Let ( ) M  denote the maximum number of disjoint bases in a matroid M . For a connected graph G , let ( ) = ( ( )) G M G   , where ( ) M G is the cycle matroid of G . The well-known spanning tree packing theorem of Nash-Williams and Tutte characterizes graphs G with ( ) G k   . Edmonds generalizes this theorem to matroids. In [1] and [2], for a matroid M with ( ) M k   , elements ( ) e E...

Abstract We conjecture that the class of frame matroids can be characterized by a sentence in monadic second-order logic matroids, and we prove there is such characterization for bicircular matroids. The proof does not depend on an excluded-minor characterization.

Erd os and Gallai showed that for any simple graph with n vertices and circumference c it holds that |E(G)| ≤ 2 (n−1)c.We extend this theorem to simple binary matroids having no F7-minor by showing that for such a matroid M with circumference c(M)≥ 3 it holds that |E(M)| ≤ 1 2 r(M)c(M). Mathematics Subject Classi cation (2000). Primary 05D15; Secondary 05B35.

This paper studies connectedness of Goetschel–Voxman fuzzy matroids (briefly, G–V fuzzy matroids), an analog of connectedness of crisp finite matroids. Based on the results of fuzzy circuits given by Goetschel and Voxman, the transitivity theorem concerning fuzzy circuits of G–V fuzzy matroids is established, and thus the useful notion of refined G–V fuzzy matroid is introduced. The connectedne...

New base exchange properties of binary and graphic matroids are derived. The graphic matroids within the class of 4-connected binary matroids are characterized by base exchange properties. Some progress with the characterization of arbitrary graphic matroids is made. Characterizing various types of matroids by base exchange properties is e.g. important in invariant theory.

Let M and N be ternary matroids having the same rank and the same ground set, and assume that every independent set in N is also independent in M . The main result of this paper proves that if M is 3-connected and N is connected and non-binary, then M = N . A related result characterizes precisely when a matroid that is obtained by relaxing a circuit-hyperplane of a ternary matroid is also tern...

In this note we introduce a sufficient condition for the OrlikSolomon algebra associated to a matroid M to be l-adic and we prove that this condition is necessary when M is binary (in particular graphic). Moreover, this result cannot be extended to the class of all matroids.

It has recently been shown that infinite matroids can be axiomatized in a way that is very similar to finite matroids and permits duality. This was previously thought impossible, since finitary infinite matroids must have non-finitary duals. In this paper we illustrate the new theory by exhibiting its implications for the cycle and bond matroids of infinite graphs. We also describe their algebr...

Mysteriously, hypergraphs that are the intersection of two matroids behave in some respects almost as well as one matroid. In the present paper we study one such phenomenon the surprising ability of the intersection of two matroids to fairly represent the parts of a given partition of the ground set. For a simplicial complex C denote by β(C) the minimal number of edges from C needed to cover th...