For a matroid, its configuration determines \({\mathcal {G}}\)-invariant. Few examples are known of pairs matroids with the same {G}}\)-invariant but different configurations. In order to produce new examples, we introduce free m-cone \(Q_m(M)\) loopless matroid M, where m is positive integer. We show that M \(Q_m(M)\), and M; so if N nonisomorphic have {G}}\)-invariant, then \(Q_m(N)\) prove a...

For a k-connected graph or matroid M , where k is a fixed positive integer, we say that a subset X of E(M) is k-removable provided M\X is k-connected. In this paper, we obtain a sharp condition on the size of a 3connected binary matroid to have a 3-removable circuit.

Given a society Γ = (M, W, K), where M , W are disjoint sets and K ⊆M×W , and a matroid T on M , a T-espousal of Γ is an injective partial function E : M → W such that E ⊆ K and domE is spanning in T. The problem of characterizing Tespousable societies is a generalization of characterizing systems of matroids with disjoint bases. Wojciechowski [15] formulated a criterion for a society Γ = (M, W...

Let E be a possibly infinite set and let M N matroids defined on E. We say that the pair {M,N} has Intersection property if share an independent I admitting bipartition IM⊔IN such spanM(IM)∪spanN(IN)=E. The Matroid Conjecture of Nash-Williams says every matroid property. conjecture is known easy to prove in case when one uniform it was shown by Bowler Carmesin implied its special where direct s...

Further work of Brylawski and Heron (see [4, p. 315]) explores other characterizations of Eulerian binary matroids. They showed, independently, that a binary matroid M is Eulerian if and only if its dual, M∗, is a binary affine matroid. More recently, Shikare and Raghunathan [5] have shown that a binary matroid M is Eulerian if and only if the number of independent sets of M is odd. This chapte...

Let F 7 denote the Fano matroid and M be a simple connected binary matroid such that every cocircuit of M has size at least d 3. We show that if M does not have an F 7-minor, M 6 = F 7 , and d = 2 f5; 6; 7; 8g, then M has a circuit of size at least minfr(M) + 1; 2dg. We conjecture that the latter result holds for all d 3.

Let M be a matroid and F the collection of all linear orderings of bases ofM. orflags of M. We define the flag matroid polytope A(F)We determine when two vertices of A(F) are adjacent, and provide a bijection between maximal chains in the lattice of flats of M and certain maximal faces of A(F).

Tropical varieties play an important role in algebraic geometry. The Bergman complex B(M) and the positive Bergman complex B+(M) of an oriented matroid M generalize to matroids the notions of the tropical variety and positive tropical variety associated to a linear ideal. Our main result is that if A is a Coxeter arrangement of type Φ with corresponding oriented matroid MΦ, then B (MΦ) is dual ...

Given an undirected graph G = (V,E) and a delta-matroid (V,F), the delta-matroid matching problem is to find a maximum cardinality matching M such that the set of the end vertices of M belongs to F . This problem is a natural generalization of the matroid matching problem to delta-matroids, and thus it cannot be solved in polynomial time in general. This paper introduces a class of the delta-ma...

In this paper, we obtain a forbidden-minor characterization of the class of binary gammoids M such that, for every pair of elements x, y of M, both the splitting matroid Mx,y and the element splitting matroid M ′ x,y are binary gammoids.