Related papers: Matroid intersection, base packing and base coveri…
We discuss several extension properties of matroids and polymatroids and their application as necessary conditions for the existence of different matroid representations, namely linear, folded linear, algebraic, and entropic…
We introduce the singular cohomology ring of a matroid which extends the Chow ring of a matroid. This is defined as the singular cohomology ring of a certain quasi-projective toric variety associated to the matroid. Using the matroidal…
Seymour's decomposition theorem is a hallmark result in matroid theory presenting a structural characterization of the class of regular matroids. Formalization of matroid theory faces many challenges, most importantly that only a limited…
Matchings and coverings are central topics in graph theory. The close relationship between these two has been key to many fundamental algorithmic and polyhedral results. For mixed graphs, the notion of matching forest was proposed as a…
Many important enumerative invariants of a matroid can be obtained from its Tutte polynomial, and many more are determined by two stronger invariants, the $\mathcal{G}$-invariant and the configuration of the matroid. We show that the same…
We describe the adjacency of vertices of the (unbounded version of the) set covering polyhedron, in a similar way to the description given by Chvatal for the stable set polytope. We find a sufficient condition for adjacency, and…
We study rank-three matroids, known as point-line configurations, and their associated matroid varieties, defined as the Zariski closures of their realization spaces. Our focus is on determining finite generating sets of defining equations…
Consider a matroid $M$ whose ground set is equipped with a labeling to an abelian group. A basis of $M$ is called $F$-avoiding if the sum of the labels of its elements is not in a forbidden label set $F$. H\"orsch, Imolay, Mizutani, Oki,…
The theory of rough sets is concerned with the lower and upper approximations of objects through a binary relation on a universe. It has been applied to machine learning, knowledge discovery and data mining. The theory of matroids is a…
We prove a common generalization of the maximal independent arborescence packing theorem of Cs. Kir\'aly and two of our earlier works about packing branchings in infinite digraphs.
In this paper, we solve a conjecture by Szigeti in [Matroid-rooted packing of arborescences, submitted], which characterizes a mixed hypergraph $\mathcal{F}=(V, \mathcal{E} \cup \mathcal{A})$ having an orientation…
The structure of the moduli space of N=1 supersymmetric gauge theories is analyzed from an algebraic geometric viewpoint. The connection between the fundamental fields of the ultraviolet theory, and the gauge invariant composite fields of…
White's conjecture predicts quadratic generators for the ideal of any matroid base polytope. We prove that White's conjecture for any matroid $M$ implies it also for any matroid $M'$, where $M$ and $M'$ differ by one basis. Our study is…
The present note is a strengthening of a recent paper by K. Takazawa and Y. Yokoi (A generalized-polymatroid approach to disjoint common independent sets in two matroids, Discrete Mathematics (2019)). For given two matroids on $E$, under…
We explore a general method based on trees of elementary submodels in order to present highly simplified proofs to numerous results in infinite combinatorics. While countable elementary submodels have been employed in such settings already,…
The Tutte polynomial for matroids is not directly applicable to polymatroids. For instance, deletion-contraction properties do not hold. We construct a polynomial for polymatroids which behaves similarly to the Tutte polynomial of a…
DeVos, Kwon, and Oum introduced the concept of branch-depth of matroids as a natural analogue of tree-depth of graphs. They conjectured that a matroid of sufficiently large branch-depth contains the uniform matroid $U_{n,2n}$ or the cycle…
Drawing parallels with hyperplane arrangements, we develop the theory of arrangements of submanifolds. Given a smooth, finite dimensional, real manifold $X$ we consider a finite collection $\mathcal{A}$ of locally flat, codimension-1…
To illustrate that the notion of convergence of submodular function sequences fits reasonably into the limit theory of graphs, we describe several classes of matroids and other submodular setfunctions for which convergence of appropriate…
We characterize the shifted simple graphs and the $3$-uniform shifted hypergraphs whose inverse image under exterior shifting is the set of bases of a matroid: those are exactly the hypergraphs whose hyperedges form an initial lex-segment.…