Let M and N be internally 4-connected binary matroids such that M has a proper N -minor, and |E(N)| ≥ 7. As part of our project to develop a splitter theorem for internally 4-connected binary matroids, we prove the following result: if M\e has no N -minor whenever e is in a triangle of M , and M/e has no N -minor whenever e is in a triad of M , then M has a minor, M ′, such that M ′ is internal...

The aim of this paper is to give insight into the behaviour of inequivalent representations of 3-connected matroids. An element x of a matroid M is fixed if there is no extension MŒ of M by an element xŒ such that {x, xŒ} is independent and MŒ is unaltered by swapping the labels on x and xŒ. When x is fixed, a representation of M0x extends in at most one way to a representation of M. A 3-connec...

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...

We consider 5the problem of covering a set of vectors of a given finite dimensional linear space (vector space) by a subspace generated by a set of vectors of minimum size. Specifically, we study the Space Cover problem, where we are given a matrix M and a subset of its columns T ; the task is to find a minimum set F of columns of M disjoint with T such that that the linear span of F contains a...

matroids are important combinatorial structures and connect close-lywith graphs. matroids and graphs were all generalized to fuzzysetting respectively. this paper tries to study  connections betweenfuzzy matroids and fuzzy graphs. for a given fuzzy graph, we firstinduce a sequence of matroids  from a sequence of crisp graph, i.e.,cuts of the fuzzy graph. a fuzzy matroid, named graph fuzzy matro...

This paper studies systems of polynomial equations that provide information about orientability of matroids. First, we study systems of linear equations over F2, originally alluded to by Bland and Jensen in their seminal paper on weak orientability. The Bland-Jensen linear equations for a matroid M have a solution if and only if M is weakly orientable. We use the Bland-Jensen system to determin...

In this paper we construct from a cographic matroid M , a pure multicomplex whose degree sequence is the h–vector of the the matroid complex of M. This result proves a conjecture of Richard Stanley [Sta96] in the particular case of cographic matroids. We also prove that the multicomplexes constructed are M–shellable, so proving a conjecture of Manoj Chari [Cha97] again in the case of cographic ...

Ingleton [8, p. 123] raised the question of characterizing the class of transversal matroids that are representable over some particular field F. He noted that when F = GF(2), this problem had already been solved by de Sousa and Welsh [6] who showed that the class of binary transversal matroids coincides with the class of graphic transversal matroids. The latter class had earlier been character...

A symmetric matroid is a matroid deened on the edge-set of some countably innnite complete graph K in a way that ranks of nite sub-graphs of K are invariant under isomorphism. Thus a symmetric ma-troid M induces on any nite graph G a uniquely determined matroid M(G). We study connectivity properties of circuits and generalized trees of symmetric matroids. We give several characterizations of a ...