Constructing dense graphs with sublinear Hadwiger number

Constructing dense graphs with sublinear Hadwig er n um b er Jacob F o x ∗ Abstract Mader asked to explicitly construct dense g raphs for which the size of the lar gest clique minor is sublinear in the num ber of v ertices. Such graphs exist as a random graph almost s urely has this prop erty . This q uestion and v ariants w ere p opular ized by Thomaso n ov er several articles . W e answer these questions b y showing how to ex plicitly constr uct s uch graphs using blow-ups of small graphs with t his prope rty . This leads to the study of a fr a ctional v ariant of the clique minor nu mber, which may b e of indep endent interest. 1 In tro duction A graph H is a minor of a graph G if H can b e obtained from a subgraph of G b y con tracting edges. Minors form an imp ortant connection b et w een graph theory , geometry , and top ology . F or example, the K urato wski-W ag ner theorem state s that a graph can b e em b ed ded in the p lane if and only if it has neither the complete graph K 5 nor the complete bip artite graph K 3 , 3 as a minor. This examp le pla y ed an imp ortan t role in the dev elopmen t of top ological graph theory , w hose masterpiece is the Rob ertson-Seymour graph minor theorem. In a series of t w en t y pap ers [15], they p ro v ed W agner’s conjecture that ev ery family of graphs closed u nder taking minors is c haracterized b y a finite list of forbidden minors. The Hadwiger numb er h ( G ) of a graph G is the order of the largest clique w h ic h is a minor of G . The famous conjecture of Hadwiger [8] states th at ev ery graph of c hromatic num b er k h as Hadwiger n um b er at least k . Hadwiger prov ed his conjecture for k ≤ 4. W agner [25] pr o ved that the case k = 5 is equiv alen t to the F our Color Theorem. In a tour de force, Rob ertson, Seymour, and T homas [16] settled the case k = 6 also us in g the F our Color Theorem. T he conjecture is s till op en for k ≥ 7. Bollob´ as, Catlin, and Erd˝ os [2] analyzed the Hadwiger num b er of random graphs. They sh o wed that a random graph G on n vertice s almost sur ely satisfies h ( G ) is asymptotic to n √ log n . Here, and throughout the pap er, all logarithms unless otherwise indicated are in base 2. Al so using the w ell kno wn fact that the c hromatic num b er of a random graph on n vertice s is almost su rely Θ( n/ log n ), they deduced that almost all graphs satisfy Hadwiger’s conjecture. Mader sho w ed that large a v erage degree is enough to i mply a large cli que minor. Precisely , f or eac h in teger t t here is a c onstan t c ( t ) suc h that ev er y graph G of a ve rage degree at least c ( t ) satisfies h ( G ) ≥ t . K ostochk a [9, 10] and Thomason [19] indep endently prov ed that c ( t ) = Θ( t √ log t ). Thomason [22] later determined the asymptotic b eha vior of c ( t ), with ran d om graph s of a particular d ensit y as extremal graphs for this problem. My ers [13] pro v ed that an y extremal grap h under certain conditions for this p roblem m u st b e q u asirandom. Random graphs hav e some remark able prop erties for w h ic h it is difficult to explicitly construct graphs with th ese prop erties. One w ell-kno wn example is Erd˝ os’ lo wer b ound on Ramsey num b ers, whic h sho ws that almost all graphs on n v ertices d o not con tain a clique or indep enden t set of order ∗ Department of Mathematics, Massac h usetts Institute of T echnolo gy , Cam bridge, MA 02139-4307. E-mail: fo x@math.mit.edu. Resea rc h sup p orted by a Simons F ellows hip and NSF grant DMS-1069197. 1 2 log n . Despite considerable attent ion o ver the last 60 years, there is n o kno wn construction (in p olynomial time) of a graph G on n vertice s for wh ic h the largest clique or indep endent set in G is of order O (log n ). Another in teresting pr op ert y of random graphs already ment ioned is that almost surely they do not con tain a clique minor of linear size. Mader asked to construct a d ense graph on n v ertices with h ( G ) = o ( n ). On e of the main motiv ations for this problem is that p ro ving in teresting up p er b ounds on the Hadwiger n u m b er of a graph app ears to b e a difficu lt problem. Thomason [20 ] sho wed th at man y of th e standard constructions of qu asirandom graphs ha v e linear clique minors and therefore cannot b e used to answer Mader’s problem. Mader’s problem and a few v ariant s were discussed b y Thomason in sev eral pap ers [20, 21, 22, 23, 24] and also by Myers [13]. Th omason [20] p osed the s tr onger problem of constructing a graph G on n v ertices for wh ic h the Hadwiger n umbers of G and its complemen t ¯ G are b oth o ( n ). He sp eculates [21] that this problem migh t b e as hard as the classica l Ramsey p roblem of finding exp licit graphs G suc h that b oth G and ¯ G con tain only small complete su bgraphs. Here we solve b oth the problems of Mader and Thomason. T o do so, it is helpful to define what an explicit construction is. W e view a graph on 2 n v ertices as a fun ction f :  { 0 , 1 } n 2  → { 0 , 1 } , where the v alue of f tells whether or n ot t w o ve rtices are adjacen t. By an explicit construction we mean that the function f is computable in p olynomial time (in n ). That is, given t w o v ertices, we can compute whether or not they are adjacen t in time p olynomial in the num b er of bits u sed to represent the vertice s. There is also a weak er notion of explicit graph whic h is sometimes used. In this v ersion, the edges of the graph can b e computed in time p olynomial in the num b er of ve rtices of the graph. Our construction whic h answe rs the p r oblems of Mader and Thomason is given in ne arly c onstant time using blow-ups of nearly constan t size graphs. W e show that if a (small) graph is dense and has relativ ely small Hadwiger n u m b er, then its blo w-u ps also h a ve this p rop erty . W e f ormally defi ne the blo w -up of a grap h as follo ws. F or graphs G and H , the lexic o g r aphic pr o duct G · H is the graph on vertex set V ( G ) × V ( H ), where ( u 1 , v 1 ) , ( u 2 , v 2 ) ∈ V ( G ) × V ( H ) are adjacen t if and only if u 1 is adj acen t to u 2 in G , or u 1 = u 2 and v 1 is adjacen t to v 2 in H . Defin e The blow-up G ( t ) = G · I t , where I t is the empt y graph on t v ertices. Define also the c omplete blow-up G [ t ] = G · K t . A graph G w e call ǫ -Hadwiger if h ( G ) ≤ ǫ | G | . W e will sho w for eac h ǫ > 0 ho w to construct a graph G on n ( ǫ ) = 2 (1+ o (1)) ǫ − 2 v ertices in time 2 (1+ o (1)) n ( ǫ ) 2 / 2 suc h that ev ery complete blow-up of G and its complemen t are ǫ -Hadwiger. Suc h blo w-u ps answer the questions of Mader and Thomason, as one can compute an y adjacency b et ween v ertices by simply lo oking at w hic h parts of the blo w-up the v ertices b elong. As the size of G dep ends only on ǫ , the time to compute whether t wo v ertices are adjacen t is nearly constan t for ǫ slo wly tending to 0. The b ounds ab o v e giv e an explicit construction of a grap h on N vertice s for whic h the Hadwiger n um b er of the graph and its complemen t is at most O ( N √ log log log N ). F or th e we ak er notion of explicit construction, in which the runn ing time is p olynomial in the num b er of ve rtices, the graph on N ve rtices w e obtain has th e p r op ert y that it and its complement has Hadwiger num b er at most O ( N √ log log N ). While these b ounds are sublinear, they do not come close to the tigh t b ound of O ( N √ log N ) which almost all graphs on N v ertices satisfy . In order to stu d y the Hadwiger num b er of a blow-up of a graph, it w ill b e helpful to d efine a fractional ve rsion of the Hadwiger num b er. This notion had indep end en tly b een in tro d uced earlier b y Seymour [18]. A br amble B for a graph G is a collection of connected subgraphs of G satisfying eac h pair B , B ′ ∈ B share a v ertex or there is an edge of G connecting B to B ′ . 2 Definition 1. The fr actional H adwiger numb er h f ( G ) of a gr aph G is the maximum h for which ther e is a b r amble B for G , and a weight func tion w : B → R ≥ 0 such that h = P B ∈B w ( B ) and for e ach vertex v , the sum of the weights of the sub g r aphs in B c ontaining v is at most 1 . Define a str ong br amble for a graph G to b e a collection of connected subgraph s of G satisfying for eac h pair B , B ′ ∈ B (with p ossibly B = B ′ ) there is an edge of G connecting B to B ′ . W e d efine the lower fr actional Hadwiger numb er h ′ f ( G ) similarly , except that B is required to b e a strong b ram ble and not a b ram ble. T h e fractional Hadwiger num b er and the lo wer fractional Hadwiger num b er are closely related. In deed, it is easy to show that if G h as an edge, then h f ( G ) / 2 ≤ h ′ f ( G ) ≤ h f ( G ). Equalit y o ccurs in the low er b ound if G is complete and th e upp er b ound if G is complete b ip artite. F or a p ositiv e in teger r , the r -inte gr al Hadwiger numb er h r ( G ) is defined the same as the fractional Hadwiger num b er, b ut all we igh ts h a ve to b e m ultiples of 1 /r . W e sim ilarly d efi ne the lo wer version h ′ r ( G ). Note th at h 1 ( G ) = h ( G ), an d if s is a m ultiple of r , th en h s ( G ) ≥ h r ( G ). It is easy to c hec k that h f ( G ) = lim r →∞ h r ( G ), and it follo ws that h f ( G ) ≥ h ( G ). The relationship b etw een the Hadwiger num b er of the blow-up of a graph and the fractional Hadwiger n um b er of the graph is demonstrated by the follo win g simp le prop osition. Prop osition 1. F or every gr aph G and p ositive inte ge r r , we have h ( G [ r ]) = r · h r ( G ) ≤ r · h f ( G ) . Essen tially the same p ro of also give s h ( G ( r )) = r · h ′ r ( G ) ≤ r · h ′ f ( G ). Th us, if we found a dens e graph G with relativ ely sm all fractional Hadwiger num b er, then th e blo w -up G [ r ] would also b e dense and hav e relativ ely sm all Hadwiger num b er. T o solv e Mader’s problem, it therefore suffices to show that there are d ense graphs G on n v ertices with fr actional Hadwiger n um b er h f ( G ) = o ( n ). It is not difficult to sho w that if h ( G ) < 4, then h f ( G ) = h ( G ). How ev er, there are p lanar graphs on n v ertices with h ( G ) = 4 and h f ( G ) = Θ( √ n ). Indeed, consid er th e √ n × √ n grid graph. The grid graph is planar an d th us has Hadwiger num b er at most 4. F or eac h i , let P i denote the indu ced path consisting of the v ertex ( i, i ) and all vertices of the grid graph directly b elo w or to th e r ight of this p oin t. Assigning eac h of these √ n paths weigh t 1 / 2, we get that the fr actional Hadwiger n um b er (and ev en th e 2-in tegral Hadwiger n um b er) of this grid graph is at least √ n/ 2. This example also sho ws that th e follo w ing upp er b ound on the fr actional Hadwiger n um b er cannot b e impr ov ed apart from the constant f actor. Theorem 2. If G is a gr aph on n vertic es, then h f ( G ) ≤ p 2 h ( G ) n. It is n atural to stud y the f ractional Hadwiger num b er of ran d om graphs. Theorem 2 imp lies that a random graph on n ve rtices almost surely has fractional Hadwiger num b er O ( n / (log n ) 1 / 4 ). W e pr o ve a muc h b etter estimate, that the fractional Hadw iger num b er of a random graph is almost surely asymptotic to its Hadw iger num b er. Bollob´ as, Catlin, and E rd˝ os [2] sho wed that the r andom graph G ( n, p ) on n v ertices w ith fixed edge pr obabilit y p almost surely h as Hadwiger n um b er asymptotic to n √ log b n , w h ere b = 1 / (1 − p ). Theorem 3. The fr actional Hadwiger numb er of a r andom gr aph is almost sur ely asymptotic al ly e qual to its Hadwiger numb er. That is, for fixe d p and al l n , almost sur ely h f ( G ( n, p )) = (1 + o (1)) n p log b n , 3 wher e b = 1 / (1 − p ) . W e conjecture that a s tronger r esult h olds, that they are in fact almost surely equal. Conjecture 1. A gr aph G on n vertic es picke d uniformly at r andom almost sur ely satisfies h ( G ) = h f ( G ) . This would imply that for most graph s G , the ratio of the Hadwiger n umber of G to the num b er of v ertices of G is the same as for its blo w-ups. Organization: In the next section, w e establish sev eral u pp er b ounds on the fractional Hadwiger n um b er, includin g Th eorems 2 and 3. In Section 3, w e u se these upp er b ound s on the fractional Hadwiger num b er to construct d ense graphs w ith sublinear Hadwiger num b er. W e fi nish with some concluding remarks. W e sometimes omit flo or and ceiling signs for clarity of presentati on. 2 Upp er b ounds on the fractional Hadwiger n um b er In this section we establish sev eral up p er b ounds on the fr actional Hadwiger n um b er of a graph. W e b egin by pr o vin g Theorem 2, wh ich state s that if G is a graph on n ve rtices, then h f ( G ) ≤ p 2 h ( G ) n . Pro of of Theorem 2: Let B b e a bram ble for a graph G on n v ertices. Let w : B → R ≥ 0 b e a weig h t function su c h that h = P B ∈B w ( B ) and f or eac h v ertex v , the sum of the weigh ts of the subgraphs in B con taining v is at most 1. It suffices to sho w that B con tains a su b collection of at least h 2 2 n v ertex-disjoin t subgraph s . Indeed, con tracting these subgraphs w e get a clique minor in G of ord er at least h 2 2 n , and pic king B and w to maximize h , we ha v e h = h f ( G ) so th at h ( G ) ≥ h f ( G ) 2 2 n or equiv alent ly h f ( G ) ≤ p 2 h ( G ) n . W e w ill prov e the desired low er b ound on the maxim u m num b er of vertex-disjoin t trees in B by in duction on n . Th e base case n = 1 clearly holds, and su p p ose the desired b ound holds for all n ′ < n . Let B 0 b e a subgraph in B with the min im um num b er of v ertices, and let t = | B 0 | . Since for eac h vertex v , the su m of the weig h ts of the s u bgraphs in B conta ining v is at most 1, summing this inequalit y o v er all v ertices yields X B ∈B w ( B ) | B | ≤ n . In particular, h = X B ∈B w ( B ) ≤ n/ | B 0 | = n/t. Delete f rom B all subgraphs con taining a v ertex in B 0 , and let B ′ b e th e resu lting su b collectio n of s u bgraphs. S ince for eac h v ertex v , the sum of the weigh ts of the trees contai ning v is at most 1, w e hav e P B ∈B ′ w ( B ) ≥ h − t . The num b er of vertic es not in B 0 is n − t . Hence, from a maxim u m sub collection of v er tex-disjoint subgraphs in B ′ and adding B 0 , w e get by indu ction at least 1 + ( h − t ) 2 2( n − t ) ≥ 1 + ( h − t ) 2 2 n = 1 + h 2 2 n (1 − t h ) 2 ≥ 1 + h 2 2 n (1 − n h 2 ) 2 ≥ 1 + h 2 2 n (1 − 2 n h 2 ) = h 2 2 n v ertex disjoin t su bgraphs in B , whic h completes the pro of. W e next establish a u s eful lemma for p ro ving Th eorem 3. This lemma extends the r esult of Bollob´ as, Catlin, and Erd˝ os [2] on the largest clique minor in a rand om graph by giving a b ound on the size of the largest clique minor in which the size of the connected sub graphs corresp onding to the v ertices of the clique are b oun ded. Recall that a clique min or in a grap h G of size t consists of t v ertex 4 disjoin t connected subsets V 1 , . . . , V t , such that for eac h p air i, j w ith i < j , there is an edge of G with one v er tex in V i and the other in V j . Defin e the br e adth of th e clique minor to b e max i | V i | . Lemma 1. L et 0 < p < 1 b e fixe d, 0 < ǫ < 1 , and define d := p (1 − ǫ ) log b n with b = 1 / (1 − p ) . Almo st sur ely, the lar gest cliqu e minor in G ( n, p ) of br e adth at most d has or der at most 4 n 1 − ǫ d ln n . Pr o of. If d < 1, th is trivially holds as there is no such nonempty clique minor of breadth at most d . Hence, we may assu me d ≥ 1. Consider a collection C = { V 1 , . . . , V h } of h = ⌈ 4 n 1 − ǫ d ln n ⌉ nonempt y v ertex sub sets eac h of size at most d . A rather cru de estimate (wh ic h is sufficient for our purp oses) on the n u m b er of suc h collec tions is that it is at most n dh . F or eac h p air V i , V j , the probabilit y there is an edge b et w een V i and V j is 1 − (1 − p ) | V i || V j | ≤ 1 − (1 − p ) d 2 ≤ e − (1 − p ) d 2 = e − n ǫ − 1 , where we used the inequalit y 1 − x ≤ e − x for 0 < x < 1 with x = (1 − p ) d 2 . By indep endence, the probabilit y that there is, for all 1 ≤ i < j ≤ h , an edge b etw een V i and V j is at most e − n ǫ − 1 ( h 2 ) . Therefore, the exp ected n um b er of clique m in ors of br eadth at most d and size at least h is at most n dh e − n ǫ − 1 ( h 2 ) = e h ( d ln n − n ǫ − 1 ( h − 1) / 2 ) = o (1) . This implies that almost s urely no su c h clique minor exists. No w we are r eady to prov e Theorem 3. Pro of of T heorem 3: Let G = G ( n, p ) b e a random graph on n v ertices with edge density p . Let b = 1 / (1 − p ) and ǫ = 4 log log n log n . Let B b e a br am b le for G . Supp ose there is a weigh t function w : B → R ≥ 0 suc h that h = P B ∈B w ( B ) and for eac h v ertex v , the su m of the w eigh ts of the subgraph s in B contai ning v is at most 1. Let B ′ denote the sub collection of subgraphs in B eac h with more d = p (1 − ǫ ) log b n vertic es, and B ′′ = B \ B ′ . W e hav e n ≥ X B ∈B w ( B ) | B | ≥ X B ∈B ′ w ( B ) | B | ≥ d X B ∈B ′ w ( B ) , where the first inequalit y follo ws from the fact that th e sum of th e w eigh ts of the su bgraphs in B con taining an y giv en vertex is at most 1. Hence, P B ∈B ′ w ( B ) ≤ n d and X B ∈B ′′ w ( B ) ≥ h − n d . W e now pick out a maximal sub collect ion of vertex-disjoin t subgraphs in B ′′ . W e can greedily do this, picking out ve rtex d isjoin t sub graphs B 1 , . . . , B s unt il there are no more sub graphs in B ′′ remaining wh ic h are vertex-disjoin t from these su b graphs. S ince the su m of the we igh t of all s ubgraphs con taining a given v er tex is at m ost 1, we must ha v e P s i =1 | B i | ≥ h − n d . S ince also | B i | ≤ d for eac h i , we ha v e s ≥ h − n/d d . On the other h and, b y Lemma 1, since B 1 , . . . , B s forms a clique minor of size s and depth at most d , almost surely s ≤ n 1 − ǫ d ln n . W e therefore get almost surely h ≤ n d + ds ≤ n d + n 1 − ǫ d 2 ln n < (1 + ǫ ) n p log b n , 5 where we use n is sufficien tly large, n ǫ = log 4 n , d = p (1 − ǫ ) log b n and the estimate 1 √ 1 − ǫ < 1 + 2 3 ǫ for ǫ < 1 / 4. As also h f ( G ) ≥ h ( G ), and almost surely h ( G ) = (1 + o (1)) n √ log b n , this estimate completes the pro of. Note that there is an edge b et ween eac h p air of connected subgraphs corresp onding to the v ertices of a clique minor. It follo ws that if G is a graph with m edges, then m ≥  h ( G ) 2  . W e finish the section with a similar upp er b ound on the fractional Hadwiger num b er. Prop osition 4. If a gr aph G has m e dges, then h f ( G ) ≤ √ 3 m + 1 . Pr o of. It is easy to s ee that we ma y assum e th at G is connected and h en ce the n u m b er of v ertices of G is at most m + 1. Let B b e a bramble for G . Su p p ose there is a weigh t fu nction w : B → R ≥ 0 suc h that h = P B ∈B w ( B ) and for eac h v ertex v , the su m of the weig h ts of the conn ected subgraph s in B con taining v is at most 1. Consider the sum S = P w ( B ) w ( B ′ ) o ver all ordered pairs of v ertex-disjoint sub grap h s in B . F or an y fixed su b graph B , the sum of th e w eigh ts of the subgraphs in B con taining at least one vertex in B is at most | B | , so the sum P B ′ w ( B ′ ) ov er all subgrap h s B ′ ∈ B disjoint from B is at least h − | B | . Therefore, S ≥ P B ∈B w ( B )( h − | B | ) = h 2 − P B ∈B w ( B ) | B | ≥ h 2 − n . F or eac h edge ( i, j ), the sum P w ( B ) w ( B ′ ) o ver all pairs of vertex-disjoin t subgraphs in B w ith i ∈ V ( B ) and j ∈ V ( B ′ ) is at most 1 since the sum of the w eights of th e su bgraphs con taining any giv en vertex is at most 1. As b etw een eac h pair of ve rtex-disjoin t sub graphs in B there is at least one edge, we therefore get S ≤ 2 m . I t follo ws h ≤ √ 2 m + n ≤ √ 3 m + 1, whic h completes the pro of. 3 Dense graphs with sublinear Hadwiger n um b er The pu rp ose of this section is to giv e the d etails for th e explicit construction of a dense graph with sublinear Hadwiger num b er. W e b egin this section by pro ving Prop osition 1, whic h states that h ( G [ r ]) = r · h r ( G ) ≤ r · h f ( G ) holds for every graph G and p ositiv e int eger r . Pro of of Prop osition 1: Let G b e a graph and G [ r ] b e th e complete blo w-up of G . Consider a maxim um clique minor in G [ r ] of order t = h ( G [ r ]) consisting of disjoin t connected ve rtex sub s ets V 1 , . . . , V t with an ed ge b et ween a vertex in V i and a vertex in V j for i 6 = j . Let B i b e the v ertex subset of G where v ∈ B i if there is a v ertex in the blo w-up of v wh ic h is also in V i . The collectio n B = { B 1 , . . . , B t } is clearly a b ram ble. Define the we igh t w ( B i ) = 1 /r for eac h i . F or eac h v ertex v of G , as V 1 , . . . , V t are vertex disjoint, at most r sets B i con tain v . Hence, the bramble B with this w eigh t function demonstrates h r ( G ) ≥ h ( G [ r ]) /r . In the other direction, consider a bramble B for G and a weigh t function w on B such that w ( B ) is a m ultiple of 1 /r for all B ∈ B and for eve ry vertex v , the sum of the w eights w ( B ) ov er all B cont aining v is at most 1. F or eac h suc h bram ble B , we pic k out r w ( B ) copies of B in th e blo w-up of B in G [ r ], suc h that all of the copies are v ertex-disjoint. W e can do th is since r w ( B ) is a n onnegativ e int eger, and f or eac h vertex v of G , th e su m of r w ( B ) ov er all B ∈ B which con tain v is at most r . These copies of th e sets in B form a clique minor in G [ r ] of order P B ∈ B r w ( B ) = r h r ( G ). Hence h ( G [ r ]) ≥ r h r ( G ), and w e h a ve p ro ved h ( G [ r ]) /r = h r ( G ). S ince h r ( G ) ≤ h f ( G ), the p ro of is complete. The follo wing th eorem sho ws h o w to fin d, for eac h 0 < ǫ, p < 1, a graph G of ed ge densit y at least p suc h that the ratio of the Hadwiger num b er of G to the num b er of v ertices of G is at most ǫ for G and its blo w-u ps. 6 Theorem 5. F or e ach 0 < ǫ, p < 1 , ther e is a gr aph G with e dge density at le ast p and h f ( G ) ≤ ǫ . In p articular, every c omplete blow-up of G has e dge-density at le ast p and i s ǫ -Hadwiger. Mor e over, for p fixe d and ǫ tending to 0 , the gr aph G has n 0 = b ǫ − 2 + o (1) vertic es with b = 1 / (1 − p ) and c an b e found in time  N pN  1+ o (1) with N =  n 0 2  . Pr o of. F rom Theorem 3, we ha v e that the r an d om graph G ( n 0 , p ) almost surely has f ractional Hadwiger n um b er (1 + o (1)) n 0 √ log b n 0 . Also, with at least constan t p ositiv e pr obabilit y , the edge d ensit y of such a random graph is at least p . F urthermore, Lemma 1 sh o w s that G ( n 0 , p ) almost surely has th e stronger prop erty that its largest clique minor of depth at most d = (1 + δ ) n 0 √ log b n 0 with δ = 4 log log n 0 log n 0 has order less than s = 4 n 1 − δ 0 d ln n 0 . This is ind eed stronger as in the p ro of of Th eorem 3, w e can b ound the fractional Hadwiger n u m b er from ab ov e by n 0 /d + ds , whic h is less th an ǫn 0 if the o (1) term in the definition of n 0 is pic ked correctly . T o sho w that a graph d o es not ha v e a clique min or of depth at most d and order s , it suffices to simply test all p ossible disjoint verte x subsets V 1 , . . . , V s with | V i | ≤ d for 1 ≤ i ≤ s , and c hec k if eac h V i is connected and there is an edge b etw een eac h V i and V j for i 6 = j . There are at most n ds 0 suc h s -tuples of s u bsets to tr y . Th us, by testing eac h graph on n 0 v ertices with ed ge densit y p for a clique min or of ord er s and depth at m ost d , w e will find suc h a graph G without a clique m inor order s and depth at most d , and this is the d esired graph G . The num b er of lab eled graphs on n 0 v ertices with edge densit y p is  N pN  with N =  n 0 2  The amount of time, roughly n ds 0 , needed to test eac h suc h graph is a low er ord er term. If w e wish to get an explicit construction of a dense graph whic h is ǫ -Hadwiger on a giv en n um b er n of v ertices, if n is not a multiple of n 0 , we can tak e a slightl y larger b lo w-u p of a sm all graph on n 0 v ertices, and simply delete a few vertice s (less th an n 0 v ertices with at most one from eac h clique in the complete b lo w-u p). The next th eorem give s a solution to Thomason’s prob lem by exp licitly constructing a dense graph, whic h is a blow-up of a sm all graph G , for whic h the Hadwiger num b er of the graph and its complemen t are b oth r elativ ely small. Theorem 6. F or al l 0 < ǫ < 1 ther e i s a gr aph G on n 0 = 2 (1+ o (1)) ǫ − 2 vertic es which c an b e f ound in time 2 (1+ o (1)) n 2 0 / 2 such that max( h f ( G ) , h f ( ¯ G )) ≤ ǫn 0 . In p articular, every c omplete blow-up of G and its c omplement ar e ǫ - Hadwiger. Pr o of. F rom Theorem 3, a graph on n 0 v ertices pic k ed uniformly at rand om almost su rely has fractional Hadwiger n um b er (1 + o (1)) n 0 √ log n 0 . F urthermore, Lemma 1 sho ws that a graph on n 0 v ertices pic ked uniformly at ran d om almost sur ely has the stronger prop erty that its largest clique minor of depth at most d = (1 + δ ) n 0 √ log n 0 with δ = 4 log log n 0 log n 0 has order less than s = 4 n 1 − δ 0 d ln n 0 . As in the pr o of of Theorem 3, w e can b ound th e fractional Hadwiger num b er from ab o ve by n 0 /d + ds , , whic h is less than ǫ n 0 if the o (1) term in the definition of n 0 is p ic ked correctly . T o sh o w that a graph and its complemen t do es not hav e a clique minor of d epth at m ost d an d order s , it suffi ces to simp ly test all p ossible d isjoin t v ertex sub sets V 1 , . . . , V s with | V i | ≤ d for 1 ≤ i ≤ s , and c hec k if eac h V i is conn ected and there is an edge b et ween eac h V i and V j for i 6 = j . There are at most n ds 0 suc h s -tup les of subsets to try . Th us, testing eac h graph on n 0 v ertices, w e find the desired graph G for which G and its complement do not con tain a clique min or order s and depth at m ost d . The num b er of graph s on n 0 v ertices is 2 ( n 0 2 ) , and the amount of time, roughly n ds 0 , n eeded to test eac h suc h graph is a lo wer order term. 7 4 Concluding remarks • W e sho w ed how to explicitly constru ct a d ense graph on n v ertices w ith Hadwiger n um b er o ( n ). Ho wev er, random graph s show that suc h graphs exist with Hadwiger n um b er O ( n √ log n ). It remains an in teresting op en p roblem to constru ct suc h grap h s. • W e conjecture that almost all graphs G satisfy h ( G ) = h f ( G ), i.e., a random graph on n ve rtices almost su r ely satisfies that its Hadwiger num b er and f ractional Hadw iger num b er are equ al. W e p ro v ed in Th eorem 3 that these num b ers are asymptotically equal for almost all graphs. This conjecture is equiv alen t to showing that almost all graphs satisfy the the ratio of the Hadwiger n u m b er to the n um b er of v ertices is equal for all blo w-u ps of the graph . • Note th at if H is a minor of G , then h f ( H ) ≤ h f ( G ). It f ollo ws that th e family F C of graphs G with h f ( G ) < C is closed un d er taking m inors. The Rob er tson-S eymour theorem imp lies that F C is c h aracterized b y a finite list of forbidden m inors. F or eac h C , what is this family? W e understand this family for C ≤ 4 as then h ( G ) = h f ( G ). • As noted b y Seymour [18], it wo uld b e interesting to prov e a fr actional analogue of Hadwiger’s conjecture, that h f ( G ) ≥ χ ( G ) for all graphs G . As h f ( G ) ≥ h ( G ), this conjecture w ould follo w from Hadwiger’s conjecture. This ma y b e hard in the case of graphs of indep endence num b er 2. F or such graphs on n v ertices, χ ( G ) ≥ n/ 2, and so Hadwiger’s conjecture would imp ly h ( G ) ≥ n/ 2, b ut the b est kno wn lo w er b ound [4] on the Hadw iger n um b er is of the form h ( G ) ≥ ( 1 3 + o (1)) n . Improving this b ound to h ( G ) ≥ ( 1 3 + ǫ ) n for some ab s olute constan t ǫ > 0 is b eliev ed to b e a c hallenging problem, and it is equiv alent to pr o vin g a similar lo w er b ou n d for h f ( G ). • Graph lifts are another in teresting op eration. An r -lift of a graph G = ( V , E ) is the graph on V × [ r ], whose ed ge set is the union of p erfect matc hings b et w een { u } × [ r ] and { v } × [ r ] for eac h edge ( u, v ) ∈ E . Drier and Linial [3] stud ied clique m inors in lifts of the complete graph K n . On e of the in teresting op en qu estions remaining h ere is whether ev ery lift of the complete graph K n has Hadwiger n um b er Ω( n ). • T reewidth is an imp ortan t graph parameter introd uced by R ob ertson and S eymour [14] in their pro of of W agner’s conjecture. A tr e e de c omp osition of a graph G = ( V , E ) is a pair ( X , T ), w here X = { X 1 , ..., X t } is a family of subsets of V , and T is a tr ee wh ose no des are the sub sets X i , satisfying the follo wing thr ee prop erties. 1. V = X 1 ∪ . . . ∪ X t . 2. F or ev ery edge ( v , w ) in the graph, there is a subs et X i that con tains b oth v and w . 3. If X i and X j b oth con tain a ve rtex v , then all no des X z of th e tree in the (un ique) path b et ween X i and X j con tain v as w ell. Rob ertson and S eymou r pr o ved that treewidth is r elated to the largest grid min or. Indeed, th ey pro v ed that for eac h r ther e is f ( r ) such th at eve ry graph with treedwidth at least f ( r ) conta ins a r × r grid minor. Th e original upp er b ound on f ( r ) w as enormous. It was later improv ed b y Rob ertson, Seymour, and Th omas [17], who sh o wed cr 2 log r ≤ f ( r ) ≤ 2 c ′ r 5 where c and c ′ are absolute constan ts. In the other direction, it is easy to sho w that an y graph whic h contai ns an r × r grid minor has treewidth at least r . Separators are another imp ortan t concept in grap h theory wh ic h ha ve many algorithmic, extremal, and enumerativ e applications. A v ertex su bset V 0 of a graph G is a sep ar ator for G if th ere is a p artition V ( G ) = V 0 ∪ V 1 ∪ V 2 suc h that | V 1 | , | V 2 | ≤ 2 n/ 3 and there are no edges w ith one v ertex in V 1 and the 8 other vertex in V 2 . A fu ndament al resu lt of Lipton and T arjan states that eve ry planar graph on n v ertices has a separator of size O ( √ n ). This result has b een generalized in many d irections, to graphs em b edded on a surface [7], graph s with a f orbidden minor [1], in tersection graphs of balls in R d , and in tersection graphs of geometric ob jects in the plane [5], [6]. Th e sep ar ation numb er of a graph G is the minim um s for which ev ery subgraph of G h as a separator of size at m ost s . The bram ble n um b er of a graph G is the minimum b suc h that for ev ery b ram ble for G ther e is a set of b vertice s for which ev ery subgraph in the bram ble conta ins at least one of these b v ertices. Tw o graph parameters are c omp ar able if one of them can b e b ounded as a fun ction of the other, and vice versa. Rob ertson and S eymou r sh o wed that tr eewidth and largest grid minor are comparable. The follo wing th eorem wh ic h we state without pr o of extends this r esu lt. It ma y b e su rprising b ecause some of these parameters app ear f r om th eir d efinitions to b e u nrelated. Theorem 7. F r actional Hadwiger numb er, r - inte gr al Hadwiger numb er for e ach r ≥ 2 , br amble num- b er, sep ar ation numb er, tr e e dwidth, and maximum grid minor size ar e al l c omp ar able. The dep end ence b etw een some of these graph parameters is not w ell u ndersto o d and improvi ng the b ound s remains an interesting op en problem. Ac knowledgemen ts: I am greatl y indebted to Noga Alon, Nati Lin ial, and P aul Seymour for h elpful con versatio ns. References [1] N. Alon, P . D. Seymour and R. Thomas, A separator theorem for nonplanar graphs, J. Amer. Math. So c. 3 (1990), 801–808. [2] B. Bollob´ as, P . A. Catlin and P . Erd˝ os, Hadwiger’s conjecture is tru e for almost ev ery graph , Eur op e an J. Comb. 1 (198 0), 195–199. [3] Y. Drier and N. Linial, Minors in lifts of graphs, R andom Structur es & Algorithms 29 (2006), 208–2 25. [4] J. F o x, Clique minors an d indep endence num b er, SIAM J . D iscr ete Math. 24 (2010 ), 1313–13 21. [5] J. F o x and J. Pa c h , Separator theorems and T ur´ an-t yp e results for planar in tersectio n graphs, A dvanc es in Math. 219 (200 8), 1070–108 0. [6] J. F o x and J. P ac h, A separator theorem for string graphs an d its applications, Combin. Pr ob ab. Comput. 19 (2010), 371–390. [7] J. R. Gilb ert, J. P . Hu tc hin son and R. E. T arjan, A sep arator theorem for graphs of b ou n ded gen u s, J. Algorithms 5 (1984), 391–403. [8] H. Hadwiger, ¨ Ub er eine Klassifik ation der Str ec kenk omplexe, Vierteljschr. Naturforsch. Ges. Z¨ urich 88 (1943), 133–1 42. [9] A. V. Kosto c h k a, T he min im um Hadwiger num b er f or graphs with a giv en m ean d egree of v ertices, Meto dy D i skr et. Anal iz. 38 (1982), 37–58. [10] A. V. Kostochk a, A lo w er b ound for the Hadwiger num b er of graphs b y their av erage degree, Combinatoric a 4 (1984 ), 307–316. 9 [11] R. J. Lipton and R. E. T arjan, A separator theorem for planar graphs, SIAM J. A ppl. Math. 36 (1979 ), 177–189. [12] G. L. Miller, S.-H T eng, W. Thurston and S. A. V a v asis, S eparators for sp here-pac kin gs and nearest neigh b orho o d graphs, J. ACM 44 (1997), 1–29. [13] J. S. My ers, Gr ap h s without large complete min ors are qu asi-random, Combin. P r ob ab. Comput. 11 (2002 ), 571–585. [14] N. Rob ertson an d P . D. Seymour, Gr aph minors I I I: Planar tr ee-width, J. Combin. The ory Ser. B 36 (1984), 49–64. [15] N. Rob ertson and P . D. Seymour, Graph minors XX. W agner’s conjecture, J. Combin. The ory Ser. B 92 (2004), 325–357. [16] N. Rob ertson, P . D. Seymour, and R. Thomas, Hadwiger’s conjecture for K 6 -free graph s, Com- binatoric a 14 (1993), 279–361 . [17] N. Rob ertson, P . D. Seymour, and R. Th omas, Quic k ly excluding a planar graph, J. Combin. The ory Ser. B 62 (1994), 323–348, [18] P . Seymour, priv ate comm u nication. [19] A. Thomason, An extremal f unction for con tractions of graphs, Math. Pr o c. Cambridge Philos. So c. 95 (1984 ), 261–265. [20] A. T h omason, Complete minors in pseudorandom graphs, R andom Structur es & Algorithms 17 (2000 ), 26–28. [21] A. Thomason, Sub divisions, linking, m inors and extremal functions. 6th Internatio nal Conference on Graph Th eory (Marseille, 200 0), 4 p p. (electronic), Electron. Notes Discrete Math., 5, Elsevier, Amsterdam, 2000 . [22] A. T h omason, The extremal function for complete minors, J . Combin. The ory Ser. B 81 (2001), 318–3 38. [23] A. Thomason, Two minor problems, Combin. Pr ob ab. Comp ut. 13 (2004) , 413–414. [24] A. Th omason, Extremal functions for graph m in ors, More sets, graphs and num b ers, 359–380, Boly ai S o c. Math. Stud., 15, Sp ringer, Berlin, 2006 . [25] K. W agner, ¨ Ub er eine Eigensc haft der eb enen Komp lexe, M athematische Annal en 114 (1937), 570–5 90. 10