Theorem For Matroids Research Articles

A Theory of Congruences and Birkhoff’s Theorem for Matroids

A congruence is defined for a matroid. This leads to suitable versions of the algebraic isomorphism theorems for matroids. As an application of the congruence theory for matroids, a version of Birkhoff’s Theorem for matroids is given which shows that every nontrivial matroid is a subdirect product of subdirectly irreducible matroids.

Improving a chain theorem for triangle-free 3-connected matroids

In 2007, Kriesell established a chain theorem for triangle-free 3-connected graphs. Any triangle-free 3-connected graph can be reduced to a double-wheel or to K3,3 by performing a sequence of simple operations without leaving the class of triangle-free 3-connected graphs. Double-wheels define the only infinite family of graphs that are irreducible with respect to these simple operations. In 2013, Lemos extended Kriesell’s theorem for matroids. In this case, there are four infinite families of irreducible matroids. In this paper, we improve these results by proving that one of Kriesell’s reduction operations can be avoided provided the number of families of irreducible matroids is increased by four.

On reid’s 3-simplicial matroid theorem

In this paper we prove the following result of Ralph Reid (which was never published nor completely proved).

Asymptotic properties of random subsets of projective spaces

A random graph on n vertices is a random subgraph of the complete graph on n vertices. By analogy with this, the present paper studies the asymptotic properties of a random submatroid ωr of the projective geometry PG(r−l, q). The main result concerns Kr, the rank of the largest projective geometry occurring as a submatroid of ωr. We show that with probability one, for sufficiently large r, Kr takes one of at most two values depending on r. This theorem is analogous to a result of Bollobás and Erdös on the clique number of a random graph. However, whereas from the matroid theorem one can essentially determine the critical exponent of ωr, the graph theorem gives only a lower bound on the chromatic number of a random graph.

Kuratowski's and Wagner's theorems for matroids

In an earlier paper we proved the following theorem, which provides a strengthening of Tutte's well-known characterization of regular (totally unimodular) matroids: A binary matroid is regular if it does not have the Fano matroid or its dual as a series-minor (parallel-minor). In this paper we prove two theorems (Theorems 5.1 and 6.1) which provide the same kind of strengthening for Tutte's characterization of the graphic matroids (i.e., bond-matroids). One interesting aspect of these theorems is the introduction of the matroids of “type R”. It turns out that these matroids are, in at least two different senses, the smallest regular matroids which are neither graphic nor cographic (Theorems 6.2 and 6.3).

Rado's theorem for polymatroids

The theorem of R. Rado (12) to which I refer by the name ‘Rado's theorem for matroids’ gives necessary and sufficient conditions for a family of subsets of a finite setYto have a transversal independent in a given matroid onY. This theorem is of fundamental importance in both transversal theory and matroid theory (see, for example, (11)). In (3) J. Edmonds introduced and studied ‘polymatroids’ as a sort of continuous analogue of a matroid. I start this paper with a brief introduction to polymatroids, emphasizing the role of the ‘ground-set rank function’. The main result is an analogue for polymatroids of Rado's theorem for matroids, which I call not unnaturally ‘Rado's theorem for polymatroids’.

A maximum-rank minimum-term-rank theorem for matroids

M. Iri has proved that the maximum rank for a pivotal system of matrices (i.e., combivalence class) equals the minimum term rank. Here this and some of Iri's related results are generalized to matroids. These generalizations are presented using a representation of matroids with (0,1)-matrices. Then, with the aid of matroid basis graphs, these generalizations are restated graph-theoretically. Finally, related results about certain uniform basis graphs are derived.

Menger's theorem for matroids

Menger's theorem for matroids

A Homotopy Theorem for Matroids, II

A Homotopy Theorem for Matroids, II