Related papers: On the Number of Circuit-cocircuit Reversal Classe…
Las Vergnas and Hamidoune studied the number of circuits needed to determine an oriented matroid. In this paper we investigate this problem and some new variants, as well as their interpretation in particular classes of matroids. We present…
In this paper we generalise the even directed cycle problem, which asks whether a given digraph contains a directed cycle of even length, to orientations of regular matroids. We define non-even oriented matroids generalising non-even…
We introduce the minor-closed, dual-closed class of multi-path matroids. We give a polynomial-time algorithm for computing the Tutte polynomial of a multi-path matroid, we describe their basis activities, and we prove some basic structural…
Oriented matroids (often called order types) are combinatorial structures that generalize point configurations, vector configurations, hyperplane arrangements, polyhedra, linear programs, and directed graphs. Oriented matroids have played a…
One characterization of binary matroids is that the symmetric difference of every pair of intersecting circuits is a disjoint union of circuits. This paper considers circuit-difference matroids, that is, those matroids in which the…
This paper studies the properties of two kinds of matroids: (a) algebraic matroids and (b) finite and infinite matroids whose ground set have some canonical symmetry, for example row and column symmetry and transposition symmetry. For (a)…
Komj\'ath, Milner, and Polat investigated when a finitary matroid admits a partition into circuits. They defined the class of ``finite matching extendable'' matroids and showed in their compactness theorem that those matroids always admit…
We use the equivariant cohomology ring of the permutohedral variety to study matroids and their invariants. Investigating the pushforward of matroid Chern classes defined by A. Berget, C. Eur, H. Spink and D. Tseng to the product space…
Following the work of Gao and Xie in [2], we state some properties of the inverse Kazhdan-Lusztig polynomial of a matroid. We also give partial answers to a conjecture that states that regular connected matroids are non-degenerate. We link…
The theory of matroids has been generalized to oriented matroids and, recently, to arithmetic matroids. We want to give a definition of "oriented arithmetic matroid" and prove some properties like the "uniqueness of orientation".
A family C of circuits of a matroid M is a linear class if, given a modular pair of circuits in C}, any circuit contained in the union of the pair is also in C. The pair (M,C) can be seen as a matroidal generalization of a biased graph. We…
Every regular matroid is associated with a sandpile group, which acts simply transitively on the set of bases in various ways. Ganguly and the second author introduced the notion of consistency to describe classes of actions that respect…
The first non-trivial case of Hadwiger's conjecture for oriented matroids reads as follows. If $\mathcal{O}$ is an $M(K_4)$-free oriented matroid, then $\mathcal{O}$ admits a NZ $3$-coflow, i.e., it is $3$-colourable in the sense of…
We introduce the notion of a quasi-matroidal class of ordered simplicial complexes: an approximation to the idea of a matroid cryptomorphism in the landscape of ordered simplicial complexes. A quasi-matroidal class contains pure shifted…
The Tutte polynomial is a fundamental invariant of graphs and matroids. In this article, we define a generalization of the Tutte polynomial to oriented graphs and regular oriented matroids. To any regular oriented matroid $N$, we associate…
Matroids generalize the familiar notion of linear dependence from linear algebra. Following a brief discussion of founding work in computability and matroids, we use the techniques of reverse mathematics to determine the logical strength of…
In his seminal 1983 paper, Jim Lawrence introduced lopsided sets and featured them as asymmetric counterparts of oriented matroids, both sharing the key property of strong elimination. Moreover, symmetry of faces holds in both structures as…
We provide a short proof of a conic version of the colorful Carath\'eodory theorem for oriented matroids. Holmsen's extension of the colorful Carath\'eodory theorem to oriented matroids (Advances in Mathematics, 2016) already encompasses…
Building on a recent characterization of tope graphs of Complexes of Oriented Matroids (COMs), we tackle and generalize several classical problems in Oriented Matroids (OMs), Lopsided Sets (aka ample set systems), and partial cubes via…
For all positive integers $t$ exceeding one, a matroid has the cyclic $(t-1,t)$-property if its ground set has a cyclic ordering $\sigma$ such that every set of $t-1$ consecutive elements in $\sigma$ is contained in a $t$-element circuit…