Forcing unbalanced complete bipartite minors

European Journal of Combinatorics 26 (2005) 75–81

www.elsevier.com/locate/ejc

Forcing unbalanced complete bipartite minors

Daniela Kuhna, Deryk Osthusb

aInstitut fur Mathematik, Freie Universität Berlin, Arnimallee 2–6, D-14195 Berlin, GermanybInstitut fur Informatik, Humboldt-Universität zu Berlin, Unter den Linden 6, D-10099 Berlin, Germany

Received 13 August 2003; received in revised form 4 February 2004; accepted 6 February 2004

Available online 4 March 2004

Abstract

Myers conjectured that for every integers there exists a positive constantC such that for allintegerst every graph of average degree at leastCt contains aKs,t minor. We prove the followingstronger result: for every 0< ε < 10−16 there exists anumbert0 = t0(ε) such that for all integerst ≥ t0 ands ≤ ε7t/ log t every graph of average degree at least(1 + ε)t contains aKs + Kt minor(and thus also aKs,t minor). The bounds are essentially the best possible.© 2004 Elsevier Ltd. All rights reserved.

1. Introduction

Let d(s) be the smallest number such that every graph of average degree greaterthan d(s) contains the complete graphKs as minor. The existence ofd(s) was firstproved by Mader [6]. Kostochka [4] and Thomason [12] independently showed that theorder of magnitude ofd(s) is s

√logs. Later, Thomason [13] was able to prove that

d(s) = (α + o(1))s√

logs, whereα = 0.638. . . is an explicit constant. Here the lowerbound ond(s) is provided by random graphs. In fact, Myers [8] proved that allextremalgraphs are essentially disjoint unions of pseudo-random graphs.

Recently, Myers and Thomason [10] extended the results of [13] from complete minorsto H minors for arbitrary dense (and large) graphsH . The extremal function has the sameform asd(s), except thatα ≤ 0.638. . . is now an explicit parameter depending onH andsis replaced by the order ofH . They raised the question of what happens for sparse graphsH . Onepartial result in this direction was obtained by Myers [9]: he showed that everygraph of average degree at leastt + 1 contains aK2,t minor. This is best possible as heobserved that for all positiveε there are infinitely many graphs of average degree at least

E-mail addresses:[email protected] (D. Kühn), [email protected] (D. Osthus).

0195-6698/$ - see front matter © 2004 Elsevier Ltd. All rights reserved.doi:10.1016/j.ejc.2004.02.002

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76 D. Kuhn, D. Osthus / European Journal of Combinatorics 26 (2005) 75–81

t +1−ε which do not contain aK2,t minor. (These examples also show that random graphsare not extremal in this case.) More generally, Myers [9] conjectured that for fixeds theextremal function for aKs,t minor islinear in t :

Conjecture 1 (Myers). Given s∈ N, there exists a positive constant C such that for allt ∈ N every graph of average degree at least Ct contains a Ks,t minor.

Here we prove the following strengthened version of this conjecture. (It implies thatasymptotically the influence of the number of edges on the extremal function is negligible.)

Theorem 2. For every0 < ε < 10−16 there exists anumber t0 = t0(ε) such that for allintegers t≥ t0 and s≤ ε6t/ log t every graph of average degree at least(1 + ε)t containsa Ks,t minor.

Since Ks,s+t contains Ks + K t as a minor,Theorem 2immediately implies thefollowing:

Corollary 3. For every0 < ε < 10−16 there exists anumber t0 = t0(ε) such that for allintegers t≥ t0 and s≤ ε7t/ log t every graph of average degree at least(1 + ε)t containsKs + K t as a minor.

(HereKs + K t denotes the graph which is obtained fromKs,t by adding all edges betweenthe vertices in the vertex class of sizes.) Theorem 2and Corollary 3 are essentiallybest possible in two ways. Firstly, the complete graphKs+t−1 shows that up to theerror termεt the bound on the average degree cannot be reduced. Secondly, as we willsee inProposition 10(applied with α := 1/3), Theorem 2(and thus alsoCorollary 3)breaks down if we try to sets ≥ 18t/ log t . Moreover,Proposition 10also implies that ift/ log t = o(s) then even a linear average degree (as inConjecture 1) no longer suffices toforce aKs,t minor.

The case wheres = ct for some constant 0< c ≤ 1 and where we are looking for aKs,t minor is covered by the results of Myers and Thomason [10]. The extremal function

in this case is(α 2√

c1+c + o(1))r

√logr whereα = 0.638. . . again andr = s + t .

For thecase whens is much smaller than logarithmic int , Kostochka and Prince [5]obtained more precise upper and lower bounds on the average degree required to forceKs + K t as a minor. (They proved these results slightly later but independently of us.)

Theorem 4 ([5]). Let s and t be positive integers with t> (180s log2 s)1+6s log2 s. Thenevery graph G of average degree at least t+3s contains Ks + K t as a minor. On the otherhand, there are infinitely many graphs of average degree at least t+ 3s − 5

√s which do

not have a Ks + K t minor.

This note is organized as follows. We first proveTheorem 2for graphs whoseconnectivity is linear in their order (Lemma 9). We then use the ideas of Thomason [13] toextendthe result to arbitrary graphs.

2. Notation and tools

We write e(G) for the number of edges of a graphG, |G| for its order andd(G) :=2e(G)/|G| for its average degree. We denote the degree of a vertexx ∈ G by dG(x) and

D. Kuhn, D. Osthus / European Journal of Combinatorics 26 (2005) 75–81 77

the set of its neighbours byNG(x). If P = x1 . . . x� is a path and 1≤ i ≤ j ≤ �, we writexi Pxj for its subpathxi . . . x j .

We say that a graphH is a minor of G if for every vertex h ∈ H there is a setCh ⊆ V(G) such that all Ch are disjoint, eachG[Ch] is connected andG contains aCh–Ch′ edge wheneverhh′ is an edge inH . Ch is called thebranch set correspondingto h.

We will use the followingresult of Mader [7].

Theorem 5. Every graph G contains a�d(G)/4�-connected subgraph.

Givenk ∈ N, we say that a graphG is k-linked if |G| ≥ 2k and for every 2k distinctvertices x1, . . . , xk and y1, . . . , yk of G there exist disjoint pathsP1, . . . , Pk suchthatPi joins xi to yi . Jung as well as Larman and Mani independently proved that everysufficiently highly connected graph isk-linked. Later, Bollobas and Thomason [2] showedthat a connectivity linear ink suffices. Simplifying the argument in [2], Thomas and Wollan[11] recently obtained an even better bound:1

Theorem 6. Every16k-connected graph is k-linked.

Similarly as in [13], given positive numbersd andk, we shallconsider the classGd,k ofgraphs defined by

Gd,k := {G : |G| ≥ d, e(G) > d|G| − kd}.We say that a graphG is minor-minimal inGd,k if G belongs toGd,k but no proper minorof G does. The following lemma states some properties of the minor-minimal elements ofGd,k. The proof is simple, its counterpart for digraphs can be found in [13, Section 2]. (Thefirst property follows by counting the number of edges of the complete graph on(2−ε)dvertices.)

Lemma 7. Given0 < ε < 1/2, d ≥ 2/ε and 1/d ≤ k ≤ εd/2, every minor-minimalgraph inGd,k satisfies the followingproperties:

(i) |G| ≥ (2 − ε)d,(ii) e(G) ≤ d|G| − kd + 1,(iii) every edge of G lies in more than d− 1 triangles,(iv) G is �k�-connected.

We will also use the following easy fact, see [13, Lemma 4.2] for a proof.

Lemma 8. Suppose that x and y are distinct vertices of a k-connected graph G. Then Gcontains at least k2/4|G| internally disjoint x–y paths of length at most2|G|/k.

3. Proof of Theorem 2

The strategy of the proof ofTheorem 2is as follows. It is easily seen that to proveTheorem 2for all graphs of average degree at least(1 + ε)t =: d, it suffices to consider

1 After completing this paper we learned that the 16k in Theorem 6was further improved to 12k byKawarabayashi, Kostochka and Yu [3]. Very recently, this was improved again by Thomas and Wollan [11]to 10k.


only those graphsG which are minor-minimal in the classGd/2,k for some suitablek. Inparticular, togetherwith Lemma 7this implies that we only have to deal withk-connectedgraphs. Ifd (and soalsok) is linear in the order ofG, then asimple probabilistic argumentgives us the desiredKs,t minor (Lemma 9). In the other case we use that byLemma 7eachvertex ofG together with its neighbourhood induces a dense subgraph ofG. We apply thisto find 10 disjointK10s,�d/9� minors which we combine to aKs,t minor.

Lemma 9. For all 0 < ε, c < 1 there exists a number k0 = k0(ε, c) such that for eachinteger k ≥ k0 every k-connected graph G whose order n satisfies k≥ cn contains aKs,t minor where t := �(1 − ε)n� and s := �c4εn/(32 logn)�. Moreover, the branchsets corresponding to the vertices in the vertex class of Ks,t of size t can be chosen to besingletons whereas all the other branch sets can be chosen to have size at most8 logn/c2.

Proof. Throughout the proof we assume thatk (and thus alson) is sufficiently large com-pared with bothε andc for our estimates to hold. Puta := 4 logs/c. Successively chooseas vertices ofG uniformly at random without repetitions. LetC1 be the set of the firstaof these vertices, letC2 be the set of the nexta vertices and so on up toCs. Let C be theunion of all theCi . Given i ≤ s, we call a vertexx ∈ G − C good for i if x has at leastone neighbour inCi . Moreover, we say thatx is goodif it is good for everyi ≤ s. Thus

P(x is not good fori ) ≤(

1 − dG(x) − as

n

)a

≤ e−a(k−as)/n ≤ e−ac/2

and sox is not good with probability at mostse−ac/2 < ε/2. Therefore the expected num-ber of good vertices outsideC is at least(1− ε/2)|G − C|. Hence there exists an outcomeC1, . . . , Cs for which at least(1 − ε/2)|G − C| vertices inG − C are good.

We now extend alltheseCi to disjoint connected subgraphs ofG as follows. Let us startwith C1. Fix a vertexx1 ∈ C1. For eachx ∈ C1\{x1} we in turn applyLemma 8to findanx–x1 path of length at most 2n/k ≤ 2/c which is internally disjoint from all the pathschosen previously and which avoidsC2 ∪ · · · ∪ Cs. SinceLemma 8guarantees at leastk2/4n ≥ as · 2/c short paths between a given pair of vertices, we are able to extend eachCi in turn to a connected subgraph in this fashion. Denote the graphs thus obtained fromC1, . . . , Cs by G1, . . . , Gs. Thus all theGi are disjoint.

Note that at most 2as/c good vertices lie in someGi . Thus at least(1− ε/2)|G − C| −2as/c ≥ (1 − ε)n good vertices avoid all theGi . Hence G contains aKs,t minor as re-quired. (The good vertices avoiding allGi correspond to the vertices ofKs,t in the vertexclass of sizet . The branch sets corresponding to the vertices ofKs,t in the vertex class ofsizes are the vertex sets ofG1, . . . , Gs.) �

Proof of Theorem 2 . Let d := (1+ ε)t ands := ε6d/ logd. Throughout the proof weassume thatt (and thus alsod) is sufficiently large compared withε for our estimates tohold. We have to show that every graph of average degree at leastd contains aKs,t minor.Putk := �εd/4�. SinceGd/2,k contains all graphs of average degree at leastd, it sufficesto show that every graphG which is minor-minimal inGd/2,k contains aKs,t minor. Letn := |G|. As iseasily seen, (i) and (iv) ofLemma 7together withLemma 9imply that wemay assume thatd ≤ n/600. (Lemma 9is applied withc := ε/2400 and withε replacedby ε/3.) Let X be the set of all those vertices ofG whose degree is at most 2d. Since by


Lemma 7(ii) the average degree ofG is at mostd, it follows that|X| ≥ n/2. Let us firstprove the following claim.

Either G contains a Ks,t minor or G contains 10 disjoint�3d/25�-connectedsubgraphs G1, . . . , G10 suchthat 3d/25 ≤ |Gi | ≤ 3d for each i≤ 10.

Choose a vertexx1 ∈ X and let G′1 denote the subgraph ofG induced byx1 and its

neighbourhood. Then|G′1| = dG(x1) + 1 ≤ 2d + 1. Since byLemma 7(iii) each edge

betweenx1 andNG(x1) lies in at leastd/2−1 triangles, it follows that the minimum degreeof G′

1 is at leastd/2 − 1. ThusTheorem 5implies thatG′1 contains a�3d/25�-connected

subgraph. TakeG1 to be this subgraph. PutX1 := X\V(G1) and letX′1 be the set of all

those vertices inX1 which have at leastd/500 neighbours inG1.Suppose first that|X′

1| ≥ |X|/10. In this case we will find aKs,t minor in G. Sincethe argument is similar to theproof of Lemma 9, we only sketch it. Seta := 104 logs.This time, we choose thea-element setsC1, . . . , Cs randomly insideV(G1). Sinceeveryvertex inX′

1 has at leastd/500 neighbours inG1, the probability that the neighbourhood

of a given vertexx ∈ X′1 avoids someCi is at mostse−a/(3×103) < ε. So the expected

number of such bad vertices inX′1 is at mostε|X′

1|. Thus for some choice ofC1, . . . , Cs

there are at least(1 − ε)|X′1| ≥ (1 − ε)n/20 ≥ t vertices inX′

1 which have a neighbour ineachCi . Sincethe connectivity ofG1 is linear in its order, we may again applyLemma 8to make theCi into disjoint connected subgraphs ofG1 by adding suitable short paths fromG1. This shows thatG contains aKs,t minor.

Thus we may assume that at least|X1| − |X|/10 ≥ 9|X|/10− 3d > 0 vertices inX1have at mostd/500 neighbours inG1. Choose such a vertexx2. Let G′

2 be the subgraphof G induced byx2 and all its neighbours outsideG1. Since byLemma 7(iii) every edgeof G lies in at leastd/2 − 1 triangles, it follows that the minimum degree ofG′

2 is at leastd/2 − 1 − d/500 > 12d/25. Again, we takeG2 to be a�3d/25�-connected subgraph ofG′

2 obtained byTheorem 5.We now put X2 := X1\(X′

1 ∪ V(G2)) and defineX′2 to be the set of all those vertices

in X2 which have at leastd/500 neighbours inG2. If |X′2| ≥ |X|/10, then as before,

we can find aKs,t minor in G. If |X′2| ≤ |X|/10 we defineG3 in a similar way asG2.

Continuing in this fashion proves the claim. (Note that when choosingx10 we still have|X9| − |X|/10 ≥ |X|/10− 9 · 3d > 0 vertices at our disposal sincen ≥ 600d.)

Apply Lemma 9with c := 1/25 to eachGi to find aK10s,�d/9� minor. LetCi1, . . . , Ci

s,

Di1, . . . , Di

9s denote the branch sets corresponding to the vertices of theK10s,�d/9� in thevertex class of size 10s. By Lemma 9we may assume that allCi

j and all Dij have size

at most 8· 252 log |Gi | ≤ 105 logd and that all the branch sets corresponding to theremaining vertices ofK10s,�d/9� are singletons. LetTi ⊆ V(Gi ) denote the union of allthese singletons. LetC be the union of allCi

j , let D be the union of allDij and letT be the

union of allTi .We will now use these 10K10s,�d/9� minors to form aKs,t minor in G. Recall that by

Lemma 7(iv) the graphG is �εd/4�-connected and so byTheorem 6it is εd/64-linked.Thus there exists a setP of 9s disjoint paths inG such that for all i ≤ 9 and all j ≤ s thesetCi

j is joined toCi+1j by one of these paths and such that no path fromP contains an

inner vertex inC ∪ D. (To see this, use thatεd/64 ≥ 100s · 105 logd ≥ |C ∪ D|.)


The paths inP can meetT in many vertices. But we can reroute them such that everynew path contains at most two vertices from eachTi . For every pathP ∈ P in turn we willdo this as follows. IfP meetsT1 in more than two vertices, lett andt ′ denote the first andthe last vertex fromT1 on P. Choose some setD1

j and replace the subpatht Pt′ by some

path betweent andt ′ whose interior lies entirely inG[D1j ]. (This is possible sinceG[D1

j ]is connected and since botht andt ′ have a neighbour inD1

j .) Proceed similarly if the path

thus obtained still meets some otherTi . Then continue withthe next path fromP . (ThesetsDi

j used for the rerouting are chosen to be distinct for different paths.) Note that thepaths thus obtained are still disjoint sinceD was avoidedby all the paths inP .

We now have found ourKs,t minor. Each vertex lying in the vertex class of sizes ofthe Ks,t corresponds to a set consisting ofC1

j ∪ · · · ∪ C10j together with the (rerouted)

paths joining these sets. For the remaining vertices of theKs,t we can take all thevertices inT which are avoided by the (rerouted) paths. There are at leastt such verticessince these paths contain at most 20· 9s vertices fromT and |T | − 180s ≥ 10d/9 −180s ≥ t . �

The following proposition shows that the bound ons in Theorem 2is essentially thebest possible. Its proof is an adaption of a well-known argument of Bollobás, Catlin andErdos [1].

Proposition 10. There exists an integer n0 such that for each integer n≥ n0 and eachnumberα > 0 there is a graph G of order n and with average degree at least n/2 whichdoes not have a Ks,t minor with s := �2n/α logn� and t := �αn�.

Proof. Let p := 1−1/e.Throughout the proof we assume thatn is sufficiently large for ourestimates to hold. Consider a random graphGp of ordern which isobtained by includingeach edge with probabilityp independently of all other edges. We will show that withpositive probabilityGp is as required in the proposition. Clearly, with probability> 3/4the averagedegree ofGp is at leastn/2. Hence it suffices to show that with probability atmost 1/2 thegraphGp will have the property that its vertex setV(Gp) can be partitionedinto disjoint setsS1, . . . , Ss andT1, . . . , Tt suchthat Gp contains an edge between everypair Si , Tj (1 ≤ i ≤ s, 1 ≤ j ≤ t). Call such a partition ofV(Gp) admissible. Thus wehave to show that the probability thatGp has an admissible partition is≤ 1/2. Let us firstestimate the probability that a given partitionP is admissible:

P(P is admissible) =∏i, j

(1 − (1 − p)|Si ||Tj |

)≤ exp

−

∑i, j

(1 − p)|Si ||Tj |

≤ exp

−st

∏i, j

(1 − p)|Si ||Tj |(st)−1

≤ exp(−st(1 − p)n2(st)−1

)

≤ exp

(− 2n2

logn· n− 1

2

)≤ exp(−n

43 ).


(The first expression in the second line followssince the arithmetic mean is at least as largeas the geometric mean.) Since the number of possible partitions is at mostnn, it followsthat the probability thatGp has an admissible partition is at mostnn · e−n4/3

< 1/2, asrequired. �

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Forcing unbalanced complete bipartite minors