aa r X i v : . [ m a t h . C O ] A ug Constructing dense graphs with sublinear Hadwiger number
Jacob Fox ∗ Abstract
Mader asked to explicitly construct dense graphs for which the size of the largest clique minoris sublinear in the number of vertices. Such graphs exist as a random graph almost surely hasthis property. This question and variants were popularized by Thomason over several articles.We answer these questions by showing how to explicitly construct such graphs using blow-ups ofsmall graphs with this property. This leads to the study of a fractional variant of the clique minornumber, which may be of independent interest.
A graph H is a minor of a graph G if H can be obtained from a subgraph of G by contracting edges.Minors form an important connection between graph theory, geometry, and topology. For example,the Kuratowski-Wagner theorem states that a graph can be embedded in the plane if and only if ithas neither the complete graph K nor the complete bipartite graph K , as a minor. This exampleplayed an important role in the development of topological graph theory, whose masterpiece is theRobertson-Seymour graph minor theorem. In a series of twenty papers [15], they proved Wagner’sconjecture that every family of graphs closed under taking minors is characterized by a finite list offorbidden minors.The Hadwiger number h ( G ) of a graph G is the order of the largest clique which is a minor of G .The famous conjecture of Hadwiger [8] states that every graph of chromatic number k has Hadwigernumber at least k . Hadwiger proved his conjecture for k ≤
4. Wagner [25] proved that the case k = 5is equivalent to the Four Color Theorem. In a tour de force, Robertson, Seymour, and Thomas [16]settled the case k = 6 also using the Four Color Theorem. The conjecture is still open for k ≥ G on n vertices almost surely satisfies h ( G ) is asymptotic to n √ log n . Here, andthroughout the paper, all logarithms unless otherwise indicated are in base 2. Also using the wellknown fact that the chromatic number of a random graph on n vertices is almost surely Θ( n/ log n ),they deduced that almost all graphs satisfy Hadwiger’s conjecture.Mader showed that large average degree is enough to imply a large clique minor. Precisely, for eachinteger t there is a constant c ( t ) such that every graph G of average degree at least c ( t ) satisfies h ( G ) ≥ t . Kostochka [9, 10] and Thomason [19] independently proved that c ( t ) = Θ( t √ log t ). Thomason [22]later determined the asymptotic behavior of c ( t ), with random graphs of a particular density asextremal graphs for this problem. Myers [13] proved that any extremal graph under certain conditionsfor this problem must be quasirandom.Random graphs have some remarkable properties for which it is difficult to explicitly constructgraphs with these properties. One well-known example is Erd˝os’ lower bound on Ramsey numbers,which shows that almost all graphs on n vertices do not contain a clique or independent set of order ∗ Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139-4307. E-mail:[email protected]. Research supported by a Simons Fellowship and NSF grant DMS-1069197. n . Despite considerable attention over the last 60 years, there is no known construction (inpolynomial time) of a graph G on n vertices for which the largest clique or independent set in G is oforder O (log n ).Another interesting property of random graphs already mentioned is that almost surely they donot contain a clique minor of linear size. Mader asked to construct a dense graph on n vertices with h ( G ) = o ( n ). One of the main motivations for this problem is that proving interesting upper boundson the Hadwiger number of a graph appears to be a difficult problem. Thomason [20] showed thatmany of the standard constructions of quasirandom graphs have linear clique minors and thereforecannot be used to answer Mader’s problem.Mader’s problem and a few variants were discussed by Thomason in several papers [20, 21, 22, 23,24] and also by Myers [13]. Thomason [20] posed the stronger problem of constructing a graph G on n vertices for which the Hadwiger numbers of G and its complement ¯ G are both o ( n ). He speculates[21] that this problem might be as hard as the classical Ramsey problem of finding explicit graphs G such that both G and ¯ G contain only small complete subgraphs.Here we solve both the problems of Mader and Thomason. To do so, it is helpful to define whatan explicit construction is. We view a graph on 2 n vertices as a function f : (cid:0) { , } n (cid:1) → { , } , wherethe value of f tells whether or not two vertices are adjacent. By an explicit construction we meanthat the function f is computable in polynomial time (in n ). That is, given two vertices, we cancompute whether or not they are adjacent in time polynomial in the number of bits used to representthe vertices. There is also a weaker notion of explicit graph which is sometimes used. In this version,the edges of the graph can be computed in time polynomial in the number of vertices of the graph.Our construction which answers the problems of Mader and Thomason is given in nearly constant time using blow-ups of nearly constant size graphs. We show that if a (small) graph is dense and hasrelatively small Hadwiger number, then its blow-ups also have this property.We formally define the blow-up of a graph as follows. For graphs G and H , the lexicographicproduct G · H is the graph on vertex set V ( G ) × V ( H ), where ( u , v ) , ( u , v ) ∈ V ( G ) × V ( H ) areadjacent if and only if u is adjacent to u in G , or u = u and v is adjacent to v in H . DefineThe blow-up G ( t ) = G · I t , where I t is the empty graph on t vertices. Define also the complete blow-up G [ t ] = G · K t .A graph G we call ǫ -Hadwiger if h ( G ) ≤ ǫ | G | . We will show for each ǫ > G on n ( ǫ ) = 2 (1+ o (1)) ǫ − vertices in time 2 (1+ o (1)) n ( ǫ ) / such that every complete blow-up of G and its complement are ǫ -Hadwiger. Such blow-ups answer the questions of Mader and Thomason, asone can compute any adjacency between vertices by simply looking at which parts of the blow-up thevertices belong. As the size of G depends only on ǫ , the time to compute whether two vertices areadjacent is nearly constant for ǫ slowly tending to 0.The bounds above give an explicit construction of a graph on N vertices for which the Hadwigernumber of the graph and its complement is at most O ( N √ log log log N ). For the weaker notion of explicitconstruction, in which the running time is polynomial in the number of vertices, the graph on N verticeswe obtain has the property that it and its complement has Hadwiger number at most O ( N √ log log N ).While these bounds are sublinear, they do not come close to the tight bound of O ( N √ log N ) which almostall graphs on N vertices satisfy.In order to study the Hadwiger number of a blow-up of a graph, it will be helpful to define afractional version of the Hadwiger number. This notion had independently been introduced earlier bySeymour [18]. A bramble B for a graph G is a collection of connected subgraphs of G satisfying eachpair B, B ′ ∈ B share a vertex or there is an edge of G connecting B to B ′ .2 efinition 1. The fractional Hadwiger number h f ( G ) of a graph G is the maximum h for which thereis a bramble B for G , and a weight function w : B → R ≥ such that h = P B ∈B w ( B ) and for eachvertex v , the sum of the weights of the subgraphs in B containing v is at most . Define a strong bramble for a graph G to be a collection of connected subgraphs of G satisfying foreach pair B, B ′ ∈ B (with possibly B = B ′ ) there is an edge of G connecting B to B ′ . We define the lower fractional Hadwiger number h ′ f ( G ) similarly, except that B is required to be a strong brambleand not a bramble. The fractional Hadwiger number and the lower fractional Hadwiger number areclosely related. Indeed, it is easy to show that if G has an edge, then h f ( G ) / ≤ h ′ f ( G ) ≤ h f ( G ).Equality occurs in the lower bound if G is complete and the upper bound if G is complete bipartite.For a positive integer r , the r -integral Hadwiger number h r ( G ) is defined the same as the fractionalHadwiger number, but all weights have to be multiples of 1 /r . We similarly define the lower version h ′ r ( G ). Note that h ( G ) = h ( G ), and if s is a multiple of r , then h s ( G ) ≥ h r ( G ). It is easy to checkthat h f ( G ) = lim r →∞ h r ( G ), and it follows that h f ( G ) ≥ h ( G ).The relationship between the Hadwiger number of the blow-up of a graph and the fractionalHadwiger number of the graph is demonstrated by the following simple proposition. Proposition 1.
For every graph G and positive integer r , we have h ( G [ r ]) = r · h r ( G ) ≤ r · h f ( G ) . Essentially the same proof also gives h ( G ( r )) = r · h ′ r ( G ) ≤ r · h ′ f ( G ).Thus, if we found a dense graph G with relatively small fractional Hadwiger number, then theblow-up G [ r ] would also be dense and have relatively small Hadwiger number. To solve Mader’sproblem, it therefore suffices to show that there are dense graphs G on n vertices with fractionalHadwiger number h f ( G ) = o ( n ).It is not difficult to show that if h ( G ) <
4, then h f ( G ) = h ( G ). However, there are planar graphson n vertices with h ( G ) = 4 and h f ( G ) = Θ( √ n ). Indeed, consider the √ n × √ n grid graph. Thegrid graph is planar and thus has Hadwiger number at most 4. For each i , let P i denote the inducedpath consisting of the vertex ( i, i ) and all vertices of the grid graph directly below or to the right ofthis point. Assigning each of these √ n paths weight 1 /
2, we get that the fractional Hadwiger number(and even the 2-integral Hadwiger number) of this grid graph is at least √ n/
2. This example alsoshows that the following upper bound on the fractional Hadwiger number cannot be improved apartfrom the constant factor.
Theorem 2. If G is a graph on n vertices, then h f ( G ) ≤ p h ( G ) n. It is natural to study the fractional Hadwiger number of random graphs. Theorem 2 implies that arandom graph on n vertices almost surely has fractional Hadwiger number O ( n/ (log n ) / ). We provea much better estimate, that the fractional Hadwiger number of a random graph is almost surelyasymptotic to its Hadwiger number. Bollob´as, Catlin, and Erd˝os [2] showed that the random graph G ( n, p ) on n vertices with fixed edge probability p almost surely has Hadwiger number asymptotic to n √ log b n , where b = 1 / (1 − p ). Theorem 3.
The fractional Hadwiger number of a random graph is almost surely asymptotically equalto its Hadwiger number. That is, for fixed p and all n , almost surely h f ( G ( n, p )) = (1 + o (1)) n p log b n , here b = 1 / (1 − p ) . We conjecture that a stronger result holds, that they are in fact almost surely equal.
Conjecture 1.
A graph G on n vertices picked uniformly at random almost surely satisfies h ( G ) = h f ( G ) . This would imply that for most graphs G , the ratio of the Hadwiger number of G to the numberof vertices of G is the same as for its blow-ups. Organization:
In the next section, we establish several upper bounds on the fractional Hadwigernumber, including Theorems 2 and 3. In Section 3, we use these upper bounds on the fractionalHadwiger number to construct dense graphs with sublinear Hadwiger number. We finish with someconcluding remarks. We sometimes omit floor and ceiling signs for clarity of presentation.
In this section we establish several upper bounds on the fractional Hadwiger number of a graph. Webegin by proving Theorem 2, which states that if G is a graph on n vertices, then h f ( G ) ≤ p h ( G ) n . Proof of Theorem 2:
Let B be a bramble for a graph G on n vertices. Let w : B → R ≥ bea weight function such that h = P B ∈B w ( B ) and for each vertex v , the sum of the weights of thesubgraphs in B containing v is at most 1. It suffices to show that B contains a subcollection of atleast h n vertex-disjoint subgraphs. Indeed, contracting these subgraphs we get a clique minor in G of order at least h n , and picking B and w to maximize h , we have h = h f ( G ) so that h ( G ) ≥ h f ( G ) n or equivalently h f ( G ) ≤ p h ( G ) n . We will prove the desired lower bound on the maximum numberof vertex-disjoint trees in B by induction on n . The base case n = 1 clearly holds, and suppose thedesired bound holds for all n ′ < n .Let B be a subgraph in B with the minimum number of vertices, and let t = | B | . Since foreach vertex v , the sum of the weights of the subgraphs in B containing v is at most 1, summing thisinequality over all vertices yields X B ∈B w ( B ) | B | ≤ n. In particular, h = X B ∈B w ( B ) ≤ n/ | B | = n/t. Delete from B all subgraphs containing a vertex in B , and let B ′ be the resulting subcollectionof subgraphs. Since for each vertex v , the sum of the weights of the trees containing v is at most 1,we have P B ∈B ′ w ( B ) ≥ h − t . The number of vertices not in B is n − t . Hence, from a maximumsubcollection of vertex-disjoint subgraphs in B ′ and adding B , we get by induction at least1 + ( h − t ) n − t ) ≥ h − t ) n = 1 + h n (1 − th ) ≥ h n (1 − nh ) ≥ h n (1 − nh ) = h n vertex disjoint subgraphs in B , which completes the proof.We next establish a useful lemma for proving Theorem 3. This lemma extends the result ofBollob´as, Catlin, and Erd˝os [2] on the largest clique minor in a random graph by giving a bound onthe size of the largest clique minor in which the size of the connected subgraphs corresponding to thevertices of the clique are bounded. Recall that a clique minor in a graph G of size t consists of t vertex4isjoint connected subsets V , . . . , V t , such that for each pair i, j with i < j , there is an edge of G withone vertex in V i and the other in V j . Define the breadth of the clique minor to be max i | V i | . Lemma 1.
Let < p < be fixed, < ǫ < , and define d := p (1 − ǫ ) log b n with b = 1 / (1 − p ) .Almost surely, the largest clique minor in G ( n, p ) of breadth at most d has order at most n − ǫ d ln n .Proof. If d <
1, this trivially holds as there is no such nonempty clique minor of breadth at most d .Hence, we may assume d ≥
1. Consider a collection C = { V , . . . , V h } of h = ⌈ n − ǫ d ln n ⌉ nonemptyvertex subsets each of size at most d . A rather crude estimate (which is sufficient for our purposes)on the number of such collections is that it is at most n dh . For each pair V i , V j , the probability thereis an edge between V i and V j is1 − (1 − p ) | V i || V j | ≤ − (1 − p ) d ≤ e − (1 − p ) d = e − n ǫ − , where we used the inequality 1 − x ≤ e − x for 0 < x < x = (1 − p ) d . By independence, theprobability that there is, for all 1 ≤ i < j ≤ h , an edge between V i and V j is at most e − n ǫ − ( h ).Therefore, the expected number of clique minors of breadth at most d and size at least h is at most n dh e − n ǫ − ( h ) = e h ( d ln n − n ǫ − ( h − / ) = o (1) . This implies that almost surely no such clique minor exists.Now we are ready to prove Theorem 3.
Proof of Theorem 3:
Let G = G ( n, p ) be a random graph on n vertices with edge density p .Let b = 1 / (1 − p ) and ǫ = 4 log log n log n . Let B be a bramble for G . Suppose there is a weight function w : B → R ≥ such that h = P B ∈B w ( B ) and for each vertex v , the sum of the weights of the subgraphsin B containing v is at most 1.Let B ′ denote the subcollection of subgraphs in B each with more d = p (1 − ǫ ) log b n vertices, and B ′′ = B \ B ′ . We have n ≥ X B ∈B w ( B ) | B | ≥ X B ∈B ′ w ( B ) | B | ≥ d X B ∈B ′ w ( B ) , where the first inequality follows from the fact that the sum of the weights of the subgraphs in B containing any given vertex is at most 1. Hence, P B ∈B ′ w ( B ) ≤ nd and X B ∈B ′′ w ( B ) ≥ h − nd . We now pick out a maximal subcollection of vertex-disjoint subgraphs in B ′′ . We can greedilydo this, picking out vertex disjoint subgraphs B , . . . , B s until there are no more subgraphs in B ′′ remaining which are vertex-disjoint from these subgraphs. Since the sum of the weight of all subgraphscontaining a given vertex is at most 1, we must have P si =1 | B i | ≥ h − nd . Since also | B i | ≤ d for each i , we have s ≥ h − n/dd . On the other hand, by Lemma 1, since B , . . . , B s forms a clique minor of size s and depth at most d , almost surely s ≤ n − ǫ d ln n . We therefore get almost surely h ≤ nd + ds ≤ nd + n − ǫ d ln n < (1 + ǫ ) n p log b n , n is sufficiently large, n ǫ = log n , d = p (1 − ǫ ) log b n and the estimate √ − ǫ < ǫ for ǫ < /
4. As also h f ( G ) ≥ h ( G ), and almost surely h ( G ) = (1 + o (1)) n √ log b n , this estimate completesthe proof.Note that there is an edge between each pair of connected subgraphs corresponding to the verticesof a clique minor. It follows that if G is a graph with m edges, then m ≥ (cid:0) h ( G )2 (cid:1) . We finish the sectionwith a similar upper bound on the fractional Hadwiger number. Proposition 4.
If a graph G has m edges, then h f ( G ) ≤ √ m + 1 .Proof. It is easy to see that we may assume that G is connected and hence the number of vertices of G is at most m + 1. Let B be a bramble for G . Suppose there is a weight function w : B → R ≥ suchthat h = P B ∈B w ( B ) and for each vertex v , the sum of the weights of the connected subgraphs in B containing v is at most 1.Consider the sum S = P w ( B ) w ( B ′ ) over all ordered pairs of vertex-disjoint subgraphs in B . Forany fixed subgraph B , the sum of the weights of the subgraphs in B containing at least one vertex in B is at most | B | , so the sum P B ′ w ( B ′ ) over all subgraphs B ′ ∈ B disjoint from B is at least h − | B | .Therefore, S ≥ P B ∈B w ( B )( h − | B | ) = h − P B ∈B w ( B ) | B | ≥ h − n . For each edge ( i, j ), the sum P w ( B ) w ( B ′ ) over all pairs of vertex-disjoint subgraphs in B with i ∈ V ( B ) and j ∈ V ( B ′ ) is at most1 since the sum of the weights of the subgraphs containing any given vertex is at most 1. As betweeneach pair of vertex-disjoint subgraphs in B there is at least one edge, we therefore get S ≤ m . Itfollows h ≤ √ m + n ≤ √ m + 1, which completes the proof. The purpose of this section is to give the details for the explicit construction of a dense graph withsublinear Hadwiger number. We begin this section by proving Proposition 1, which states that h ( G [ r ]) = r · h r ( G ) ≤ r · h f ( G )holds for every graph G and positive integer r . Proof of Proposition 1:
Let G be a graph and G [ r ] be the complete blow-up of G . Consider amaximum clique minor in G [ r ] of order t = h ( G [ r ]) consisting of disjoint connected vertex subsets V , . . . , V t with an edge between a vertex in V i and a vertex in V j for i = j . Let B i be the vertexsubset of G where v ∈ B i if there is a vertex in the blow-up of v which is also in V i . The collection B = { B , . . . , B t } is clearly a bramble. Define the weight w ( B i ) = 1 /r for each i . For each vertex v of G , as V , . . . , V t are vertex disjoint, at most r sets B i contain v . Hence, the bramble B with thisweight function demonstrates h r ( G ) ≥ h ( G [ r ]) /r .In the other direction, consider a bramble B for G and a weight function w on B such that w ( B ) is amultiple of 1 /r for all B ∈ B and for every vertex v , the sum of the weights w ( B ) over all B containing v is at most 1. For each such bramble B , we pick out rw ( B ) copies of B in the blow-up of B in G [ r ],such that all of the copies are vertex-disjoint. We can do this since rw ( B ) is a nonnegative integer,and for each vertex v of G , the sum of rw ( B ) over all B ∈ B which contain v is at most r . These copiesof the sets in B form a clique minor in G [ r ] of order P B ∈ B rw ( B ) = rh r ( G ). Hence h ( G [ r ]) ≥ rh r ( G ),and we have proved h ( G [ r ]) /r = h r ( G ). Since h r ( G ) ≤ h f ( G ), the proof is complete.The following theorem shows how to find, for each 0 < ǫ, p <
1, a graph G of edge density at least p such that the ratio of the Hadwiger number of G to the number of vertices of G is at most ǫ for G and its blow-ups. 6 heorem 5. For each < ǫ, p < , there is a graph G with edge density at least p and h f ( G ) ≤ ǫ . Inparticular, every complete blow-up of G has edge-density at least p and is ǫ -Hadwiger. Moreover, for p fixed and ǫ tending to , the graph G has n = b ǫ − + o (1) vertices with b = 1 / (1 − p ) and can be foundin time (cid:0) NpN (cid:1) o (1) with N = (cid:0) n (cid:1) .Proof. From Theorem 3, we have that the random graph G ( n , p ) almost surely has fractional Hadwigernumber (1 + o (1)) n √ log b n . Also, with at least constant positive probability, the edge density of such arandom graph is at least p . Furthermore, Lemma 1 shows that G ( n , p ) almost surely has the strongerproperty that its largest clique minor of depth at most d = (1 + δ ) n √ log b n with δ = 4 log log n log n hasorder less than s = 4 n − δ d ln n . This is indeed stronger as in the proof of Theorem 3, we can boundthe fractional Hadwiger number from above by n /d + ds , which is less than ǫn if the o (1) term inthe definition of n is picked correctly. To show that a graph does not have a clique minor of depth atmost d and order s , it suffices to simply test all possible disjoint vertex subsets V , . . . , V s with | V i | ≤ d for 1 ≤ i ≤ s , and check if each V i is connected and there is an edge between each V i and V j for i = j .There are at most n ds such s -tuples of subsets to try.Thus, by testing each graph on n vertices with edge density p for a clique minor of order s anddepth at most d , we will find such a graph G without a clique minor order s and depth at most d ,and this is the desired graph G . The number of labeled graphs on n vertices with edge density p is (cid:0) NpN (cid:1) with N = (cid:0) n (cid:1) The amount of time, roughly n ds , needed to test each such graph is a lower orderterm.If we wish to get an explicit construction of a dense graph which is ǫ -Hadwiger on a given number n of vertices, if n is not a multiple of n , we can take a slightly larger blow-up of a small graph on n vertices, and simply delete a few vertices (less than n vertices with at most one from each clique inthe complete blow-up).The next theorem gives a solution to Thomason’s problem by explicitly constructing a dense graph,which is a blow-up of a small graph G , for which the Hadwiger number of the graph and its complementare both relatively small. Theorem 6.
For all < ǫ < there is a graph G on n = 2 (1+ o (1)) ǫ − vertices which can be found intime (1+ o (1)) n / such that max( h f ( G ) , h f ( ¯ G )) ≤ ǫn . In particular, every complete blow-up of G andits complement are ǫ -Hadwiger.Proof. From Theorem 3, a graph on n vertices picked uniformly at random almost surely has fractionalHadwiger number (1 + o (1)) n √ log n . Furthermore, Lemma 1 shows that a graph on n vertices pickeduniformly at random almost surely has the stronger property that its largest clique minor of depth atmost d = (1 + δ ) n √ log n with δ = 4 log log n log n has order less than s = 4 n − δ d ln n . As in the proof ofTheorem 3, we can bound the fractional Hadwiger number from above by n /d + ds , , which is lessthan ǫn if the o (1) term in the definition of n is picked correctly. To show that a graph and itscomplement does not have a clique minor of depth at most d and order s , it suffices to simply test allpossible disjoint vertex subsets V , . . . , V s with | V i | ≤ d for 1 ≤ i ≤ s , and check if each V i is connectedand there is an edge between each V i and V j for i = j . There are at most n ds such s -tuples of subsetsto try.Thus, testing each graph on n vertices, we find the desired graph G for which G and its complementdo not contain a clique minor order s and depth at most d . The number of graphs on n vertices is2( n ), and the amount of time, roughly n ds , needed to test each such graph is a lower order term.7 Concluding remarks • We showed how to explicitly construct a dense graph on n vertices with Hadwiger number o ( n ).However, random graphs show that such graphs exist with Hadwiger number O ( n √ log n ). It remains aninteresting open problem to construct such graphs. • We conjecture that almost all graphs G satisfy h ( G ) = h f ( G ), i.e., a random graph on n verticesalmost surely satisfies that its Hadwiger number and fractional Hadwiger number are equal. We provedin Theorem 3 that these numbers are asymptotically equal for almost all graphs. This conjecture isequivalent to showing that almost all graphs satisfy the the ratio of the Hadwiger number to thenumber of vertices is equal for all blow-ups of the graph. • Note that if H is a minor of G , then h f ( H ) ≤ h f ( G ). It follows that the family F C of graphs G with h f ( G ) < C is closed under taking minors. The Robertson-Seymour theorem implies that F C ischaracterized by a finite list of forbidden minors. For each C , what is this family? We understandthis family for C ≤ h ( G ) = h f ( G ). • As noted by Seymour [18], it would be interesting to prove a fractional analogue of Hadwiger’sconjecture, that h f ( G ) ≥ χ ( G ) for all graphs G . As h f ( G ) ≥ h ( G ), this conjecture would follow fromHadwiger’s conjecture. This may be hard in the case of graphs of independence number 2. For suchgraphs on n vertices, χ ( G ) ≥ n/
2, and so Hadwiger’s conjecture would imply h ( G ) ≥ n/
2, but the bestknown lower bound [4] on the Hadwiger number is of the form h ( G ) ≥ ( + o (1)) n . Improving thisbound to h ( G ) ≥ ( + ǫ ) n for some absolute constant ǫ > h f ( G ). • Graph lifts are another interesting operation. An r -lift of a graph G = ( V, E ) is the graph on V × [ r ], whose edge set is the union of perfect matchings between { u } × [ r ] and { v } × [ r ] for eachedge ( u, v ) ∈ E . Drier and Linial [3] studied clique minors in lifts of the complete graph K n . Oneof the interesting open questions remaining here is whether every lift of the complete graph K n hasHadwiger number Ω( n ). • Treewidth is an important graph parameter introduced by Robertson and Seymour [14] in theirproof of Wagner’s conjecture. A tree decomposition of a graph G = ( V, E ) is a pair (
X, T ), where X = { X , ..., X t } is a family of subsets of V , and T is a tree whose nodes are the subsets X i , satisfyingthe following three properties.1. V = X ∪ . . . ∪ X t .2. For every edge ( v, w ) in the graph, there is a subset X i that contains both v and w .3. If X i and X j both contain a vertex v , then all nodes X z of the tree in the (unique) path between X i and X j contain v as well.Robertson and Seymour proved that treewidth is related to the largest grid minor. Indeed, theyproved that for each r there is f ( r ) such that every graph with treedwidth at least f ( r ) contains a r × r grid minor. The original upper bound on f ( r ) was enormous. It was later improved by Robertson,Seymour, and Thomas [17], who showed cr log r ≤ f ( r ) ≤ c ′ r where c and c ′ are absolute constants.In the other direction, it is easy to show that any graph which contains an r × r grid minor hastreewidth at least r .Separators are another important concept in graph theory which have many algorithmic, extremal,and enumerative applications. A vertex subset V of a graph G is a separator for G if there is a partition V ( G ) = V ∪ V ∪ V such that | V | , | V | ≤ n/ V and the8ther vertex in V . A fundamental result of Lipton and Tarjan states that every planar graph on n vertices has a separator of size O ( √ n ). This result has been generalized in many directions, to graphsembedded on a surface [7], graphs with a forbidden minor [1], intersection graphs of balls in R d , andintersection graphs of geometric objects in the plane [5], [6]. The separation number of a graph G isthe minimum s for which every subgraph of G has a separator of size at most s .The bramble number of a graph G is the minimum b such that for every bramble for G there is aset of b vertices for which every subgraph in the bramble contains at least one of these b vertices.Two graph parameters are comparable if one of them can be bounded as a function of the other,and vice versa. Robertson and Seymour showed that treewidth and largest grid minor are comparable.The following theorem which we state without proof extends this result. It may be surprising becausesome of these parameters appear from their definitions to be unrelated. Theorem 7.
Fractional Hadwiger number, r -integral Hadwiger number for each r ≥ , bramble num-ber, separation number, treedwidth, and maximum grid minor size are all comparable. The dependence between some of these graph parameters is not well understood and improvingthe bounds remains an interesting open problem.
Acknowledgements:
I am greatly indebted to Noga Alon, Nati Linial, and Paul Seymour for helpfulconversations.
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