the pairui, wiwill be used to ‘glue’Pi to the (almost) triangulation corresponding toDi−1. As the nextstep, we apply the Blow-up lemma to find a spanning copyP2 of H1 if D2 is of type I, orofH2 if it is of type II, in the subgraph ofG′ corresponding toR[D2] such that the verticesof one facial 4-cycle lie alternately inS′

2 andT2, the vertices of the other facial 4-cycle liealternately inU2 andW2 and such that every exceptional vertexv assigned toD2 is joinedto all vertices on some facial triangle ofP2. (Again, these facial triangles are disjoint fordistinct such verticesv.) Let x2S, y

2S ∈ S′

2 andx2T , y

2T ∈ T2 be the vertices of the first facial

4-cycleC2ST and letx2U , y

2U ∈ U2 andx2W, y

2W ∈ W2 be the vertices of the other facial

4-cycleC2UW . As, by definition ofS′1, each ofx

1S, y

1S has at least(d − 3ε)4L′ neighbours

in U2, we may also require thatx2U is joined tox1S andy

2U is joined toy

1S . (To achieve this,

we restrict the image ofx2U to the neighbourhood ofx1S in U2 and the image ofy

2U to the

neighbourhood ofy1S in U2.) Furthermore, by definition ofS′1, T1, U2 andW2, bothx1 and

y1 are joined to all vertices ofC1ST andC2UW . Thusx1 andy1 may be used to ‘glue’P1 and

P2 together in order to obtain a planar graph which is a triangulation apart from one facial4-cycle, namelyC2ST (Fig. 4).We may continue in this fashion to obtain a spanning triangulation. Indeed, forPk′ we

again choose a copy ofH3 such that the vertices on the unique facial 4-cycleCk′UW of Pk′ lie

alternately inUk′ andWk′ and such that one of the two vertices fromUk′ onCk′UW is joined

D. Kühn et al. / Journal of Combinatorial Theory, Series B 95 (2005) 263–282 281

P2P1

y1

x1

Fig. 5. Gluing two quadrangulationsP1 andP2.

to xk′−1S while the other one is joined toyk

′−1S . Thus if we gluePk′ into the planar graph

constructed in the previous step, we obtain a triangulationT. As each exceptional vertexvis joined to all vertices on some facial triangle ofT and all these are distinct, we can addthe exceptional vertices toT to obtain a triangulation containing all vertices ofG. �

Proof of Theorem 5 (Sketch). The proof proceeds in a similar way as that of Theorem 4except for a fewmodifications (and simplifications) which we describe below.Wemay nowassume that the reducedgraphRhasevenorderandcontainsaHamiltonpathP(insteadof thesquare of aHamilton path).We partitionP into |P |/2 := k′ independent edgesD1, . . . , Dk′ .We then adjust the clusters such that each edgeDi corresponds to a(2ε, d − 2ε)-super-regular subgraph ofG′. A calculation similar to (5) shows that for every pairDi,Di+1 thereis a vertexai ∈ R which is joined to both a vertexsi ∈ Di and a vertexui+1 ∈ Di+1. Wechoose two verticesxi, yi ∈ V ′(ai) which have many common neighbours in bothV ′(si)andV ′(ui+1). Finally, we apply the Blow-up lemma to obtain spanning quadrangulationsPi of the subgraphs ofG′ corresponding to theDi which are ‘glued together’ into a singlequadrangulationP using the verticesxi andyi (Fig. 5). These quadrangulations are chosenso that every exceptional vertexv is joined to two opposite vertices on some facial 4-cyclewhere these 4-cycles are disjoint for distinct exceptional verticesv. So all the exceptionalvertices can be added toP to obtain a spanning quadrangulation ofG. �

As remarked towards the end of Section 1, the planar graphs guaranteed by Theorems2–5 can be constructed in polynomial time: both the Regularity lemma and the Blow-up

282 D. Kühn et al. / Journal of Combinatorial Theory, Series B 95 (2005) 263–282

lemma can be implemented in polynomial time (see[1,15]). As the order of the reducedgraph is constant, the remaining steps can also be carried out in polynomial time.

Acknowledgments

Wewould like to thank Béla Bollobás,Matthias Jüngel andCarstenThomassen for usefuldiscussions.

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