AA GENERALISATION OF UNIFORM MATROIDS
GEORGE DRUMMOND
Abstract.
A matroid is uniform if and only if it has no minorisomorphic to U , ⊕ U , and is paving if and only if it has nominor isomorphic to U , ⊕ U , . This paper considers, more generally,when a matroid M has no U k,k ⊕ U ,(cid:96) -minor for a fixed pair of positiveintegers ( k, (cid:96) ) . Calling such a matroid ( k, (cid:96) ) -uniform , it is shown thatthis is equivalent to the condition that every rank- ( r ( M ) − k ) flat of M has nullity less than (cid:96) . Generalising a result of Rajpal, we prove thatfor any pair ( k, (cid:96) ) of positive integers and prime power q , only finitelymany simple cosimple GF ( q ) -representable matroids are ( k, (cid:96) ) -uniform.Consequently, if Rota’s Conjecture holds, then for every prime power q ,there exists a pair ( k q , (cid:96) q ) of positive integers such that every excludedminor of GF ( q ) -representability is ( k q , (cid:96) q ) -uniform. We also determineall binary (2 , -uniform matroids and show the maximally -connectedmembers to be Z \ t, AG (4 , , AG (4 , ∗ and a particular self-dual ma-troid P . Combined with results of Acketa and Rajpal, this completesthe list of binary ( k, (cid:96) ) -uniform matroids for which k + (cid:96) ≤ . Introduction
A matroid M is paving if every rank- ( r ( M ) − flat is independent, orequivalently, if M | H is uniform for every hyperplane H of M . Thus, in anatural sense, paving matroids are close to uniform. In this paper, we gen-eralise this observation and describe a two-parameter property of matroidsthat captures just how close to uniform a given matroid is.For positive integers k and (cid:96) , we define a matroid to be ( k, (cid:96) ) -uniform if ithas no minor isomorphic to U k,k ⊕ U ,(cid:96) . It is easy to show that a matroid is (1 , -uniform precisely if it is uniform and is (2 , -uniform precisely if it ispaving. It is also evident that all matroids are ( k, (cid:96) ) -uniform for some ( k, (cid:96) ) pair and that if M is ( k, (cid:96) ) -uniform, then it is ( k (cid:48) , (cid:96) (cid:48) ) -uniform for all k (cid:48) ≥ k and (cid:96) (cid:48) ≥ (cid:96) . Furthermore, an easy duality argument shows that a matroid is ( k, (cid:96) ) -uniform if and only if its dual is ( (cid:96), k ) -uniform. We will use all thesefacts freely. The first main result of this paper, proved in Section 2, concernsrepresentability of ( k, (cid:96) ) -uniform matroids. Theorem 1.1.
For every pair ( k, (cid:96) ) of positive integers and every primepower q , only finitely many simple cosimple GF ( q ) -representable matroidsare ( k, (cid:96) ) -uniform. Mathematics Subject Classification.
Key words and phrases. matroids, uniform, paving, flats, binary. a r X i v : . [ m a t h . C O ] F e b G.DRUMMOND
Note that both the simple and cosimple requirements in this theorem arenecessary, as the uniform matroids U ,n and U n − ,n are representable overevery field for all n ≥ . The following is an interesting corollary of thistheorem: Corollary 1.2.
For every prime power q , the set of excluded minors for GF ( q ) -representability is finite if and only if for some fixed pair ( k q , (cid:96) q ) ofpositive integers, every such excluded minor is ( k q , (cid:96) q ) -uniform. To illustrate, for q ≤ , every excluded minor of GF ( q ) -representabilityis (2 , -uniform, that is, paving. As Geelen, Gerards and Whittle have an-nounced a proof of Rota’s Conjecture [2], it would seem that such ( k q , (cid:96) q ) pairs exist for all q . If well behaved, these bounds may offer improved meth-ods for explicitly determining the excluded minors of GF ( q ) -representability.By applying duality to the lists of binary (2 , -uniform and (3 , -uniformmatroids of Acketa [1] and Rajpal [6] respectively, one may explicitly list allbinary (1 , -uniform and (1 , -uniform matroids. These results concernbinary ( k, (cid:96) ) -uniform matroids such that k + (cid:96) ≤ . We complete this picturein Sections 3 and 4 by determining the binary (2 , -uniform matroids. Themost difficult part of the characterisation is in establishing the followingresult, the proof of which appears in Section 4: Theorem 1.3.
The -connected binary (2 , -uniform matroids are preciselythe -connected minors of Z \ t, P , AG (4 , , and AG (4 , ∗ . Here, Z \ t is the tipless binary -spike, AG (4 , is the rank- affinegeometry, and P is the rank- binary matroid represented by the matrix ofFigure 1. It is easily seen that P is self-dual and that P / \ ∼ = M ( W ) .Moreover, by pivoting, one can show that P / ∼ = Z . A further descriptionof P is given in Section 4. The terminology throughout will follow Oxley [5]unless otherwise specified. Figure 1.
A binary representation of P .The nullity of a set X in a matroid M is | X | − r M ( X ) . We conclude thissection by proving a characterisation of ( k, (cid:96) ) -uniform matroids in terms ofnullity of flats that will be treated as an alternate definition. Proposition 1.4.
A matroid M is ( k, (cid:96) ) -uniform if and only if every rank- ( r ( M ) − k ) flat of M has nullity less than (cid:96) . GENERALISATION OF UNIFORM MATROIDS 3
Proof.
Suppose first that M is not ( k, (cid:96) ) -uniform. Then M has an indepen-dent set X and coindependent set Y such that M/X \ Y ∼ = U k,k ⊕ U ,(cid:96) and r M ( X ) = r ( M ) − k . Letting Z denote the (cid:96) loops of M/X \ Y , every elementof Z must be in the closure of X in M . Thus, cl M ( X ) is a rank- ( r ( M ) − k ) flat of M with nullity at least (cid:96) .For the converse, suppose M has a rank- ( r ( M ) − k ) flat F of nullity atleast (cid:96) . Contracting any basis for F achieves a rank- k matroid with at least (cid:96) loops. An appropriate restriction then yields a U k,k ⊕ U ,(cid:96) -minor. (cid:3) Finiteness over GF ( q ) In this section, we prove Theorem 1.1 by showing that, for any ( k, (cid:96) ) pairand prime power q , there exists a constant f ( k, (cid:96), q ) bounding the rank ofany simple cosimple ( k, (cid:96) ) -uniform GF ( q ) -representable matroid. We willrequire the following result, a rewording of [6, Proposition 8]. Lemma 2.1.
For k ≥ , let M be a simple ( k, -uniform matroid such that M is GF ( q ) -representable and r ∗ ( M ) > q . Then there is a constant g ( k, q ) such that r ( M ) ≤ g ( k, q ) . We will also make use of the following easy lemma.
Lemma 2.2. If M is a matroid of rank r ≥ , then M is simple if and onlyif it is ( r − , -uniform. Theorem 1.1 is then a direct consequence of the following:
Proposition 2.3.
For every pair ( k, (cid:96) ) of positive integers and every primepower q , there is a constant f ( k, (cid:96), q ) such that if M is a simple cosimple GF ( q ) -representable ( k, (cid:96) ) -uniform matroid, then r ( M ) ≤ f ( k, (cid:96), q ) . More-over, if k ≥ , then f ( k, (cid:96), q ) ≤ max { f ( k − , (cid:96) + 1 , q ) , f (1 , (cid:96), q ) + ( k − } . Proof.
Fixing q , we perform induction on k . The base case k = 1 we splitinto two parts. Firstly, if k = (cid:96) = 1 , then M is uniform and r ( M ) ≤ q − , asotherwise M has a U q,q +2 minor, a contradiction to GF ( q ) -representability.If k = 1 and (cid:96) ≥ , then by the dual of Lemma 2.1, either r ( M ) ≤ q or r ∗ ( M ) ≤ g ( (cid:96), q ) . In the latter case, since M is cosimple, M ∗ is a restrictionof P G ( g ( (cid:96), q ) − , q ) . It follows that r ( M ) ≤ ( q g ( (cid:96),q ) − / ( q − − g ( (cid:96), q ) .Thus, f (1 , (cid:96), q ) exists for all (cid:96) ≥ .Now suppose k ≥ and that f ( k (cid:48) , (cid:96) (cid:48) , q ) exists for all k (cid:48) < k , (cid:96) (cid:48) ≥ . If M is also ( k − , (cid:96) + 1) -uniform, then r ( M ) ≤ f ( k − , (cid:96) + 1 , q ) by induction.Otherwise, M has a rank- ( r ( M ) − k + 1) flat F with nullity at least (cid:96) + 1 .As M is ( k, (cid:96) ) -uniform, M | F is (1 , (cid:96) ) -uniform. By duality, ( M | F ) ∗ is ( (cid:96), -uniform and, as r ∗ ( M | F ) ≥ (cid:96) +1 , it follows Lemma 2.2 that M | F is cosimple.Thus, M | F is a simple cosimple binary (1 , (cid:96) ) -uniform matroid. By induction, r ( M | F ) ≤ f (1 , (cid:96), q ) and hence r ( M ) ≤ f (1 , (cid:96), q ) + ( k − . We conclude that f ( k, (cid:96), q ) exists and f ( k, (cid:96), q ) ≤ max { f ( k − , (cid:96) +1 , q ) , f (1 , (cid:96), q )+( k − } . (cid:3) G.DRUMMOND
The next two results are extracted from [1] and [6] respectively.
Lemma 2.4. f (2 , ,
2) = f (1 , ,
2) = 4 . Lemma 2.5. f (3 , ,
2) = 5 and f (1 , ,
2) = 11 . It follows Proposition 2.3, that a simple cosimple binary (2 , -uniform ma-troid has rank at most . A more helpful bound, that will be instrumentalin our determination of the binary (2 , -uniform matroids is the following: Lemma 2.6.
Let M be a simple cosimple binary matroid. If M is (2 , -uniform, then min { r ( M ) , r ∗ ( M ) } ≤ .Proof. If M is (1 , -uniform, then M ∗ is (3 , -uniform and r ∗ ( M ) ≤ byLemma 2.5. Otherwise, M has a hyperplane H of nullity at least . Thematroid M | H is simple and (1 , -uniform. Furthermore, its dual is (2 , -uniform and has rank at least . Thus, by Lemma 2.2, M | H is cosimple. Itfollows Lemma 2.4 that r ( M | H ) ≤ and, consequently, r ( M ) ≤ . (cid:3) The remaining two sections will make frequent use of the fact that amatroid M is (2 , -uniform if and only if the union of any pair of circuits of M has rank at least r ( M ) − .3. The (2 , -uniform matroids that are not -connected In this section we describe all (2 , -uniform matroids which are not -connected and explicitly list those that are binary. The following resultscontain some redundancy but have been chosen for their clarity and to em-phasise links to paving matroids. A matroid M is sparse paving if both M and M ∗ are paving, or equivalently, if M is both (2 , and (1 , -uniform. Proposition 3.1.
Let M be a disconnected matroid. Then M is (2 , -uniform if and only if (i) M or M ∗ is paving; or (ii) M ∼ = M p ⊕ U , or M ∼ = M ∗ p ⊕ U , , where M p is a paving matroid; or (iii) M ∼ = M p ⊕ U , , where M p is a sparse paving matroid.Proof. The disconnected matroids of type (i), (ii) and (iii) are easily seen tobe (2 , -uniform. To see that there are no others, let M be a disconnected (2 , -uniform matroid. If M has a loop l , then M \ l is certainly paving and(ii) holds. Otherwise, by duality, we may assume that M has no loops orcoloops. It follows that if r ( M ) ≤ or r ∗ ( M ) ≤ , then (i) holds. Hence,we may also assume that r ( M ) , r ∗ ( M ) ≥ . Now, if every component of M has rank, corank at least two, then each component contains at least twocircuits and the union of any two such circuits has rank less than r ( M ) − , acontradiction to the (2 , -uniform property. Thus, up to duality, M has atleast one rank- component M . If | E ( M ) | ≥ , then by the (2 , -uniformproperty, r ( M ) ≤ , a contradiction. Thus, M ∼ = U , . It then follows easilyfrom the (2 , -uniform property that M \ E ( M ) is both (2 , -uniform and (1 , -uniform. In particular, (iii) is satisfied. (cid:3) GENERALISATION OF UNIFORM MATROIDS 5
We follow Oxley [5] in using P ( M , M ) to denote the parallel connectionof matroids M and M across some common basepoint. Proposition 3.2.
Let M be a connected matroid that is not -connected.Then M is (2 , -uniform if and only if (i) M or M ∗ is paving; or (ii) M or M ∗ has rank and no parallel class of size more than two; or (iii) M has a parallel or series pair { p, p (cid:48) } such that M \ p/p (cid:48) is sparsepaving; or (iv) M = P ( N, U , ) \ p , where N is a connected matroid such that N/p and N ∗ /p are paving.Proof. It is straightforward to show that all matroids of type (i)-(iv) are (2 , -uniform. To see that this list is complete, let M be a connected (2 , -uniform matroid that is not -connected. If M has rank or corank at most , then it is easily seen to satisfy (i) or (ii). Thus, we may assume that r ( M ) , r ∗ ( M ) ≥ . Suppose now that, up to duality, M has a parallel pair { p, p (cid:48) } and let N = M \ p/p (cid:48) . If there exists a circuit C of N of rank at most r ( N ) − , then as C or C ∪ p (cid:48) is a circuit of M , it follows that C ∪ { p, p (cid:48) } contains two circuits of M whose union has rank at most r ( N ) − r ( M ) − .Similarly, if there exists a pair of circuits C , C of N such that r N ( C ∪ C ) ≤ r ( N ) − , then C ∪ C ∪ { p, p (cid:48) } contains two circuits of M whose unionhas rank at most r ( M ) − . Both situations contradict the fact that M is (2 , -uniform. Hence, N is sparse paving and (iii) holds. Otherwise, M has no parallel or series pairs and we may assume that M = P ( M , M ) \ p ,for some connected matroids M , M each having at least three elementsand rank, corank at least two. If r ( M ) , r ( M ) ≥ , then by the (2 , -uniform property, each of M \ p and M \ p contains at most one circuit. Aseach M i is connected, it follows that for i ∈ { , } , M i \ p is a circuit and r ∗ ( M ) ≤ , a contradiction. Thus, without loss of generality, r ( M ) = 2 and r ( M ) = r ( M ) + 1 . If | E ( M ) | ≥ , then E ( M ) − p contains two trianglesof M , and by the (2 , -uniform property, r ( M ) ≤ , a contradiction. Thus, M ∼ = U , . Now let T = E ( M ) − p . By the (2 , -uniform property, r M ( C ∪ T ) ≥ r ( M ) − r ( M ) for every circuit C of M . It follows thatevery circuit of M containing p must have rank at least r ( M ) − and everycircuit avoiding p has rank at least r ( M ) − . Thus, M /p is paving. Alsoby the (2 , -uniform property, every pair of circuits of M \ p must span. Weconclude that M /p and M ∗ /p = ( M \ p ) ∗ are paving and that (iv) holds. (cid:3) Restricting our attention to binary matroids, we may ignore case (iv) ofProposition 3.2 as such matroids have a U , -minor. We then achieve thefollowing list by combining Propositions 3.1 and 3.2 with Acketa’s list [1]of binary paving matroids. Note that, as M ( W ) , F , F ∗ and AG (3 , havetransitive automorphism groups, any parallel connections of these matroidsand U , are free of reference to a specific basepoint. The matroid S isisomorphic to the unique non-tip deletion of the binary -spike Z . G.DRUMMOND
Corollary 3.3.
The following matroids and their duals are all the binary (2 , -uniform matroids that are not -connected. (i) The matroids of rank at most other than U , , U , , U , , U , ; (ii) the non-simple rank- binary matroids with at most one loop; (iii) the looples, non-simple rank- binary matroids with every parallel classof size at most ; (iv) M p ⊕ U , and M p ⊕ U , , for M p in { M ( W ) , F , F ∗ , AG (3 , } ; (v) P ( Z , U , ) \ t and P ( S , U , ) \ t , where t is the tip of Z ; (vi) P ( F , U , ) \ p and P ( AG (3 , , U , ) \ p ; and (vii) P ( M p , U , ) for M p in { M ( W ) , F , F ∗ , AG (3 , } . The -connected binary (2 , -uniform matroids In this section we prove Theorem 1.3, and in doing so, complete the deter-mination of the binary (2 , -uniform matroids. We also remark that two ofthe important matroids of this section, P and L , arise as graft matroids.A graft [7] is a pair ( G, γ ) where G is a graph and γ is a subset of V ( G ) thought of as the coloured vertices. The associated graft matroid is the vec-tor matroid of the matrix obtained by adjoining the incidence vector of theset γ to the vertex-edge incidence matrix of G . We follow [3] in using P to denote the simple binary extension of M ( W ) represented by the matrixof Figure 2. This is isomorphic to the graft of W in which the hub vertexand three of the four rim vertices are coloured. By considering the repre-sentation of the matroid P given in Figure 1, we see that P arises as asingle-element coextension of P . In fact, it is routine (if tedious) to verifythat P is the -sum of P and F across any of the four triangles of P other than { , , } and { , , } . Up to isomorphism, there are two othersimple binary extensions of M ( W ) , namely M ( K \ e ) and M ∗ ( K , ) . Figure 2.
A binary representation of P and P as a graft of W .In proving Theorem 1.3, we will require the following characterisation ofbinary matroids with no M ( W ) -minor due to Oxley [3, Theorem 2.1]. Here Z r is the rank- r binary spike with tip t and y is some non-tip element of Z r . Lemma 4.1.
Let M be a binary matroid. Then M is -connected and hasno M ( W ) minor if and only if (i) M ∼ = Z r , Z ∗ r , Z r \ y, or Z r \ t for some r ≥ ; or (ii) M ∼ = U , , U , , U , , U , , U , , or U , . GENERALISATION OF UNIFORM MATROIDS 7
The flats of the rank- r binary spike are very well behaved and the straight-forward proof of the following is omitted. Lemma 4.2.
For r ≥ , Z r and Z r \ y are (2 , -uniform if and only if r ≤ and Z r \ t is (2 , -uniform if and only if r ≤ . Now consider the rank- binary affine geometry AG (4 , . As its rank- flats are all isomorphic to U , , this matroid is certainly (2 , -uniform.Viewing AG (4 , as the deletion of a hyperplane H from the projectivegeometry P G (4 , , we see that every element of H is in a triangle withtwo elements of AG (4 , . It follows that any rank- binary extension of AG (4 , must have a rank- flat of nullity at least and hence fail to be (2 , -uniform. Furthermore, by Lemma 2.6, AG (4 , has no binary (2 , -uniform coextensions. Thus, AG (4 , is a maximal binary (2 , -uniformmatroid. The next lemma concerning binary affine matroids will be used inthe proof of Theorem 1.3. Lemma 4.3.
Let M be a simple rank- binary extension of M ( K , ) . Then M is (2 , -uniform if and only if M is affine.Proof. If M is a simple rank- binary affine matroid, then it is a restriction of AG (4 , and thus is (2 , -uniform. For the other direction, let M be a simplerank- binary extension of M ( K , ) that is (2 , -uniform. By uniqueness ofbinary representation, M may be represented by a binary matrix whose firstnine columns are the representation of M ( K , ) given in Figure 3. Let e label an extension column. It is easily seen that if the last entry of column e is zero, then e is in a triangle with two elements of M ( K , ) . But everypair of elements of M ( K , ) are in a circuit of size four. Thus, if column e ends in zero, then e is in a rank- flat of M of nullity at least . Thisis a contradiction to the (2 , -uniform property. We conclude that everyextension column ends in and that, consequently, M is affine. (cid:3) Figure 3.
Binary and graphic representations for M ( K , ) .Two of the four non-isomorphic simple rank- binary single-element ex-tensions of M ( K , ) are affine. These are the well-known regular matroid R and a matroid that we name L , a representation for which is given inFigure 4. In [7], R is identified as the graft matroid of K , in which every G.DRUMMOND
Figure 4.
A binary representation of L and L as a graftof K , .vertex is coloured. We remark here that L is the graft matroid of K , inwhich all but two vertices, both in the same partition, are coloured.In our final step before proving Theorem 1.3, we determine the binary (2 , -uniform coextensions of M ( K \ e ) and P ; geometric representationsof which are given in Figure 8. Lemma 4.4.
The sets of non-isomorphic binary (2 , -uniform coextensionsof M ( K \ e ) and P , respectively, are { L } and { P , L } .Proof. Let M be a binary (2 , -uniform matroid with a subset X ⊆ E ( M ) such that M/X ∼ = N for N in { M ( K \ e ) , P } . By uniqueness of binary rep-resentation, we may assume that M/X is represented by the binary matrix A given in Figure 5, where α ∈ { , } depends on N . e e e e e e e e e α Figure 5.
Matrix A . M [ A ] is isomorphic to M ( K \ e ) when α = 0 and P when α = 1 , respectively.The set H = { e , e , e , e , e , e } is a hyperplane of M [ A ] regardless of α .As M is (2 , -uniform, it follows that H ∪ X is a hyperplane of M of nullity . Moreover, M | H ∪ X is (1 , -uniform and ( M | H ∪ X ) ∗ is (2 , -uniform byduality. Thus, by Lemma 2.2, ( M | H ∪ X ) ∗ is a rank- simple matroid with | X | + 6 elements. It follows that | X | = 1 . By appropriate row operations,one then sees that M may be represented by the × binary matrix B asgiven in Figure 6. It remains to determine the coefficients β , . . . , β .A representation for ( M | H ∪ x ) ∗ is given in Figure 7. As this must besimple, we deduce that β = β = 1 and β = 1 − α . To determine β and β , we consider the hyperplane H (cid:48) = cl M ( { e , e , e } ) of M [ A ] . If α = 0 ,then the hyperplane H (cid:48) ∪ x of M contains e and by an identical argumentto before, β = β = 1 . We conclude that if N ∼ = M ( K \ e ) , then M ∼ = L . GENERALISATION OF UNIFORM MATROIDS 9 e e e e e e e e e x α
00 0 1 0 0 1 1 0 1 00 0 0 1 0 0 1 1 0 00 0 0 0 β β β β β Figure 6.
Matrix B . M [ B ] /x is isomorphic to M ( K \ e ) when α = 0 and P when α = 1 , respectively. e e e e e e x β β α β e e e x e e β β Figure 7.
Matrices representing ( M | H ∪ x ) ∗ and M | H (cid:48) ∪ x .Otherwise N ∼ = P , α = 1 and H (cid:48) = { e , e , e , e , e } . Then M | H (cid:48) ∪ x is represented by the rank- matrix of Figure 7. As this matroid must be (1 , -uniform, it follows that either β = β = 1 , in which case M ∼ = L , orprecisely one of { β , β } is zero, in which case, M ∼ = P . (cid:3) e e e e M ( K \ e ) e e e e P Figure 8.
Geometric representations of M ( K \ e ) and P .We now conclude the paper by proving Theorem 1.3. Proof of Theorem 1.3.
We first observe that a matroid is a binary -connected (2 , -uniform matroid if and only if its dual is also. In particular, both AG (4 , and AG (4 , ∗ are minor-maximal such matroids. To complete our list, let M be a minor-maximal binary -connected (2 , -uniform matroid. If r ( M ) ≤ , or r ∗ ( M ) ≤ , then M is a minor of either AG (4 , or AG (4 , ∗ ,a contradiction to maximality. Thus, r ( M ) , r ∗ ( M ) ≥ . Switching to thedual if necessary, we may then assume by Lemma 2.6 that r ( M ) = 5 .If M has no M ( W ) minor, then by Lemma 4.1, M is isomorphic to one of Z r , Z ∗ r , Z r \ t, Z r \ y for some r ≥ and, by Lemma 4.2, M ∼ = Z \ t . Otherwise,we may assume that M does possess an M ( W ) -minor. Then, as r ( M ) = 5 , M is an extension of a single-element coextension N of M ( W ) . As M ( W ) is self-dual, the matroid N ∗ is a binary (2 , -uniform single-element exten-sion of M ( W ) . These are just the simple binary extensions of M ( W ) ,namely M ( K \ e ) , P and M ∗ ( K , ) . Thus, N ∈ { M ∗ ( K \ e ) , P ∗ , M ( K , ) } .If N ∼ = M ( K , ) , then by Lemma 4.3, M must be affine and thus, by maxi-mality, M ∼ = AG (4 , . Otherwise, N ∈ { M ∗ ( K \ e ) , P ∗ } , in which case, bythe dual of Lemma 4.4, M is isomorphic to either P or L ∗ . But, as L isaffine, L ∗ is a minor of AG (4 , ∗ . We conclude by maximality that, in thiscase, M ∼ = P . The theorem then follows by duality. (cid:3) Acknowledgements
The author thanks Professors Charles Semple and James Oxley for readingthe early drafts of this paper and for their invaluable suggestions towardsimproving the exposition.
References [1] Acketa, D., On binary paving matroids,
Discrete Math. (1988), 109–110.[2] Geelen, J., Gerards, B., Whittle, G., Solving Rota’s conjecture, Notices Amer. Math.Soc. , (2014), 736–743.[3] Oxley, J.G., The binary matroids with no 4-wheel minor., Trans. Amer. Math. Soc. (1987), 63–75[4] Oxley, J.G, Ternary paving matroids,
Discrete Math. (1991), 77–86.[5] Oxley, J., Matroid Theory,
Second edition, Oxford University Press, New York, 2011.[6] Rajpal, S., On binary k -paving matroids and Reed-Muller codes, Discrete Math. (1998), 191–200.[7] Seymour, P.D., Decomposition of regular matroids,
J. Combin. Theory Ser. B (1980), 305–359. School of Mathematics and Statistics, University of Canterbury, Christchurch,New Zealand
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