On perturbations of highly connected dyadic matroids
OON PERTURBATIONS OF HIGHLY CONNECTEDDYADIC MATROIDS
KEVIN GRACE AND STEFAN H. M. VAN ZWAM
Abstract.
Geelen, Gerards, and Whittle [3] announced the fol-lowing result: let q = p k be a prime power, and let M be a properminor-closed class of GF( q ) -representable matroids, which does notcontain PG( r − , p ) for sufficiently high r . There exist integers k, t such that every vertically k -connected matroid in M is a rank- ( ≤ t ) perturbation of a frame matroid or the dual of a frame matroid over GF( q ) . They further announced a characterization of the pertur-bations through the introduction of subfield templates and frametemplates.We show a family of dyadic matroids that form a counterexam-ple to this result. We offer several weaker conjectures to replacethe ones in [3], discuss consequences for some published papers,and discuss the impact of these new conjectures on the structureof frame templates. Introduction
Robertson and Seymour profoundly transformed graph theory withtheir Graph Minors Theorem [18]. Geelen, Gerards, and Whittle are ontrack to do the same for matroid theory with their Matroid StructureTheory for matroids representable over a finite field (see, e.g. [4]). Thetheorem they intend to prove is the following:
Conjecture 1.1 (Matroid Structure Theorem, rough idea) . Let F be afinite field, and let M be a proper minor-closed class of F -representablematroids. If M ∈ M is sufficiently large and has sufficiently highbranch-width, then M has a tree-decomposition, the parts of which cor-respond to mild modifications of matroids representable over a propersubfield of F , or to mild modifications of frame matroids and their du-als. Date : September 30, 2018.1991
Mathematics Subject Classification. a r X i v : . [ m a t h . C O ] J un KEVIN GRACE AND STEFAN H. M. VAN ZWAM
The words “tree-decomposition”, “parts”, “correspond to”, and “mildmodifications” need (a lot of) elaboration, and hide over 15 years ofvery hard work. Whittle [22] described the proof of Rota’s Conjecture,which has the Matroid Structure Theorem as a major ingredient, asfollows:“It’s a little bit like discovering a new mountain – we’vecrossed many hurdles to reach a new destination and wehave returned scratched, bloodied and bruised from thearduous journey – we now need to create a pathway soothers can reach it.”In this paper we will only focus on the last part of Conjecture 1.1.Geelen, Gerards, and Whittle announced without proof a theoremabout that part [3, Theorem 3.1] that we will repeat here as Conjecture1.2. First, we require some definitions. An F - represented matroid (orsimply represented matroid if the field is understood from the context)is a matroid with a fixed class of representation matrices over F thatare row equivalent up to column scaling and removal of zero rows. A represented frame matroid is a matroid with a representation matrix A that has at most two nonzero entries per column. The matroidswe will be working with in Section 3 are dyadic and therefore ternary.Since ternary matroids are uniquely GF(3) -representable [2], we willnot make any distinction between matroids and represented matroidsin that case. We also extend this convention to binary matroids, par-ticularly complete graphic matroids, since every binary matroid that isrepresentable over some field F is uniquely F -representable.A matroid (or represented matroid) is vertically k -connected if, forevery separation ( A, B ) of order less than k , one of A and B spans E ( M ) . If a matroid (or represented matroid) M is vertically k -connected,then M ∗ is cyclically k -connected. A rank- ( ≤ t ) perturbation of a rep-resented matroid M is the represented matroid obtained by adding amatrix of rank at most t to the representation matrix of M . Conjecture 1.2 ([3, Theorem 3.1]) . Let F be a finite field and let M be a proper minor-closed class of F -represented matroids. Then thereexist k, t ∈ Z + such that each vertically k -connected member of M isa rank- ( ≤ t ) perturbation of an F -represented matroid N , such thateither (i) N is a represented frame matroid, (ii) N ∗ is a represented frame matroid, or (iii) N is confined to a proper subfield of F . ERTURBED DYADIC MATROIDS 3
In this paper we present a counterexample to this conjecture. Inparticular, we build a family of dyadic matroids that are vertically k -connected for any desired k , and not a bounded-rank perturbation ofeither a represented frame matroid or the dual of a represented framematroid. The construction starts with a cyclically k -connected graph G , modifies it at a number of vertices that grows with | V ( G ) | , anddualizes the resulting matroid. We detail the construction, and proveits key properties, in Section 3.Vertical connectivity and cographic matroids are not very compat-ible notions. Because of this, our examples, which are very sparse,arise only in situations where one might expect the second outcomeof the conjecture to hold. For this reason, the forthcoming proof ofthe Matroid Structure Theorem itself is not jeopardized, and versionsof Conjecture 1.2 can be recovered. In Section 4, we provide severalsuch conjectures. Included in Section 4 is Section 4.3, where we discussconsequences for [10] and [15]. In Section 5, we discuss consequencesfor the notion of frame templates , introduced in [3] to describe theperturbations in more detail.2. Preliminaries
Unexplained notation and terminology will generally follow Oxley [16].One exception is that we denote the vector matroid of a matrix A by M ( A ) , rather than M [ A ] . The following characterization of the dyadicmatroids was shown by Whittle in [23]. Theorem 2.1.
A matroid is dyadic if and only if it is representableover both
GF(3) and
GF(5) . We will need some definitions and results related to bounded-rankperturbations of represented matroids. The next three definitions arefrom [3].
Definition 2.2.
Let M and M be F -represented matroids on a com-mon ground set. Then M is a rank- ( ≤ t ) perturbation of M if thereexist matrices A and P such that M ( A ) = M , the rank of P is atmost t , and M ( A + P ) = M . Definition 2.3.
Let M and M be F -represented matroids on groundset E . If there is some F -represented matroid M on ground set E ∪ { e } such that M = M \ e and M = M/e , then M is an elementary lift of M , and M is an elementary projection of M .Note that an elementary lift of a represented matroid M ( A ) can beobtained by appending a row to A . KEVIN GRACE AND STEFAN H. M. VAN ZWAM
Definition 2.4.
Let M and M be F -represented matroids on a com-mon ground set. We denote by dist( M , M ) the minimum number ofelementary lifts and elementary projections needed to transform M into M , and we denote by pert( M , M ) the smallest integer t suchthat M is a rank-( ≤ t ) perturbation of M .The following observation will be quite useful; in particular, we useit to prove Lemma 2.6 below. Remark . Suppose that M = M ( A ) is a rank- ( ≤ t ) perturbationof M = M ( A ) . Let P be the matrix of rank at most t such that A + P = A . Let { v , v , . . . , v t } be a basis for the row space of P . Note that neither A , P , nor A + P need have full row rank. If r = r ( M ) , then we may assume that P has r + t rows. If a i,j ∈ F forall i, j , let a i, v + a i, v + . . . + a i,t v t be the i -th row of P . Then M canbe obtained by contracting C from the represented matroid obtainedfrom the following matrix. Cv ... − Iv t a , . . . a ,t A ... . . . ...a r + t, . . . a r + t,t Lemma 2.6 appears in [3] as Lemma 2.1; however, no proof was givenin [3]. We will need it to prove our main result, so we give a proof here.
Lemma 2.6 ([3, Lemma 2.1]) . If M and M are F -represented ma-troids on the same ground set, then pert( M , M ) ≤ dist( M , M ) ≤ M , M ) . Proof.
A rank- ( ≤ t ) perturbation of a represented matroid M can beobtained by successively adding t rank- matrices to some matrix A with M ( A ) = M . Therefore, we can prove this result inductively byconsidering the behavior of elementary lifts, elementary projections,and rank- perturbations. An elementary lift of M can be obtainedby adding the rank- matrix (cid:20) v (cid:21) , for some vector v , to the matrix (cid:20) · · · A (cid:21) , which represents M . Thus, every elementary lift of arepresented matroid is also a rank- perturbation of the representedmatroid. Now, since M is a rank-( ≤ t ) perturbation of M if and only ERTURBED DYADIC MATROIDS 5 if M is a rank-( ≤ t ) perturbation of M and since M is an elementarylift of M if and only if M is an elementary projection of M , wealso have that every elementary projection of a represented matroidis a rank- perturbation of the represented matroid. The converse ofthese statements is not true in general; however, we will show thatevery rank- perturbation of a represented matroid can be obtained byperforming an elementary lift followed by an elementary projection.Suppose that M is a rank- perturbation of M . By Remark 2.5,there are vectors v and w and a matrix A with M ( A ) = M such that M is obtained from the matrix A = (cid:20) v − A w (cid:21) by contractingthe element represented by the last column. The represented matroidobtained from A (cid:48) = (cid:20) vA (cid:21) is an elementary lift of M . Since M is obtained from M ( A ) by contracting the element represented by thelast column, M is an elementary projection of M ( A (cid:48) ) .The fact that a rank- perturbation can be obtained by at least oneelementary lift or projection implies that pert( M , M ) ≤ dist( M , M ) .The fact that a rank- perturbation can be obtained by at most two ele-mentary lifts and projections implies that dist( M , M ) ≤ M , M ) . (cid:4) In order to prove our main result, we will need some lemmas regard-ing duality. The first lemma is an easy corollary of Lemma 2.6. Infact, the following lemma still holds if t is replaced by t , but that bestpossible result is not necessary for our purposes. Lemma 2.7.
Suppose that M is a rank- ( ≤ t ) perturbation of M .Then M ∗ is a rank- ( ≤ t ) perturbation of M ∗ .Proof. By Lemma 2.6 and duality of elementary lifts and elementaryprojections, we have pert( M ∗ , M ∗ ) ≤ dist( M ∗ , M ∗ ) = dist( M , M ) ≤ M , M ) ≤ t. (cid:4) Let ε ( M ) = | si ( M ) | ; that is, ε ( M ) is the number of rank- flats of M . The next lemma, proved by Nelson and Walsh [14], gives a boundon ε ( M ) , when M is the dual of a frame matroid. We use this andLemma 2.9 to prove Lemma 2.10 below. Lemma 2.8 ([14, Lemma 6.2]) . If M ∗ is a frame matroid, then ε ( M ) ≤ r ( M ) . Although [14] was based on [3], the previous lemma was proved in-dependently of [3] and remains accurate for that reason.
KEVIN GRACE AND STEFAN H. M. VAN ZWAM
Lemma 2.9. If M is a rank-( ≤ t ) perturbation of a GF( q ) -representedmatroid N , then ε ( M ) ≤ q t ε ( N ) + (cid:80) t − i =0 q i .Proof. We proceed by induction on t . If t = 0 , then M = N , andthe result is clear. Now suppose the result holds for rank-( ≤ t (cid:48) ) per-turbations for all t (cid:48) < t . Since M is a rank-( ≤ t ) perturbation of N there is some represented matroid M (cid:48) such that M (cid:48) is a rank-( ≤ t − )perturbation of N and M is a rank-( ≤ ) perturbation of M (cid:48) . Thus,there are matrices A and P such that M (cid:48) = M ( A ) , the rank P is ,and M = M ( A + P ) . We will show that the nonloop elements in arank- flat of M (cid:48) become members of at most q distinct rank- flats in M . Let { a , a , . . . , a q − , a q } be the elements of GF( q ) , with a q = 0 ,and let v be a nonzero column in A indexed by an element in a rank- flat F of M (cid:48) . Then the nonloop elements of F are each represented bya column a i v for some i such that ≤ i ≤ q − . Similarly, let w bea nonzero column of P . Then every column of P is represented by acolumn a i w for some i such that ≤ i ≤ q . Thus, every element in F that is not a loop in M (cid:48) will be represented in A + P by a column of theform a i v + a j w = a i ( v + a − i a j w ) , where ≤ i ≤ q − and ≤ j ≤ q .There are q distinct possible values for a − i a j ; therefore, the elementsof F that are not loops in M (cid:48) are in at most q distinct rank- flats in M . Moreover, after P is added to A , loops in M (cid:48) will be representedby columns of the form a j w . This accounts for one additional rank- flat in M . Thus, ε ( M ) ≤ qε ( M (cid:48) ) + 1 . By the induction hypothesis,we have ε ( M ) ≤ q ( q t − ε ( N ) + (cid:80) t − i =0 q i ) + 1 = q t ε ( N ) + (cid:80) t − i =0 q i , whichproves the result. (cid:4) Lemma 2.10.
Let t be a positive integer, and let F = GF( q ) . Thenthere are finitely many integers r such that the complete graphic matroid M ( K r +1 ) is a rank-( ≤ t ) perturbation of the dual of an F -representedframe matroid.Proof. Suppose M is a rank-( ≤ t ) perturbation of an F -representedmatroid N , and let r = r ( M ) . Combining the previous two lemmas, wehave ε ( M ) ≤ q t (3 r ( N ))+ (cid:80) t − i =0 q i . Since M is a rank-( ≤ t ) perturbationof N , we have r ( M ) ≤ r ( N ) + t . Therefore, ε ( M ) ≤ q t ( r − t ) + (cid:80) t − i =0 q i . Since q and t are constant, this expression is less than (cid:0) r +12 (cid:1) = ε ( M ( K r +1 )) for all sufficiently large r . (cid:4) Notation 2.11.
Let g ( q, t ) be the least value n such that for all n (cid:48) ≥ n , the complete graphic matroid M ( K n (cid:48) ) is not represented by anyrepresented matroid that is a rank-( ≤ t ) perturbation of the dual of arepresented frame matroid over GF( q ) . ERTURBED DYADIC MATROIDS 7
Lemma 2.10 can be restated as saying that g ( q, t ) is finite for everyprime power q and positive integer t .As stated in the introduction, our construction makes use of highlycyclically connected graphs. The next two results allow us to specifysome additional details about these graphs. The following lemma seemsto be fairly well known; however, we were unable to find an explicitproof in the literature. For the sake of completeness, we state the resultand provide a proof, obtained by combining some older results. Lemma 2.12.
For every positive integer k , there is a cyclically k -connected cubic graph.Proof. There is a cubic Cayley graph of girth g ≥ k . (See, for example,Biggs [1] or Jajcay and Širáň [11, Theorem 2.1]. In particular, [11] con-tains a nice summary of various related results.) Since Cayley graphsare vertex-transitive, a result of Nedela and Škoviera [13, Theorem 17]states that such a graph has cyclic connectivity g . (cid:4) Thomassen [21, Corollary 3.2] showed the following.
Theorem 2.13.
There is a function ξ such that a graph G with min-imum degree at least and girth at least ξ ( n ) has a minor isomorphicto K n . Finally, we clarify some notation and terminology, following [14].These will be used in Sections 4 and 5. If F is a field and A ∩ B = ∅ , thenwe identify the vector space F A × F B with F A ∪ B . If U ⊆ F E and X ⊆ E ,then U [ X ] is the set of vectors consisting of all coordinate projectionsof u onto X for u ∈ U . If Γ ⊆ F , then Γ U = { γu | γ ∈ Γ , u ∈ U } .If U and W are additive subgroups of F E , then U and W are skew if U ∩ W = { } . 3. The Construction
Our construction involves repeated use of the generalized parallelconnection of a matroid with copies of M ( K ) over a copy of M ( K ) represented in a specific way. The next two results specify that repre-sentation. Both results are easily checked, so we state them withoutproof. Lemma 3.1.
The following matrix represents M ( K ) over all fields ofcharacteristic other than : KEVIN GRACE AND STEFAN H. M. VAN ZWAM
Figure 1.
A signed-graphic representation of M ( K ) v i x yz x yz Figure 2.
Changing G to G (cid:48) A = − − − . Lemma 3.2.
The signed graph shown in Figure 1, with negative edgesprinted in bold, represents M ( K ) . This representation of M ( K ) has been encountered before, for ex-ample in [25, 9, 20]. Definition 3.3.
Let G be a cubic graph, and R ⊆ V ( G ) . For eachvertex v i of R , perform the operation of altering the graph on the left inFigure 2 to become the signed graph on the right, with negative edgesprinted in bold. Let G (cid:48) be the signed graph that results from performingthis operation on every vertex in R . Note that G (cid:48) contains | R | copies ofthe signed-graphic representation of M ( K ) described in Lemma 3.2.Let X , X , . . . , X | R | be the edge sets of these representations of M ( K ) .For each X i , take the generalized parallel connection of M ( G (cid:48) ) with acopy of M ( K ) over X i . Delete the X i , and call the resulting matroidthe ornamentation of ( G, R ) , denoted by Or ( G, R ) . Each X i is a modular flat of M ( K ) , which is uniquely representable over anyfield. Therefore, these generalized parallel connections are well defined. ERTURBED DYADIC MATROIDS 9
Lemma 3.4.
For any cubic graph G , with R ⊆ V ( G ) , the ornamenta-tion Or ( G, R ) is dyadic and has M ( G ) as a minor.Proof. It is well-known that signed-graphic matroids are dyadic (see,for example, [24, Lemma 8A.3]). The construction of Or ( G, R ) involvesgeneralized parallel connections of a signed graph with copies of M ( K ) over a common representation of M ( K ) . Thus, a result of Mayhew,Whittle, and Van Zwam [12, Theorem 3.1] implies that Or ( G, R ) isdyadic.Note that Or ( G, R ) is the result of taking | R | copies of the submatrix [ − I ] of the signed incidence matrix of G , with columns indexed bythe set { i , i , i } , and altering each of them to become a copy of thefollowing matrix: i i i d i e i f i g i − − − . One can check that in the vector matroid of the above submatrix,deleting g i and contracting { d i , e i , f i } results in the vector matroid of [ − I ] . Therefore, M ( G ) is a minor of Or ( G, R ) . (cid:4) Definition 3.5.
In the matrix given in the proof of the previous lemma,let F i = { d i , e i , f i , g i } . We will call each F i a gadget .We are now ready to prove our main result. Theorem 3.6.
For every k, t ∈ Z + , there exists a vertically k -connecteddyadic matroid that is not a rank- ( ≤ t ) perturbation of either a framematroid or the dual of a frame matroid.Proof. Let g and ξ be the functions given in Notation 2.11 and Theorem2.13, respectively. We must define several constants that will be usedthroughout this proof. First, let d > (cid:98) log ( k ) (cid:99) +1 − , and let c ≥ t + 20(3 t ) . We define m = max { c (3 d + 3) + 7 , g (3 , t ) } . Finally, let h = max { k + 1 , ξ ( m ) } . Let G be a cyclically h -connected cubic graph. Such a graph existsby Lemma 2.12. This implies that G has girth at least h ≥ ξ ( m ) . ByTheorem 2.13, G has a minor H isomorphic to K m . Let C and D bethe sets of edges such that G/C \ D = H . Each vertex of H is obtainedby contracting all edges in a subtree of G \ D . Thus, there is a function φ : V ( G ) → V ( H ) such that φ ( w ) = v if w is a vertex in the subtreeof G \ D that is contracted to result in v . Claim 3.6.1.
There is a set R = { v , v , . . . , v c } ⊆ V ( G ) of size c and c +1 pairwise disjoint sets { a , b , c } , { a , b , c } , . . . , { a c , b c , c c } ⊆ V ( H ) that are also disjoint from φ ( R ) such that (1) the members of R are pairwise a distance of at least d from eachother in G , and (2) for each integer i with ≤ i ≤ c , if T i is the subtree of G \ D that is contracted to obtain φ ( v i ) , then there is a set of vertices { a (cid:48) i , b (cid:48) i , c (cid:48) i } ⊆ V ( T i ) (possibly some or all of a (cid:48) i , b (cid:48) i , and c (cid:48) i areequal to v i ) and three internally disjoint subpaths of T i from v i to a (cid:48) i , b (cid:48) i , and c (cid:48) i such that a (cid:48) i , b (cid:48) i , and c (cid:48) i are neighbors in G \ D of some vertex in φ − ( a i ) , φ − ( b i ) , and φ − ( c i ) , respectively.Proof. Suppose { v , a , b , c } , . . . , { v k − , a k − , b k − , c k − } and { a , b , c } were chosen to satisfy (1) and (2), with k maximal. Also suppose, fora contradiction, that k − < c . Since G is cubic, there are at most (cid:80) d − i =0 i < d vertices in G whose distance from some v i is less than d .Thus, after choosing { v , . . . , v k − } ⊆ V ( G ) and { a , b , c } , { a , b , c } , . . . , { a k − , b k − , c k − } ⊆ V ( H ) , there are at least m − ( k − d +3) − > m − c (3 d + 3) − ≥ vertices w in V ( H ) − ( { a , b , c } ∪{ φ ( v ) , a , b , c } ∪ . . . ∪ { φ ( v ) , a , b , c } ) such that every vertex in φ − ( w ) is at a distance of at least d from each member of { v , . . . , v k − } .(In the expression m − ( k − d + 3) − , the +3 comes from the sets { a i , b i , c i } for i > , and the − comes from { a , b , c } .) Choose one ofthese vertices w to be φ ( v k ) , and let three of the others be { a k , b k , c k } .Since each of a k , b k , and c k is a neighbor of φ ( v k ) in H , there must bevertices { a (cid:48) k , b (cid:48) k , c (cid:48) k } ⊆ V ( T k ) that are neighbors in G \ D of some vertexin φ − ( a k ) , φ − ( b k ) , and φ − ( c k ) , respectively. If a (cid:48) k = b (cid:48) k = c (cid:48) k , thenlet v k = a (cid:48) k = b (cid:48) k = c (cid:48) k . If two of { a (cid:48) k , b (cid:48) k , c (cid:48) k } are equal, say a (cid:48) k = b (cid:48) k ,then let v k = a (cid:48) k = b (cid:48) k . Since T k is a tree, there must be a path in T k that joins v k to c (cid:48) k . Now suppose a (cid:48) k (cid:54) = b (cid:48) k (cid:54) = c (cid:48) k . Since T k is a tree,there must be a path P in T k from a (cid:48) k to b (cid:48) k . Similarly, there must be apath P (cid:48) that joins c (cid:48) k to some vertex in P . Let v k be the vertex wherethese two paths meet. In each of these cases, we have three subpaths of T k that satisfy (2). Moreover, R = { v , . . . , v k } also satisfies (1) sinceevery vertex in T k is at a distance of at least d from each member of ERTURBED DYADIC MATROIDS 11 { v , . . . , v k − } . This contradicts the maximality of k and proves theclaim. (cid:3) Claim 3.6.2.
Every circuit of Or ( G, R ) contains either the edge set ofa cycle of G or the edge set of a path in G between two vertices in R .Proof. Suppose for a contradiction that C is a circuit of Or ( G, R ) thatcontains neither the edge set of a cycle of G nor the edge set of a pathin G joining vertices in R . Then C ∩ E ( G ) must consist of the edgesets of vertex-disjoint subtrees S , S , . . . , S n of G such that no S i con-tains more than one vertex in R . Thus, C ⊆ ( ∪ ci =1 F i ) ∪ ( ∪ ni =1 E ( S i )) .However, we will show by induction on | ∪ ni =1 E ( S i ) | , that ( ∪ ci =1 F i ) ∪ ( ∪ ni =1 E ( S i )) is an independent set. Since no pair of gadgets is repre-sented by submatrices whose sets of rows intersect, ∪ ci =1 F i is an inde-pendent set in Or ( G, R ) . Thus, the result holds when | ∪ ni =1 E ( S i ) | = 0 .Now, consider ( ∪ ci =1 F i ) ∪ ( ∪ ni =1 E ( S i )) where | ∪ ni =1 E ( S i ) | = k > andthe result holds for | ∪ ni =1 E ( S i ) | < k . Delete a pendant edge e insome S i . By the induction hypothesis, ( ∪ ci =1 F i ) ∪ ( ∪ ni =1 E ( S i )) − { e } is an independent set in Or ( G, R ) . Since e is a pendant edge in some S i , it must be a coloop in ( Or ( G, R )) | (( ∪ ci =1 F i ) ∪ ( ∪ ni =1 E ( S i ))) . Thus, ( ∪ ci =1 F i ) ∪ ( ∪ ni =1 E ( S i )) is an independent set in Or ( G, R ) . By contra-diction, this proves the claim. (cid:3) Let M be the dual matroid of Or ( G, R ) , and let λ M and λ G be theconnectivity functions of M and M ( G ) , respectively. Then, by duality, λ Or ( G,R ) = λ M . Claim 3.6.3.
The matroid Or ( G, R ) is cyclically k -connected.Proof. Suppose for a contradiction that ( X, Y ) is a cyclic k (cid:48) -separationof Or ( G, R ) , where k (cid:48) < k . Let A ∪ B = E ( G ) , with A ⊆ X and B ⊆ Y . Since M ( G ) has cyclic connectivity k > k (cid:48) , it has no cyclic k (cid:48) -separation. Therefore, one of A or B , say A , has no cycles. However,since ( X, Y ) is a cyclic k (cid:48) -separation, X and Y each contain a circuitof Or ( G, R ) . Since A , and therefore X , contain no edge set of a cycleof G , we see from Claim 3.6.2 that X , and therefore A , contain theedge set of a path in G joining vertices in R . By Claim 3.6.1, this pathhas length at least d . This path must be contained in some componentof G [ A ] with edge set A . If a cubic graph either is disconnected orhas a cut vertex, then both sides of the separation must contain cycles.Therefore, G is a connected graph with no cut vertices. Let B = E ( G ) − A , and let A = A − A . Suppose G [ B ] is not connected.Then, since G [ A ] is a tree, there is a unique path in G [ A ] from onecomponent of G [ B ] to another. This implies that G has a cut vertex.Thus, we deduce that G [ B ] is connected. Let r G be the rank function of M ( G ) . Since B is the disjoint unionof B and A , we have r G ( B ) ≤ r G ( B ) + r G ( A ) . Moreover, since G [ A ] is a component of G [ A ] , we have r G ( A ) = r G ( A ) − r G ( A ) . Therefore, λ G ( A ) ≤ r G ( A ) − r G ( A ) + r G ( B ) + r G ( A ) − r G ( E ( G )) = λ G ( A ) . Let W be the set of vertices of the vertex boundary between A and B .We have λ G ( A ) = r G ( A ) + r G ( B ) − r G ( E ( G )) = | V ( G [ A ]) | − | V ( G [ B ]) | − − ( | V ( G ) | −
1) = | W | − . Thus, we have | W | − λ G ( A ) ≤ λ G ( A ) ≤ λ M ( X ) < k (cid:48) . Therefore, | W | < k (cid:48) + 1 .Note that, since G is cubic and G [ A ] contains no cycle, G [ A ] isa cubic tree whose set of leaves is W . We now claim that no vertexof G [ A ] is at a distance greater than (cid:98) log ( k (cid:48) ) (cid:99) + 1 from W . Sup-pose for a contradiction that v is such a vertex. Therefore, there are (cid:98) log ( k (cid:48) ) (cid:99) +1 ) > log ( k (cid:48) ) ) = 3 h vertices at distance (cid:98) log ( k (cid:48) ) (cid:99) + 2 from v in G [ A ] . This implies that | W | > k (cid:48) , contradicting the facts that | W | < k (cid:48) + 1 and k (cid:48) is a positive integer.Therefore, each vertex of G [ A ] is a distance of at most (cid:98) log ( k (cid:48) ) (cid:99) + 1 from W . Thus, since G is cubic, an upper bound for | A | is (cid:98) log ( k (cid:48) ) (cid:99) (cid:88) i =0 i ) = 3(2 (cid:98) log ( k (cid:48) ) (cid:99) +1 − ≤ (cid:98) log ( k ) (cid:99) +1 − < d, contradicting the fact that G [ A ] must have a path of length at least d .Thus, Or ( G, R ) has no cyclic k (cid:48) -separation for k (cid:48) < k and is thereforecyclically k -connected. (cid:3) For each v i ∈ R , let L i consist of the three edges in H that join φ ( v i ) to the vertices in { a i , b i , c i } . Let D (cid:48) consist of all the edges in H incident with a vertex in φ ( R ) other than the edges in some L i . Claim 3.6.4.
Consider c copies of the submatrix − − − ofthe signed incidence matrix of K m − c , where each of these submatriceshas rows indexed by some { a i , b i , c i } . Then Or ( G, R ) has a minor N that is the vector matroid of the matrix obtained from the signed inci-dence matrix of K m − c by altering each of these submatrices to becomethe following matrix, where the bottom row is a new row added to the ERTURBED DYADIC MATROIDS 13 original matrix. Here, F i = { d i , e i , f i , g i } is a gadget. d i e i f i g i a i b i − c i − − Proof.
Recall that C and D are the sets of edges such that G/C \ D = H ∼ = K m . Then N = ( Or ( G, R )) /C \ ( D ∪ D (cid:48) ) / ∪ ci =1 L i . Informally, N is the result of “gluing” each F i onto the set { a i , b i , c i } of vertices in K m − c . (cid:3) Call the resulting matrix J so that N = M ( J ) . Note that J has m rows and r ( N ) = m − . Let N + be a rank- ( ≤ t ) perturbation of N . Claim 3.6.5.
For some s ≤ t , there are vectors w (cid:48)(cid:48) , . . . , w (cid:48)(cid:48) s and asubmatrix J (cid:48) of J such that N + has a minor isomorphic to the vectormatroid of w (cid:48)(cid:48) ...w (cid:48)(cid:48) s J (cid:48) and such that J (cid:48) contains at least t ) copies of the submatrix d i e i f i g i a i b i c i that represents a gadget.Proof. By Remark 2.5, N + is the result of contracting C (cid:48) from thevector matroid of the matrix below, where ∆ is some arbitrary ternarymatrix. C (cid:48) w ... − Iw t J ∆0 Let V ∆ be the set of row indices of a basis for the rowspace of ∆ .Delete from N all elements represented by columns with nonzero entriesin V ∆ , along with all gadgets containing such an element. This isequivalent to deleting vertices from the complete graph K m − c that wasused to construct N , as well as any gadgets glued onto these vertices.Thus, we are still left with a complete graph with gadgets glued ontoit. Moreover, since | V ∆ | ≤ t , we have at least c − t ≥ t ) gadgets remaining. Since V ∆ is a basis for the rowspace of ∆ , we mayperform row operations to obtain the following matrix, where J (cid:48) is asubmatrix of J , where each w (cid:48) i is a coordinate projection of w i , andwhere ∆ (cid:48) = ∆[ V ∆ , C (cid:48) ] . C (cid:48) w (cid:48) ... − Iw (cid:48) t J (cid:48)
00 ∆ (cid:48)
For each element of V ∆ , we contract one element of C (cid:48) , pivoting onan entry in the row of ∆ (cid:48) . We obtain the following matrix. C (cid:48)(cid:48) w (cid:48) ... Qw (cid:48) t J (cid:48) By contracting C (cid:48)(cid:48) , we obtain the desired matrix, with s = 2 t −| C (cid:48)(cid:48) | . (cid:3) ERTURBED DYADIC MATROIDS 15 F , t ) F , b a c F , F , t ) F t ) , F t ) , t ) . . . . . . . . . . . .. . .. . . . . .. . .... ... Figure 3.
A representation of the matroid M ( J (cid:48)(cid:48) ) inClaim 3.6.6 a a , a , a t ) , a , a , a t ) , a , t ) a , t ) a t ) , t ) . . . ... . . . . . . Figure 4.
The tree T a with its vertex labels, used inClaim 3.6.6Recall that { a , b , c } , { a , b , c } , . . . , { a c , b c , c c } are the sets ofvertices of H ∼ = K m onto which the gadgets are “glued”. Figure 3,shows a representation of a restriction of M ( J (cid:48) ) where each shadedtriangle represents a gadget F i with vertices a i , b i , and c i positioned atthe top, left, and bottom respectively. Consider the subtree of K m − c obtained by deleting from the matroid represented in Figure 3 all ofthe gadgets as well as all vertices b i and c i . Call this tree T a . Figure 4shows T a ; the trees T b and T c are defined similarly. Claim 3.6.6.
The matrix J (cid:48) has a submatrix J (cid:48)(cid:48) , with the same numberof rows as J (cid:48) , such that M ( J (cid:48)(cid:48) ) is represented by Figure 3.Proof. Partition the set of gadgets into t ) subsets F , . . . , F t ) ,each of size at least t ) . This is possible since t ) = (5)(3 t )(4)(3 t ) . F F F F F F F F Figure 5.
A representation of the matroid M ( J (cid:48)(cid:48)(cid:48) ) inClaim 3.6.7Let F i,j = { d i,j , e i,j , f i,j , g i,j } be the j -th gadget in F i , and let it be gluedonto the vertices { a i,j , b i,j , c i,j } .Consider the submatrix J (cid:48)(cid:48) of J (cid:48) consisting of the columns indexedby the union of E ( T a ) , E ( T b ) , and E ( T c ) with the union F of all of thegadgets. One can see that M ( J (cid:48)(cid:48) ) can be represented by Figure 3. (cid:3) Claim 3.6.7.
There are ternary matrices U and J (cid:48)(cid:48)(cid:48) such that M ( J (cid:48)(cid:48)(cid:48) ) can be represented by Figure 5 and such that N + has a minor N (cid:48) rep-resented by the following matrix. F F · · · F U U · · · UJ (cid:48)(cid:48)(cid:48) Proof.
Let W = w (cid:48)(cid:48) ...w (cid:48)(cid:48) s . Since E ( T a ) ∪ E ( T b ) ∪ E ( T c ) is anindependent set in M ( J (cid:48)(cid:48) ) , we may perform row operations so that theportion of W with columns indexed by E ( T a ) ∪ E ( T b ) ∪ E ( T c ) becomesthe zero matrix. Thus, we have the following matrix. E ( T a ) E ( T b ) E ( T c ) F W (cid:48) J (cid:48)(cid:48) The portion of W (cid:48) whose columns are indexed by the elements of agadget is an s × ternary matrix; therefore, there are s ≤ t possi-ble such matrices. Since each F i contains at least t ) gadgets, thepigeonhole principle implies that each F i contains five gadgets whose ERTURBED DYADIC MATROIDS 17 corresponding submatrices of W (cid:48) are equal. Again by the pigeonholeprinciple, since there are t ) sets F i , there is a set of four F i suchthat each contains at least five gadgets such that all 20 of the gadgetscorrespond to equal submatrices of W (cid:48) .Delete all of the other c − t − gadgets. All of the remaininggadgets come from four F i which we can relabel as F , F , F , and F . In addition, delete all but one gadget from each of F , F , and F , cosimplify the resulting matroid, and contract the remaining edgesincident with either a , b , or c . In the resulting matroid, there areeight remaining gadgets which we relabel as F , F , . . . , F . The ele-ments of each F i we relabel as { d i , e i , f i , g i } , and we relabel the verticesonto which F i is glued as a i , b i , and c i . This matroid is the desired ma-troid N (cid:48) . In Figure 5, again each shaded triangle represents a gadget F i with vertices a i , b i , and c i positioned at the top, left, and bottomrespectively. (cid:3) Claim 3.6.8.
Regardless of U , the matroid N (cid:48) is not a frame matroid.Proof. Let P consist of all edges joining a gadget F i to a gadget F i +1 ,for ≤ i ≤ , and for ≤ i ≤ , let α i , β i , and γ i be the edges thatjoin a i , b i , and c i to a , b , and c , respectively. Now let N (cid:48)(cid:48) be thesimplification of N (cid:48) /P/ { α , β , γ } / ( ∪ i =3 i =1 { e i , f i , g i } ) / { d , e , f , g } . Inthe case where U is the zero matrix, N (cid:48)(cid:48) is the generalized parallelconnection of M ( K ) with the ternary Dowling geometry of rank ;that is, N (cid:48)(cid:48) is the vector matroid of the following matrix, where thelast six columns come from the gadgets F , . . . , F . d d d d e f g − − − In the Appendix, we show how the mathematics software systemSageMath was used to show that, regardless of U , the matroid N (cid:48)(cid:48) ,and therefore N (cid:48) , are not signed-graphic matroids. The computationswere carried out in Version 8.0 of SageMath [19], in particular makinguse of the matroids component [17]. We used the CoCalc (formerlySageMathCloud) online interface.Indeed, since U has four columns, its rank is at most 4. Therefore, wemay assume that U has at most four rows. There are 16 possible basesfor M ( U ) – one of size 0, four of size 1, six of size 2, four of size 3, andone of size 4. For each of these bases, we checked all possible matrices U where the basis indexed an identity matrix, unless the resultingmatroid M ( U ) contained a basis that was already checked. In eachcase, N (cid:48)(cid:48) was found not to be signed-graphic. A ternary matroid is aframe matroid if and only if it is a signed-graphic matroid. Therefore, N (cid:48) is not a frame matroid. (cid:3) Recall that M ∗ = Or ( G, R ) . Claim 3.6.9.
The matroid M is not a rank- ( ≤ t ) perturbation of thedual of a frame matroid.Proof. Suppose otherwise. Then by Lemma 2.7, Or ( G, R ) is a rank- ( ≤ t ) perturbation of a frame matroid. The class of matroids that arerank- ( ≤ t ) perturbations of a frame matroid is minor-closed. There-fore, by Claim 3.6.4, N is a rank- ( ≤ t ) perturbation of a frame ma-troid. However, by Claims 3.6.7 and 3.6.8, this is impossible. (cid:3) Claim 3.6.10.
The matroid M is not a rank- ( ≤ t ) perturbation of aframe matroid.Proof. Suppose otherwise. Then by Lemma 2.7, Or ( G, R ) is a rank- ( ≤ t ) perturbation of the dual of a frame matroid. Recall that,by Theorem 2.13, G contains a minor isomorphic to K m . Therefore,by Lemma 3.4, Or ( G, R ) has a minor isomorphic to M ( K m ) . But,since m ≥ g (3 , t ) , Lemma 2.10 implies that M ( K m ) , and therefore Or ( G, R ) , are not rank- ( ≤ t ) perturbations of the dual of a framematroid. (cid:3) By Lemma 3.4, Or ( G, R ) , is dyadic. Since the class of dyadic ma-troids is closed under duality, M is dyadic also. By Claim 3.6.3 andduality, M is vertically k -connected. Claims 3.6.9 and 3.6.10 show that M is not a rank- ( ≤ t ) perturbation of either a frame matroid or thedual of a frame matroid. This completes the proof of the theorem. (cid:4) Corollary 3.7.
The family of matroids given in Theorem 3.6 is a coun-terexample to Conjecture 1.2.Proof.
By Theorem 3.6, for every k, t ∈ Z + , there exists a vertically k -connected dyadic matroid that is not a rank- ( ≤ t ) perturbation ofeither a frame matroid or the dual of a frame matroid. Thus, neither(i) nor (ii) of Conjecture 1.2 is satisfied. Moreover, since the matroidsgiven by Theorem 3.6 are dyadic, they are representable over GF(3) which has no proper subfield. Therefore, if F = GF(3) (or any primefield of odd order, for that matter), then the matroids given by Theorem3.6 do not satisfy (iii) of Conjecture 1.2 either. (cid:4) ERTURBED DYADIC MATROIDS 19
Remark . Our construction relies heavily on a non-standard framematroid representation of M ( K ) , and involves a notion of 4-sums.Each gadget is -separating in our construction. The following resultby Zaslavsky [25] shows that 5-sums and higher cannot be encounteredin an analogous way. Theorem 3.9 ([25, Proposition 5A]) . Let Ω be a biased graph such thatthe frame matroid of Ω is isomorphic to M ( K m ) for m ≥ . Then Ω is isomorphic to either ( K m , ∅ ) or Φ (cid:48) m − , where the latter is the biasedgraph obtained by adding an edge e in parallel with an edge of K m ,taking the unbalanced cycles to be the collection of cycles through e ,and contracting e in the resulting biased graph. This makes us cautiously optimistic that our construction cannot begeneralized to have “gadgets” with arbitrary connectivity.We believe that the subfield case, as stated by Geelen, Gerards, andWhittle [3], does not need to be modified.4.
Discussion
Frame templates.
In Subsection 4.2, we will offer some updatedconjectures to replace Conjecture 1.2. Some of the hypotheses will bestated in terms of frame templates. Geelen, Gerards, and Whittle in-troduced the notions of subfield templates and frame templates in [3].A template is a concise description of certain perturbations of repre-sented matroids. We will recall several definitions concerning frametemplates which essentially can be found in [3] as well as [10] and [14].Let A be a matrix over a field F . Then A is a frame matrix ifeach column of A has at most two nonzero entries. Let F × denote themultiplicative group of F , and let Γ be a subgroup of F × . A Γ -framematrix is a frame matrix A such that: • Each column of A with a nonzero entry contains a 1. • If a column of A has a second nonzero entry, then that entry is − γ for some γ ∈ Γ .A frame template over F is a tuple Φ = (Γ , C, X, Y , Y , A , ∆ , Λ) such that the following hold :(i) Γ is a subgroup of F × .(ii) C , X , Y and Y are disjoint finite sets.(iii) A ∈ F X × ( C ∪ Y ∪ Y ) .(iv) Λ is a subgroup of the additive group of F X and is closed underscaling by elements of Γ . The authors of [3] divided our set X into two separate sets which they called X and D . Their set X can be absorbed into Y , therefore we omit it. (v) ∆ is a subgroup of the additive group of F C ∪ Y ∪ Y and is closedunder scaling by elements of Γ .Let Φ = (Γ , C, X, Y , Y , A , ∆ , Λ) be a frame template. Let B and E be finite sets, and let A (cid:48) ∈ F B × E . We say that A (cid:48) respects Φ if thefollowing hold:(i) X ⊆ B and C, Y , Y ⊆ E .(ii) A (cid:48) [ X, C ∪ Y ∪ Y ] = A .(iii) There exists a set Z ⊆ E − ( C ∪ Y ∪ Y ) such that A (cid:48) [ X, Z ] = 0 ,each column of A (cid:48) [ B − X, Z ] is a unit vector, and A (cid:48) [ B − X, E − ( C ∪ Y ∪ Y ∪ Z )] is a Γ -frame matrix.(iv) Each column of A (cid:48) [ X, E − ( C ∪ Y ∪ Y ∪ Z )] is contained in Λ .(v) Each row of A (cid:48) [ B − X, C ∪ Y ∪ Y ] is contained in ∆ .The structure of A (cid:48) is shown below. Z Y Y CX columns from Λ 0 A Γ -frame matrix unit columns rows from ∆ Now, suppose that A (cid:48) respects Φ and that A ∈ F B × E satisfies thefollowing conditions:(i) A [ B, E − Z ] = A (cid:48) [ B, E − Z ] .(ii) For each i ∈ Z there exists j ∈ Y such that the i -th column of A is the sum of the i -th and the j -th columns of A (cid:48) .We say that such a matrix A conforms to Φ .Let M be an F -represented matroid. We say that M conforms to Φ if there is a matrix A conforming to Φ such that M is isomorphic to M ( A ) /C \ Y . We denote by M (Φ) the set of F -represented matroidsthat conform to Φ .4.2. Updated conjectures.
We offer several updated conjectures.We will state them as hypotheses for easy reference in other papers.We will give both a “perturbation version” and a “template version”of each hypothesis. The template version of Conjecture 1.2 follows asConjecture 4.1. Since a template gives a description of certain types ofperturbations, Conjecture 4.1 is false, as well as Conjecture 1.2. If M is a represented matroid, we denote by (cid:102) M the matroid (in the usualsense) that arises from M . For a field F of characteristic p (cid:54) = 0 , wedenote the prime subfield of F by F p . Conjecture 4.1 ([3, Theorem 4.2]) . Let F be a finite field, let m be apositive integer, and let M be a minor-closed class of F -represented ma-troids. Then there exist k ∈ Z + and frame templates Φ , . . . , Φ s , Ψ , . . . , Ψ t such that ERTURBED DYADIC MATROIDS 21 • M contains each of the classes M (Φ ) , . . . , M (Φ s ) , • M contains the duals of the represented matroids in each of theclasses M (Ψ ) , . . . , M (Ψ t ) , and • if M is a simple vertically k -connected member of M and (cid:102) M has no P G ( m − , F p ) -minor, then either M is a member of atleast one of the classes M (Φ ) , . . . , M (Φ s ) , or M ∗ is a memberof at least one of the classes M (Ψ ) , . . . , M (Ψ t ) . Since every represented matroid conforming to a particular templateis a bounded-rank perturbation of a represented frame matroid, ourconstruction disproves Conjecture 4.1 also.The next hypothesis replaces the requirement of vertical connectivitywith the stronger requirement of Tutte connectivity.
Hypothesis 4.2.
Let F be a finite field, and let M be a proper minor-closed class of F -represented matroids. There exist constants k, t ∈ Z + such that each k -connected member of M is a rank- ( ≤ t ) perturbationof an F -represented matroid N , such that either (1) N is a represented frame matroid, (2) N ∗ is a represented frame matroid, or (3) N is confined to a proper subfield of F . The template version of Hypothesis 4.2 follows.
Hypothesis 4.3.
Let F be a finite field, let m be a positive integer, andlet M be a minor-closed class of F -represented matroids. Then thereexist k ∈ Z + and frame templates Φ , . . . , Φ s , Ψ , . . . , Ψ t such that (1) M contains each of the classes M (Φ ) , . . . , M (Φ s ) , (2) M contains the duals of the represented matroids in each of theclasses M (Ψ ) , . . . , M (Ψ t ) , and (3) if M is a simple k -connected member of M with at least k elements and (cid:102) M has no P G ( m − , F p ) -minor, then either M is amember of at least one of the classes M (Φ ) , . . . , M (Φ s ) , or M ∗ is a member of at least one of the classes M (Ψ ) , . . . , M (Ψ t ) . Taken together, the next two hypotheses revise Conjecture 1.2 bypairing the condition of vertical connectivity with its natural match ofhaving a large clique minor and by pairing the dual condition of cyclicconnectivity with the property of having a large coclique minor.
Hypothesis 4.4.
Let F be a finite field, and let M be a proper minor-closed class of F -represented matroids. There exist constants k, t, n ∈ Z + such that each vertically k -connected member of M containing aminor isomorphic to M ( K n ) is a rank- ( ≤ t ) perturbation of an F -represented matroid N , such that either (1) N is a represented frame matroid, or (2) N is confined to a proper subfield of F . Hypothesis 4.5.
Let F be a finite field, and let M be a proper minor-closed class of F -represented matroids. There exist constants k, t, n ∈ Z + such that each cyclically k -connected member of M containing aminor isomorphic to M ∗ ( K n ) is a rank- ( ≤ t ) perturbation of an F -represented matroid N , such that either (1) N ∗ is a represented frame matroid, or (2) N is confined to a proper subfield of F . Now we give the template version of Hypotheses 4.4 and 4.5.
Hypothesis 4.6.
Let F be a finite field, let m be a positive integer, andlet M be a minor-closed class of F -represented matroids. Then thereexist k, n ∈ Z + and frame templates Φ , . . . , Φ s , Ψ , . . . , Ψ t such that (1) M contains each of the classes M (Φ ) , . . . , M (Φ s ) , (2) M contains the duals of the represented matroids in each of theclasses M (Ψ ) , . . . , M (Ψ t ) , (3) if M is a simple vertically k -connected member of M with an M ( K n ) -minor but no P G ( m − , F p ) -minor, then M is a mem-ber of at least one of the classes M (Φ ) , . . . , M (Φ s ) , and (4) if M is a cosimple cyclically k -connected member of M withan M ∗ ( K n ) -minor but no P G ( m − , F p ) -minor, then M ∗ is amember of at least one of the classes M (Ψ ) , . . . , M (Ψ t ) . Consequences.
We conclude Section 4 by detailing the conse-quences of our results on previously published papers that were writ-ten based on [3]. It should be noted that our results in [10] remaincorrect insofar as they regard templates themselves, as opposed to theapplications of Conjecture 4.1. This means that all of the proofs in [10,Section 3] remain valid. In particular, the determination of the mini-mal nontrivial binary frame templates [10, Theorem 3.19] is accurate.(However, [10, Definition 3.20] is no longer useful to prove the resultsin [10, Section 4].)A significant portion of our work in [10], our forthcoming work, andthe work of Nelson and Walsh [14] involves the use of the structuretheory of Geelen, Gerards, and Whittle to obtain results about theextremal functions (also called growth rate functions) of classes of rep-resented matroids. We will prove that these results are true subjectto Hypothesis 4.6. The extremal function for a minor-closed class M ,denoted by h M ( r ) , is the function whose value at an integer r ≥ is given by the maximum number of elements in a simple representedmatroid in M of rank at most r . We will make use of several results ERTURBED DYADIC MATROIDS 23 in the literature. The first of these is the Growth Rate Theorem ofGeelen, Kung, and Whittle [6, Theorem 1.1].
Theorem 4.7 (Growth Rate Theorem) . If M is a nonempty minor-closed class of matroids, then there exists c ∈ R such that either: (1) h M ( r ) ≤ cr for all r , (2) (cid:0) r +12 (cid:1) ≤ h M ( r ) ≤ cr for all r and M contains all graphicmatroids, (3) there is a prime-power q such that q r − q − ≤ h M ( r ) ≤ cq r for all r and M contains all GF( q ) -representable matroids, or (4) M contains all simple rank-2 matroids. If (2) of the previous theorem holds for M , then M is quadraticallydense . If M is a simple rank- r matroid in M such that ε ( M ) = h M ( r ) ,then we call M an extremal matroid of M .The proof of the Growth Rate Theorem was based on work in [8]and [5]. Specifically, [8] contains the following result. Theorem 4.8 ([8, Theorem 1.1]) . For any finite field F and any graph G , there exists an integer c such that, if M is an F -represented matroidwith no M ( G ) -minor, then ε ( M ) ≤ cr ( M ) . Geelen and Nelson proved the next result. (In fact, their result is abit more detailed, but the following result follows from theirs.)
Theorem 4.9 ([7, Theorem 6.1]) . Let M be a quadratically denseminor-closed class of matroids and let p ( x ) be a real quadratic polyno-mial with positive leading coefficient. If h M ( n ) > p ( n ) for infinitelymany n ∈ Z + , then for all integers r, s ≥ there exists a vertically s -connected matroid M ∈ M satisfying ε ( M ) > p ( r ( M )) and r ( M ) ≥ r . The next result is from Nelson and Walsh [14]. It is accurate sinceit was proved independently of [3].
Lemma 4.10 ([14, Lemma 2.2]) . Let F be a finite field, let f ( x ) bea real quadratic polynomial with positive leading coefficient, and let k ∈ N . If M is a restriction-closed class of F -represented matroidsand if, for all sufficiently large n , the extremal function of M at n isgiven by f ( n ) , then for all sufficiently large r , every rank- r matroid M ∈ M with ε ( M ) = f ( r ) is vertically k -connected. Nelson and Walsh [14] define a pair of templates Φ , Φ (cid:48) to be equivalent if M (Φ) = M (Φ (cid:48) ) . This is the definition we will use in the remainderof this paper, although it should be noted that we defined a differentnotion of equivalence in [10]. A pair of templates that are equivalentin the sense of [14] are also equivalent in the sense of [10], but the converse is not true. Moreover, Nelson and Walsh gave Definition 4.11and proved Lemma 4.12 below. Lemma 4.12 is a result about templatesin their own right, rather than applications of Conjecture 4.1, and istrue for that reason. Definition 4.11.
A frame template
Φ = (Γ , C, X, Y , Y , A , ∆ , Λ) over F is Y -reduced if ∆[ C ] = Γ( F Cp ) and ∆[ Y ∪ Y ] = { } , and there is apartition ( X , X ) of X for which F X p ⊆ Λ[ X ] and Λ[ X ] = { } . Wewill call the partition X = X ∪ X the reduction partition of Φ . Lemma 4.12 ([14, Lemma 5.5]) . Every frame template is equivalentto a Y -reduced frame template. The next lemma is an easy observation.
Lemma 4.13.
Every frame template is equivalent to a Y -reduced frametemplate such that no column of A [ X, Y ] is contained in Λ .Proof. By Lemma 4.12, every template is equivalent to some Y -reducedtemplate Φ = (Γ , C, X, Y , Y , A , ∆ , Λ) . Note that the role of Y inmatroids conforming to Φ is to construct the set Z . Every element of Z indexes a column constructed by placing a column of A [ X, Y ] ontop of an identity column. If such a column is made from a column of A [ X, Y ] that is a copy of an element of Λ , then the column can alsobe obtained in E − ( Z ∪ C ∪ Y ∪ Y ) by choosing an identity columnfor the portion of the column coming from the Γ -frame matrix. Thus,that element of Y is unnecessary, and a template equivalent to Φ canbe obtained from Φ by removing that element of Y . (cid:4) We will now show that, for all sufficiently large ranks, the extremalfunction for the set of matroids conforming to a frame template is givenby a quadratic polynomial. We will call a largest simple matroid of agiven rank that conforms to a template an extremal matroid of thetemplate.
Lemma 4.14.
Let
Φ = (Γ , C, X, Y , Y , A , ∆ , Λ) be a Y -reduced frametemplate, with reduction partition X = X ∪ X , such that no columnof A [ X, Y ] is contained in Λ . Let | (cid:98) Y | denote the number of columnsof A [ X, Y ] that are not contained in Λ . Let | (cid:98) Λ | denote the maximumnumber of nonzero elements of Λ that pairwise are not scalar multiplesof each other. And let t denote the difference between | X | and the rankof the matrix A [ X , C ∪ Y ∪ Y ] . If r ≥ | C | + | X | − t + 2 , then thesize of a rank- r extremal matroid of Φ is ar + br + c , where a = 12 | Γ || Λ | , ERTURBED DYADIC MATROIDS 25 b = 12 | Γ || Λ | (2 | C | + 2 t − | X | −
1) + | Λ | + | Y | , and c = 12 ( | C | + t − | X | )[ | Γ || Λ | ( | C | + t − | X | −
1) + 2 | Λ | + 2 | Y | ] + | (cid:98) Λ | + | (cid:98) Y | . Proof.
An extremal matroid M of Φ is obtained by contracting C anddeleting Y from the vector matroid of some matrix A that conforms to Φ . Let r C = r M ( A ) ( C ) . Then r ( M ( A ) \ Y ) = r + r C , and the numberof rows of A is r + r C + t . We wish to calculate the largest possible sizeof a simple matroid of the form M ( A ) \ Y , where A conforms to Φ andwhere r ( M ( A ) \ Y /C ) = r . Since A has r + r C + t rows, the numberof rows of the Γ -frame submatrix of A is r + r C + t − | X | , which weabbreviate as n . Let n ≥ . Thus, A has at least | X | + 1 rows andrank at least | X | − t + 1 , and r ≥ | X | − r C − t + 1 .In E − ( Z ∪ C ∪ Y ∪ Y ) , there are | Γ || Λ | (cid:0) n (cid:1) distinct possible columnswhere the Γ -frame matrix has two nonzero entries per column. Thereare | Λ | n distinct possible columns where the Γ -frame matrix has onenonzero entry per column. And there are | (cid:98) Λ | distinct possible nonzerocolumns where the Γ -frame matrix is a zero column because includingall of the elements of Λ would result in a matroid that is not simple.The size of Z is at most | Y | n since there are that many possibledistinct possible columns.The entire sets C and Y are always contained in M ( A ) , but ifany columns of A [ X, Y ] are contained in Λ , then the correspond-ing element of E − ( Z ∪ C ∪ Y ∪ Y ) must be deleted in order forthe matroid to be simple. Therefore, adding together the elements of E − ( Z ∪ C ∪ Y ∪ Y ) , the elements of Z , and the elements of C ∪ Y ,we see that ε ( M ( A ) \ Y ) = | Γ || Λ | (cid:18) n (cid:19) + | Λ | n + | (cid:98) Λ | + | Y | n + | C | + | (cid:98) Y | . If C = ∅ , then M ( A ) \ Y = M ( A ) \ Y /C . Keeping in mind that n = r + r C + t − | X | , some arithmetic shows that this proves the resultin the case where C = ∅ . Thus, we now assume C (cid:54) = ∅ .One can see that ε ( M ( A ) \ Y ) increases as r C increases, since n = r + r C + t −| X | . Thus, to achieve maximum density, we should take C tobe independent, if possible. This can easily be achieved since ∆[ C ] =Γ( F Cp ) . Thus, we take r C to be equal to | C | and n = r + | C | + t − | X | .In fact, let A [ B − X, C ] be equal to I | C | I | C | I | C | · · · · · · n ≥ | C | + 2 and, therefore, r = n − | C | + | X | − t ≥ | C | + | X | − t + 2 . Claim 4.14.1.
For every pair { e, f } ⊆ E − ( C ∪ Y ) , the set C ∪ { e, f } is independent in M ( A ) .Proof. Note that every column of A [ B − X, E − ( C ∪ Y )] has at mosttwo nonzero entries. So the columns of A [ B − X, E − ( C ∪ Y )] labeledby e and f have nonzero entries in at most four rows. We proceedby induction on | C | . Suppose | C | = 1 . Since the single column of A [ B − X, C ] has five nonzero entries, there is a unit row in A [ B − X, C ∪{ e, f } ] whose nonzero entry is in C . This implies that r ( C ∪ { e, f } ) = r ( { e, f } ) + 1 . Since e and f are not parallel elements, C ∪ { e, f } mustbe independent.Now suppose | C | > . Then there are at least | C | ≥ unit rowsin A [ B − X, C ] . Since e and f have nonzero entries in at most fourrows, this implies that there is a unit row in A [ B − X, C ∪ { e, f } ] withits nonzero entry in a column labeled by some element c ∈ C . Thus r ( C ∪ { e, f } ) = r (( C − c ) ∪ { e, f } ) + 1 . By the induction hypothesis, ( C − c ) ∪ { e, f } is independent. Therefore, C ∪ { e, f } is independentalso. (cid:3) Claim 4.14.1 implies that, when C is contracted, the resulting ma-troid is still simple. Thus, ε ( M ) = ε ( M ( A ) \ Y ) − | C | = | Γ || Λ | (cid:18) n (cid:19) + | Λ | n + | (cid:98) Λ | + | Y | n + | (cid:98) Y | . Some arithmetic, recalling that n = r + | C | + t − | X | , shows that thisimplies the result. (cid:4) Lemma 4.15.
Suppose Hypothesis 4.6 holds, and let F be a finite field.Let M be a quadratically dense minor-closed class of F -representedmatroids, and let { Φ , . . . , Φ s , Ψ , . . . , Ψ t } be the set of templates givenby Hypothesis 4.6. For all sufficiently large r , the extremal matroidsof M are the extremal matroids of the templates in some subset of { Φ , . . . , Φ s } . ERTURBED DYADIC MATROIDS 27
Proof.
Let p be the characteristic of F . Since M is a quadraticallydense minor-closed class and since ε (PG( r − , F p )) = p r − p − , for allsufficiently large r , no member of M contains PG( r − , F p ) as a minor.By Hypothesis 4.6, there are a pair of integers k, n such that everysimple vertically k -connected member of M with an M ( K n ) -minor isa member of at least one of the classes M (Φ ) , . . . , M (Φ s ) .By Lemmas 4.13 and 4.14, for every frame template Φ and for allsufficiently large r , the size of a rank- r extremal matroid of Φ is givenby a quadratic polynomial in r . Thus, for all sufficiently large r , the sizeof the largest simple rank- r matroid that conforms to some templatein { Φ , . . . , Φ s } is given by a quadratic polynomial h (cid:48)M ( r ) .By definition, h M ( r ) ≥ h (cid:48)M ( r ) . We wish to show that equality holdsfor all sufficiently large r . Suppose otherwise. Then, for infinitelymany r , we have h M ( r ) > h (cid:48)M ( r ) . Theorem 4.9, with h (cid:48)M ( r ) playingthe role of p ( n ) and with k playing the role of s , implies that, forinfinitely many r , there is a vertically k -connected rank- r matroid M r ∈M with ε ( M r ) > h (cid:48)M ( r ) . Thus, these M r do not conform to anytemplate in { Φ , . . . , Φ s } . By Hypothesis 4.6, these M r contain no M ( K n ) minor. However, by Theorem 4.8, there is an integer c suchthat ε ( M r ) ≤ cm . This contradicts the fact that ε ( M r ) > h (cid:48)M ( r ) forall r . By contradiction, we determine that h M ( r ) = h (cid:48)M ( r ) , for allsufficiently large r .Therefore, we know that, for all sufficiently large r , the extremalfunction h M ( r ) is given by a quadratic polynomial. Now, Lemma 4.10implies that, for all sufficiently large r , the rank- r extremal matroidsof M are vertically k -connected. Thus, by Hypothesis 4.6, it sufficesto show that, for all sufficiently large r , the largest simple matroidsof rank r contain M ( K n ) as a minor. Suppose otherwise. Then, forinfinitely many r , the largest simple matroids in M of rank r haveno M ( K n ) -minor. By Theorem 4.8, for infinitely many r , the largestsimple matroids in M of rank r have size at most cr , for some integer c . This contradicts the quadratic density of M . (cid:4) Because of Lemma 4.15, the results in [10] regarding extremal func-tions ( [10, Theorem 1.3], [10, Theorem 4.2], and [10, Corollary 4.10])now require Hypothesis 4.6. However, [10, Theorem 4.2] was alsoproved (independently of [3]) by Geelen and Nelson [7, Corollary 1.6].Furthermore, [10, Corollary 4.10] follows easily from [10, Theorem 4.2].Finally, we modify [10, Theorem 1.4] to require Tutte connectivity,rather than vertical connectivity.
Theorem 4.16.
If Hypothesis 4.3 holds, then there exists k ∈ Z + such that every simple, k -connected, 1-flowing matroid with at least k elements is either graphic or cographic. Another paper that relies on the faulty version of the structure the-orem is [15]. (It is stated as Hypothesis 4.1 in that paper.) The mainresult of [15] is [15, Theorem 1.1]. In order to recover this result, [15,Lemma 3.1] will need to be refined in order to deal with represented ma-troids close to the dual of a represented frame matroid. Since this caseis significantly easier than the case in which the represented matroidis close to a represented frame matroid, Nelson and Van Zwam expectthat the main result can be recovered, but elect to wait with their fixuntil the structure theorem has been proven by Geelen, Gerards, andWhittle. 5.
Refined Templates
In this section, we prove a result that will be of interest for futurework. Nelson and Walsh gave Definition 5.1 and proved Lemma 5.2below.
Definition 5.1.
A frame template
Φ = (Γ , C, X, Y , Y , A , ∆ , Λ) is reduced if there is a partition ( X , X ) of X such that • ∆ = Γ( F Cp × ∆ (cid:48) ) for some additive subgroup ∆ (cid:48) of F Y ∪ Y , • F X p ⊆ Λ[ X ] while Λ[ X ] = { } and A [ X , C ] = 0 , and • the rows of A [ X , C ∪ Y ∪ Y ] form a basis for a subspace whoseadditive subgroup is skew to ∆ . Lemma 5.2 ([14, Lemma 5.6]) . Every frame template is equivalent toa reduced frame template.
Although [14] was based on the faulty version of the structure theo-rem, Lemma 5.2 is a result about templates in their own right, ratherthan Conjecture 4.1, and is true for that reason. We will refer to thepartition X = X ∪ X given in Definition 5.1 as the reduction partition of Φ . We also introduce the following definition. Definition 5.3.
A frame template
Φ = (Γ , C, X, Y , Y , A , ∆ , Λ) is refined if it is reduced, with reduction partition X = X ∪ X , and if Y spans the matroid M ( A [ X , Y ∪ Y ]) . Remark . Each of the minimal nontrivial binary frame templatesgiven in [10] is reduced (often in trivial or vacuous ways) and, in fact,refined.
ERTURBED DYADIC MATROIDS 29
We wish to show that, for the purposes of using the Hypotheses givenin Section 4, only refined frame templates must be considered.
Lemma 5.5.
Let
Φ = (Γ , C, X, Y , Y , A , ∆ , Λ) be a reduced frametemplate that is not refined. If M ∈ M (Φ) , then E ( M ) − Y is notspanning in M .Proof. Let A be the matrix conforming to Φ such that M = M ( A ) /C \ Y .Since Φ is not refined, Y does not span M ( A [ Y ∪ Y ]) . Therefore, Y contains a cocircuit in M ( A [ Y ∪ Y ]) . In fact, since the definition ofreduced implies that A [ X , E − ( Y ∪ Y ∪ Z )] is the zero matrix, andsince every column of A [ X , Z ] is a copy of a column of A [ X , Y ] , wesee that Y contains a cocircuit in M ( A ) . This implies that Y alsocontains a cocircuit in M = M ( A ) /C \ Y . Thus, E ( M ) − Y is notspanning in M . (cid:4) Now we prove the main result of this section.
Theorem 5.6.
If Hypothesis 4.3 holds for a class M , then the constant k , and the templates Φ , . . . , Φ s , Ψ , . . . , Ψ t can be chosen so that thetemplates are refined. Moreover, If Hypothesis 4.6 holds for a class M ,then the constants k, n , and the templates Φ , . . . , Φ s , Ψ , . . . , Ψ t can bechosen so that the templates are refined.Proof. Suppose that Hypothesis 4.3 holds, and let Φ ∈ { Φ , . . . , Φ s , Ψ , . . . , Ψ t } . By Lemma 5.2, we may assume that Φ is reduced withreduction partition X = X ∪ X . Suppose for a contradiction that Φ is not refined. Choose k ≥ | Y | . If M is a k -connected representedmatroid conforming to Φ , then Lemma 5.5 implies that λ M ( Y )
We would like to thank Jim Geelen, Peter Nelson, and Geoff Whittlefor helpful discussions. In particular, Peter Nelson gave many helpfulsuggestions about the manuscript and alerted us to an error in one ofthe proofs.
References [1] N.L. Biggs, Cubic graphs with large girth, in
Combinatorial Mathematics:Proceedings of the Third International Conference , Annals of the New YorkAcademy of Sciences (1989), 56–62.[2] Thomas H. Brylawski and T.D. Lucas, Uniquely representable combinatorialgeometries, in
Teorie combinatorie (Proc. 1973 Internat. Colloq.), AcademiaNazionale dei Lincei (1976), Rome pp. 83–104.[3] Jim Geelen, Bert Gerards, and Geoff Whittle, The highly connected matroidsin minor-closed classes,
Annals of Combinatorics (2015), 107–123.[4] Jim Geelen, Bert Gerards, and Geoff Whittle, Towards a matroid-minor struc-ture theory, in Combinatorics, Complexity, and Chance , Oxford UniversityPress (2007), pp. 72–82.[5] Jim Geelen, and Kasper Kabell, Projective geometries in dense matroids,
Jour-nal of Combinatorial Theory, Series B (2009), 1–8.[6] Jim Geelen, Joseph P.S. Kung, and Geoff Whittle, Growth rates of minor-closed classes of matroids, Journal of Combinatorial Theory, Series B (2009), 420–427.[7] Jim Geelen and Peter Nelson, Matroids denser than a clique, Journal of Com-binatorial Theory, Series B (2015), 51–69.[8] Jim Geelen, and Geoff Whittle, Cliques in dense
GF( q ) -representable matroids, Journal of Combinatorial Theory, Series B (2003), 264–269.[9] A.M.H. Gerards, Graphs and Polyhedra. Binary Spaces and Cutting Planes ,CWI Tract , Stichting Mathematisch Centrum Centrum voor Wiskunde enInformatica, 1990.[10] Kevin Grace and Stefan H.M. van Zwam, Templates for binary matroids, SIAMJournal on Discrete Mathematics (2017), 254–282.[11] Robert Jajcay and Jozef Širáň, Small vertex-transitive graphs of given degreeand girth, Ars Mathematica Contemporanea (2011), 375–384. ERTURBED DYADIC MATROIDS 31 [12] Dillon Mayhew, Geoff Whittle, and Stefan H.M. van Zwam, An obstacle to adecomposition theorem for near-regular matroids,
SIAM Journal on DiscreteMathematics (2011), 271–279.[13] Roman Nedela and Martin Škoviera, Atoms of cyclic connectivity in cubicgraphs, Mathematica Slovaca (1995), 481–499.[14] Peter Nelson and Zachary Walsh, The extremal function for geometry minorsof matroids over prime fields, arXiv:1703.03755 [math.CO].[15] Peter Nelson and Stefan H.M. van Zwam, On the existence of asymptoticallygood linear codes in minor-closed classes, IEEE Transactions on InformationTheory (2015), 1153–1158.[16] James Oxley, Matroid Theory , Second Edition, Oxford University Press, NewYork, 2011.[17] Rudi Pendavingh, Stefan van Zwam, et al.
Sage Matroids Component, in-cluded in Sage Mathematics Software 5.12
Journal of Combinatorial Theory, Series B Discrete Mathematics (2007), 2187–2199.[21] Carsten Thomassen, Girth in graphs,
Journal of Combinatorial Theory, SeriesB (1983), 129–141.[22] Victoria University of Wellington, School of mathematics and Statis-tics. Researcher solves 40 year old problem. On-line press release.See http://sms.victoria.ac.nz/Main/Researchersolves40yearoldproblem, Au-gust 2013.[23] Geoff Whittle, On matroids representable over GF(3) and other fields,
Trans-actions of the American Mathematical Society (1997), 579-603.[24] Thomas Zaslavsky, Signed graphs,
Discrete Applied Mathematics (1982),47–74.[25] Thomas Zaslavsky, Biased graphs whose matroids are special binary matroids, Graphs and Combinatorics (1990), 77–93. ppendix: Verifying Claim 3.6.8 In [3]: from itertools import combinations from sage.matroids.advanced import * from itertools import productIn [4]: def DowlingGeometry(r): A = Matrix(GF(3), r, r*r) i = 0 for j in range(r): A[j,i] = 1 i += 1 for j in range(r-1): for k in range(j+1,r): A[j,i] = -1 A[k,i] = 1 i += 1 A[j,i] = -1 A[k,i] = -1 i += 1 return Matroid(A)In [5]: def vname(path, ray, index): return path + str(ray) + '_' + str(index) def ename(path, ray, index): return path + str(ray) + '_' + str(index)In [6]: def perturbed_gadget_matroid(perturb_matrix, gadgets_per_ray, num_rays=1): """ Create the matrix representing "rays" of gadgets joined by graph edges. In the case where perturb_matrix=[0 0 0 0], num_rays = 4, and gadgets_per_ray=[5,2,2,2], the result is M(J''') from Figure 5. Edges are encoded
A[vtx[vname('q',ray,i)],j] = -1 A[vtx[vname('q',ray,i+1)],j] = 1 j += 1 groundset.append(ename('q',ray,i))
A[vtx[vname('r',ray,i)],j] = -1 A[vtx[vname('r',ray,i+1)],j] = 1 j += 1 groundset.append(ename('r',ray,i)) first_gadget = 0 for ray in range(num_rays): for k in range(first_gadget, gadgets_per_ray[ray]): first_gadget = 1 elts = 'defg' for l in range(4): A[vtx[vname('p',ray,k)], j + l] = gadget_block[0, l] A[vtx[vname('q',ray,k)], j + l] = gadget_block[1, l] A[vtx[vname('r',ray,k)], j + l] = gadget_block[2, l] A[imax, j + l] = gadget_block[3, l] groundset.extend([ename(e,ray,k) for e in elts]) imax += 1 A.set_block(num_rows - perturb_matrix.nrows(), j, perturb_matrix) j += 4 return
A, groundsetIn [7]: def check_perturbation(P):
A, gs = perturbed_gadget_matroid(P, [5,2,2,2], 4) M = Matroid(gs, A) contract_set = [] contract_set.extend(['e1_1', 'f1_1', 'g1_1', 'e2_1', 'f2_1', 'g2_1', 'e3_1', 'f3_1', 'g3_1', 'p1_0', 'q2_0', 'r3_0']) contract_set.extend(['d0_1', 'e0_2', 'f0_3', 'g0_4']) contract_set.extend([ename('p',0,i) for i in range(4)]) contract_set.extend([ename('q',0,i) for i in range(4)]) contract_set.extend([ename('r',0,i) for i in range(4)]) N = (M / contract_set).simplify() return DowlingGeometry(N.rank()).has_minor(N)
In [8]: perturbation_list=[Matrix(GF(3), [[0,0,0,0]])]n [9]: bases_already_checked=[]X=set([0,1,2,3]) for B in Subsets(X,1): nonB=X.difference(B) for T in product(range(3), repeat=3): A=Matrix(GF(3), 1, 4) i=0 for b in B: A[i,b]=1 i=i+1 A[0,list(nonB)[0]]=T[0] A[0,list(nonB)[1]]=T[1] A[0,list(nonB)[2]]=T[2] if all((A.matrix_from_columns(S)).det()==0 for S in bases_already_checked): perturbation_list.append(A) bases_already_checked.append(B)n [10]: bases_already_checked=[]X=set([0,1,2,3]) for B in Subsets(X,2): nonB=X.difference(B) for T in product(range(3), repeat=4): A=Matrix(GF(3), 2, 4) i=0 for b in B: A[i,b]=1 i=i+1 A[0,list(nonB)[0]]=T[0] A[1,list(nonB)[0]]=T[1] A[0,list(nonB)[1]]=T[2] A[1,list(nonB)[1]]=T[3] if all((A.matrix_from_columns(S)).det()==0 for S in bases_already_checked): perturbation_list.append(A) bases_already_checked.append(B)n [11]: bases_already_checked=[]X=set([0,1,2,3]) for B in Subsets(X,3): nonB=X.difference(B) for T in product(range(3), repeat=3): A=Matrix(GF(3), 3, 4) i=0 for b in B: A[i,b]=1 i=i+1 A[0,list(nonB)[0]]=T[0] A[1,list(nonB)[0]]=T[1] A[2,list(nonB)[0]]=T[2] if all((A.matrix_from_columns(S)).det()==0 for S in bases_already_checked): perturbation_list.append(A) bases_already_checked.append(B)In [12]: perturbation_list.append(matrix.identity(GF(3),4))In [12]: if any(check_perturbation(perturbation_list[t]) != False for t in range(len(perturbation_list))): print "signed-graphic matroid" else : print "No perturbation is signed-graphic."In [13]: version()No perturbation is signed-graphic.Out[13]: 'SageMath version 8.0, Release Date: 2017-07-21' E-mail address : [email protected] E-mail address : [email protected]@math.lsu.edu