Related papers: Fair representation in the intersection of two mat…
We call a family of sets intersecting, if any two sets in the family intersect. In this paper we investigate intersecting families $\mathcal{F}$ of $k$-element subsets of $[n]:=\{1,\ldots, n\},$ such that every element of $[n]$ lies in the…
The basis exchange axiom has been a driving force in the development of matroid theory. However, the axiom gives only a local characterization of the relation of bases, which is a major stumbling block to further progress, and providing a…
Matroids, particularly linear ones, have been a powerful tool in parameterized complexity for algorithms and kernelization. They have sped up or replaced dynamic programming. Delta-matroids generalize matroids by encapsulating structures…
Finding a maximum cardinality common independent set in two matroids (also known as \textsc{Matroid Intersection}) is a classical combinatorial optimization problem, which generalizes several well-known problems, such as finding a maximum…
A matroid is a machine capturing linearity of mathematical objects and producing combinatorial structures. Matroid structure arises everywhere since linearity is a ubiquitous concept. One natural way to obtain matroids is by considering…
Seymour's Decomposition Theorem for regular matroids states that any matroid representable over both GF(2) and GF(3) can be obtained from matroids that are graphic, cographic, or isomorphic to R10 by 1-, 2-, and 3-sums. It is hoped that…
This paper develops matroidal analogues of classical results on matchings in abelian groups. By embedding matroid ground sets in an abelian group, we introduce base matchings between matroid bases, recover the group-theoretic setting in the…
Given a matroid together with a coloring of its ground set, a subset of its elements is called rainbow colored if no two of its elements have the same color. We show that if a binary matroid of rank $r$ is colored with exactly $r$ colors,…
An open problem in convex geometry asks whether two simplices $A,B\subseteq\mathbb{R}^d$, both containing the origin in their convex hulls, admit a polynomial-length sequence of vertex exchanges transforming $A$ into $B$ while maintaining…
Given a simple Eulerian binary matroid $M$, what is the minimum number of disjoint circuits necessary to decompose $M$? We prove that $|M| / (\operatorname{rank}(M) + 1)$ many circuits suffice if $M = \mathbb F_2^n \setminus \{0\}$ is the…
Let $K$ be an imaginary quadratic field and let $\mathcal{O}_K$ be the ring of algebraic integers of $K$. For $\alpha \in \mathcal{O}_K$ with $|\alpha| > 1$, define \[ \mathcal{D}_\alpha = \bigcup_{n=0}^\infty…
The homological information about a filtered simplicial complex over the poset of positive real numbers is often presented by a barcode which depicts the evolution of the associated Betti numbers. However, there is a wonderfully complex…
Rota's basis conjecture, open since 1989, states that if B_1, B_2, ..., B_n are n bases of a vector space of rank n, then there is an nxn grid of vectors such that the vectors in the ith row are precisely the elements of B_i and such that…
The cycles of a graph give a natural cyclic ordering to their edge-sets, and these orderings are consistent in that two edges are adjacent in one cycle if and only if they are adjacent in every cycle in which they appear together. An…
Let $M$ be a matroid defined on a finite set $E$ and $L\subset E$. $L$ is locked in $M$ if $M|L$ and $M^*|(E\backslash L)$ are 2-connected, and $min\{r(L), r^*(E\backslash L)\} \geq 2$. Locked subsets characterize nontrivial facets of the…
Given a graph $G$ and two spanning trees $T$ and $T'$ in $G$, Spanning Tree Reconfiguration asks whether there is a step-by-step transformation from $T$ to $T'$ such that all intermediates are also spanning trees of $G$, by exchanging an…
One of the major open questions in matroid theory asks whether the $h$-vector $(h_0,h_1,\ldots,h_s)$ of the broken circuit complex of a matroid $M$ satisfies the following inequalities: $$ h_0\leq h_1\leq \cdots\leq h_{\lfloor s/2\rfloor}…
Halin proved that every minimally $k$-connected graph has a vertex of degree $k$. More generally, does every minimally vertically $k$-connected matroid have a $k$-element cocircuit? Results of Murty and Wong give an affirmative answer when…
We partition in classes the set of matroids of fixed dimension on a fixed vertex set. In each class we identify two special matroids, respectively with minimal and maximal h-vector in that class. Such extremal matroids also satisfy a…
We study the algebraic matroid induced by the ideal of (r+1)-minors of a matrix of variables over a field. This is inherently connected to the bounded-rank matrix completion problem, in which the aim is to complete a partially observed rank…