Brylawski [2] has shown that loopless principal transversal matroids have critical exponent at most 2. Welsh [4] asks if a similar result holds for all transversal matroids. We answer in the affirmative by proving that all loopless transversal matroids have critical exponent at most 2. The terminology used here for matroids will in general follow Welsh 131. A rank r transversal matroid M is rep...

We prove that for every proper minor-closed class M of Fp-representable matroids, there exists a O(1)-competitive algorithm for the matroid secretary problem on M. This result relies on the extremely powerful matroid minor structure theory being developed by Geelen, Gerards and Whittle. We also note that for asymptotically almost all matroids, the matroid secretary algorithm that selects a rand...

We introduce a noncommutative binary operation on matroids, called free product. We show that this operation respects matroid duality, and has the property that, given only the cardinalities, an ordered pair of matroids may be recovered, up to isomorphism, from its free product. We use these results to give a short proof of Welsh’s 1969 conjecture, which provides a progressive lower bound for t...

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...

We consider the rank reduction problem for matroids: Given a matroid M and an integer k, find a minimum size subset of elements of M whose removal reduces the rank of M by at least k. When M is a graphical matroid this problem is the minimum k-cut problem, which admits a 2-approximation algorithm. In this paper we show that the rank reduction problem for transversal matroids is essentially at l...

The classes of binary and ternary matroids are both relatively well understood as is their intersection, the class of regular matroids. This paper considers the union M of the classes of binary and ternary matroids. M is a minor-closed class and the focus of the paper is on determining its set of excluded minors. It is conjectured here that this set of excluded minors unique matroids that are o...

In this paper, some characterizations of fuzzifying strong compactness are given, including characterizations in terms of nets and pre -subbases. Several characterizations of locally strong compactness in the framework of fuzzifying topology are introduced and the mapping theorems are obtained.

We give an excluded-minor characterization for the class of matroids M in which M\e or M/e is binary for all e in E(M). This class is closely related to the class of matroids in which every member is binary or can be obtained from a binary matroid by relaxing a circuithyperplane. We also provide an excluded-minor characterization for the second class.

Coxeter matroids are combinatorial objects associated with finite Coxeter groups; they can be viewed as subsets M of the factor set W/P of a Coxeter group W by a parabolic subgroup P which satisfy a certain maximality property with respect to a family of shifted Bruhat orders on W/P . The classical matroids of matroid theory are exactly the Coxeter matroids for the symmetric group Symn (which i...

The present paper studies fuzzy matroids in view of degree. First wegeneralize the notion of $(L,M)$-fuzzy independent structure byintroducing the degree of $M$-fuzzy family of independent $L$-fuzzysets with respect to a mapping from $L^X$ to $M$. Such kind ofdegrees is proved to satisfy some axioms similar to those satisfiedby $(L,M)$-fuzzy independent structure. ...