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...

The expansion axiom of matroids requires only the existence of some kind of independent sets, not the uniqueness of them. This causes that the base families of some matroids can be reduced while the unions of the base families of these matroids remain unchanged. In this paper, we define unique expansion matroids in which the expansion axiom has some extent uniqueness; we define union minimal ma...

The independent assignment problem (or the weighted matroid intersection problem) is extended using Dress-Wenzel’s matroid valuations, which are attached to the vertex set of the underlying bipartite graph as an additional weighting. Specifically, the problem considered is: Given a bipartite graph G = (V , V −;A) with arc weight w : A → R and matroid valuations ω and ω− on V + and V − respectiv...

Matroids were introduced by Whitney in 1935 to try to capture abstractly the essence of dependence. Matroids generalize linear dependence over vector spaces, and they also abstract the properties of graphs, in the former case they are called Vector Matroids, in the latter they are called Graphic Matroids [3]. The operation of Matroid Union was introduced by Nash-Williams in 1966. A matroid is i...

Recently, the relationship between matroids and generalized rough sets based on relations has been studied from the viewpoint of linear independence of matrices. In this paper, we reveal more relationships by the predecessor and successor neighborhoods from relations. First, through these two neighborhoods, we propose a pair of matroids, namely predecessor relation matroid and successor relatio...

This paper defines the q-analogue of a matroid and establishes several properties like duality, restriction and contraction. We discuss possible ways to define a q-matroid, and why they are (not) crypto-morphic. Also, we explain the motivation for studying q-matroids by showing that a rank metric code gives a q-matroid. This paper establishes the definition and several basic properties of q-mat...

The bases-exchange graph of a matroid is the graph whose vertices are the bases of the matroid, and two bases are connected by an edge if and only if one can be obtained from the other by the exchange of a single pair of elements. In this paper we prove that a matroid is \connected" if and only if the \restricted bases-exchange graph" (the bases-exchange graph restricted to exchanges involving ...

Motivated by the question of when the characteristic polynomial of a matroid factorizes, we study join-factorizations of broken circuit complexes and rooted complexes (a more general class of complexes). Such factorizations of complexes induce factorizations not only of characteristic polynomial but also of the Orlik-Solomon algebra of the matroid. The broken circuit complex of a matroid factor...

A new isomorphism invariant of matroids is introduced, in the form of a quasisymmetric function. This invariant • defines a Hopf morphism from the Hopf algebra of matroids to the quasisymmetric functions, which is surjective if one uses rational coefficients, • is a multivariate generating function for integer weight vectors that give minimum total weight to a unique base of the matroid, • is e...

A matroid is a notion of independence in combinatorial optimization which is closely related to computational efficiency. In particular, it is well known that the maximum of a constrained modular function can be found greedily if and only if the constraints are associated with a matroid. In this paper, we bring together the ideas of bandits and matroids, and propose a new class of combinatorial...