From the configuration of a matroid (which records the size and rank of the cyclic flats and the containments among them, but not the sets), one can compute several much-studied matroid invariants, including the Tutte polynomial and a newer, stronger invariant, the $\mathcal{G}$-invariant. To gauge how much additional information the configuration contains compared to these invariants, it is of interest to have methods for constructing matroids with different configurations but the same $\mathcal{G}$-invariant. We offer several such constructions along with tools for developing more.
We show for each positive integer $a$ that, if $\mathcal{M}$ is a minor-closed class of matroids not containing all rank-$(a+1)$ uniform matroids, then there exists an integer $c$ such that either every rank-$r$ matroid in $\mathcal{M}$ can be covered by at most $r^c$ rank-$a$ sets, or $\mathcal{M}$ contains the GF$(q)$-representable matroids for some prime power $q$ and every rank-$r$ matroid in $\mathcal{M}$ can be covered by at most $cq^r$ rank-$a$ sets. In the latter case, this determines the maximum density of matroids in $\mathcal{M}$ up to a constant factor.
For a set of matroids $\mathcal{M}$, let $ex_\mathcal{M}(n)$ be the maximum size of a simple rank-$n$ matroid in $\mathcal{M}$. We prove that, for any finite field $\mathbb{F}$, if $\mathcal{M}$ is a minor-closed class of $\mathbb{F}$-representable matroids of bounded branch-width, then $\lim_{n \rightarrow \infty} ex_\mathcal{M}(n) / n$ exists and is a rational number, $Δ$. We also show that $ex_\mathcal{M}(n) - Δn$ is periodic when $n$ is sufficiently large and that $ex_\mathcal{M}$ is achieved by a subclass of $\mathcal{M}$ of bounded path-width.
The Las Vergnas' strong map conjecture, states that any strong map of oriented matroids $f:\mathcal{M}_1\rightarrow\mathcal{M}_2$ can be factored into extensions and contractions. The conjecture is known to be false due to a construction by Richter-Gebert, he find a non-factorizable strong map $f:\mathcal{M}_1\rightarrow\mathcal{M}_2$, however in his example $\mathcal{M}_1$ is not realizable. The problem that whether there exists a non-factorizable strong map between realizable oriented matroids still remains open. In this paper we provide a counterexample to the strong map conjecture on reali…
For a simplicial complex ${\mathcal C}$ denote by $β({\mathcal C})$ the minimal number of edges from ${\mathcal C}$ needed to cover the ground set. If ${\mathcal C}$ is a matroid then for every partition $A_1, \ldots, A_m$ of the ground set there exists a set $S \in {\mathcal C}$ meeting each $A_i$ in at least $\frac{|A_i|}{β({\mathcal C})}$ elements. We conjecture that a slightly weaker result is true for the intersections of two matroids: if ${\mathcal D}={\mathcal P} \cap {\mathcal Q}$, where ${\mathcal P},{\mathcal Q}$ are matroids on the same ground set $V$ and $β({\mathcal P}), β({\mathc…
Given an $\Bbb{F}$-represented matroid $(M,ρ)$ with the ground set $[m]$, the representation $ρ$ naturally defines a hyperplane arrangement $\mathcal{A}_ρ$. We will study its parallel translates $\mathcal{A}_{ρ,{g}}$ of $\mathcal{A}_ρ$ for all ${ g}\in \mathbb{F}^m$. Its intersection semi-lattices $L(\mathcal{A}_{ρ,{ g}})$ and the characteristic polynomials $χ(\mathcal{A}_{ρ,{ g}},t)$ will be classified by the intersection lattice of the derived arrangement $\mathcal{A}_{δρ}$, which is a hyperplane arrangement associated with the derived matroid $(δM,δρ)$ and also known as the discriminantal a…
We present a strengthening of the countable Menger theorem (edge version) of R. Aharoni. Let $ D=(V,A) $ be a countable digraph with $ s\neq t\in V $ and let $\mathcal{M}=\bigoplus_{v\in V}\mathcal{M}_v $ be a matroid on $ A $ where $ \mathcal{M}_v $ is a finitary matroid on the ingoing edges of $ v $. We show that there is a system of edge-disjoint $ s \rightarrow t $ paths $ \mathcal{P} $ such that the united edge set of the paths is $ \mathcal{M} $-independent, and there is a $ C \subseteq A $ consists of one edge from each element of $ \mathcal{P} $ for which $ \mathsf{span}_{\mathcal{M}}(…
Let $\mathcal{N}$ be a set of matroids. A matroid $M$ is strictly $\mathcal{N}$-fragile if $M$ has a member of $\mathcal{N}$ as minor and, for all $e \in E(M)$, at least one of $M\backslash e$ and $M/e$ has no minor in $\mathcal{N}$. In this paper we give a structural description of the strictly $\{U_{2,5},U_{3,5}\}$-fragile matroids that have six inequivalent representations over $\mathrm{GF}(5)$. Roughly speaking, these matroids fall into two classes. The matroids without an $\{X_8, Y_8, Y_8^{*}\}$-minor are constructed, up to duality, from one of two matroids by gluing wheels onto specified…
An 'induced restriction' of a simple binary matroid $M$ is a restriction $M|F$, where $F$ is a flat of $M$. We consider the class $\mathcal{M}$ of all simple binary matroids $M$ containing neither a free matroid on three elements (which we call a 'claw'), nor a Fano plane as an induced restriction. We give an exact structure theorem for this class; two of its consequences are that the matroids in $\mathcal{M}$ have unbounded critical number, while the matroids in $\mathcal{M}$ not containing the clique $M(K_5)$ as an induced restriction have critical number at most $2$.
We give a cohomological interpretation of the Heaviside filtration on the Varchenko--Gelfand ring of a pair $(\mathcal{A},\mathcal{K})$, where $\mathcal{A}$ is a real hyperplane arrangement and $\mathcal{K}$ is a convex open subset of the ambient vector space. This builds on work of the first author, who studied the filtration from a purely algebraic perspective, as well as work of Moseley, who gave a cohomological interpretation in the special case where $\mathcal{K}$ is the ambient vector space. We also define the Gelfand--Rybnikov ring of a conditional oriented matroid, which simultaneously…
The $\mathit{growth\ rate\ function}$ for a nonempty minor-closed class of matroids $\mathcal{M}$ is the function $h_{\mathcal{M}}(n)$ whose value at an integer $n \ge 0$ is defined to be the maximum number of elements in a simple matroid in $\mathcal{M}$ of rank at most $n$. Geelen, Kabell, Kung and Whittle showed that, whenever $h_{\mathcal{M}}(2)$ is finite, the function $h_{\mathcal{M}}$ grows linearly, quadratically or exponentially in $n$ (with base equal to a prime power $q$), up to a constant factor. We prove that in the exponential case, there are nonnegative integers $k$ and $d \le \…
Orlik-Solomon algebras determine the Tutte polynomials of matroids of rank at most three
Xiamen University（厦门大学）；Qinghai Minzu University（青海民族大学）；Qinghai Institute of Applied Mathematics（青海省应用数学研究所）；Guangdong Polytechnic Normal University（广东技术师范大学）
We investigate the set of excluded minors of connectivity 2 for the class of frame matroids. We exhibit a list $\mathcal{E}$ of 18 such matroids, and show that if $N$ is such an excluded minor, then either $N \in \mathcal{E}$ or $N$ is a 2-sum of $U_{2,4}$ and a 3-connected non-binary frame matroid.
The notion of $\mathcal{H}$-matroids was introduced by U. Faigle and S. Fujishige in 2009 as a general model for matroids and the greedy algorithm. They gave a characterization of $\mathcal{H}$-matroids by the greedy algorithm. In this note, we give a characterization of some $\mathcal{H}$-matroids by rank functions.
Rigidity and reconstruction in matroids of highly connected graphs
HUN-REN-ELTE Egerváry Research Group on Combinatorial Optimization and the MTA-ELTE Momentum Matroid Optimization Research Group（匈牙利研究与教育网络-厄特沃什·罗兰大学埃格瓦里组合优化研究组和匈牙利科学院-厄特沃什·罗兰大学动量拟阵优化研究组）；HUN-REN Alfréd Rényi Institute of Mathematics（匈牙利研究与教育网络阿尔弗雷德·雷尼数学研究所）
Let $M$ be an arbitrary matroid with circuits $\mathcal{C}(M)$. We propose a definition of a derived matroid $δM$ that has as its ground set $\mathcal{C}(M)$. Unlike previous attempts of such a definition, our definition applies to arbitrary matroids, and is completely combinatorial. We prove that the rank of $δM$ is bounded from above by $|M|-r(M)$, that it is connected if and only if $M$ is connected. We compute examples including the derived matroids of uniform matroids, the Vámos matroid and the graphical matroid $M(K_4)$. We formulate conjectures relating our construction to previous defi…
In this paper we investigate a family of matroids introduced by Ardila and Billey to study one-dimensional intersections of complete flag arrangements of $\mathbb{C}^n$. The set of lattice points $P_n$ inside the equilateral triangle $S_n$ obtained by intersecting the nonnegative cone of $\mathbb{R}^3$ with the affine hyperplane $x_1 + x_2 + x_3 = n-1$ is the ground set of a matroid $\mathcal{T}_n$ whose independent sets are the subsets $S$ of $P_n$ satisfying that $|S \cap P| \le k$ for each translation $P$ of the set $P_k$. Here we study the structure of the matroids $\mathcal{T}_n$ in conne…
Given a map $\mathcal M$ on a connected and closed orientable surface, the delta-matroid of $\mathcal M$ is a combinatorial object associated to $\mathcal M$ which captures some topological information of the embedding. We explore how delta-matroids associated to dessins d'enfants behave under the action of the absolute Galois group. Twists of delta-matroids are considered as well; they correspond to the recently introduced operation of partial duality of maps. Furthermore, we prove that every map has a partial dual defined over its field of moduli. A relationship between dessins, partial dual…
For a map $\mathcal M$ cellularly embedded on a connected and closed orientable surface, the bases of its Lagrangian (also known as delta-) matroid $Δ(\mathcal M)$ correspond to the bases of a Lagrangian subspace $L$ of the standard orthogonal space $\mathbb{Q}^E\oplus\mathbb{Q}^{E^*}$, where $E$ and $E^*$ are the edge-sets of $\mathcal M$ and its dual map. The Lagrangian subspace $L$ is said to be a representation of both $\mathcal M$ and $Δ(\mathcal M)$. Furthermore, the bases of $Δ(\mathcal M)$, when understood as vertices of the hypercube $[-1,1]^n$, induce a polytope $\mathbf P(Δ(\mathcal…
In this paper, first steps are taken towards characterising lattices of cyclic flats $\mathcal{Z}(M)$ that belong to matroids $M$ that can be represented over a prescribed finite field $\mathbb{F}_q$. Two natural maps from $\mathcal{Z}(M)$ to the lattice of cyclic flats of a minor of $M$ are given. Binary matroids are characterised via their lattice of cyclic flats. It is shown that the lattice of cyclic flats of a simple binary matroid without isthmuses is atomic.
Let $M$ be a 3-connected matroid and let $\mathbb F$ be a field. Let $A$ be a matrix over $\mathbb F$ representing $M$ and let $(G,\mathcal B)$ be a biased graph representing $M$. We characterize the relationship between $A$ and $(G,\mathcal B)$, settling four conjectures of Zaslavsky. We show that for each matrix representation $A$ and each biased graph representation $(G,\mathcal{B})$ of $M$, $A$ is projectively equivalent to a canonical matrix representation arising from $G$ as a gain graph over $\mathbb F^+$ or $\mathbb F^\times$ realizing $\mathcal{B}$. Further, we show that the projectiv…
In this work we present an algorithm to construct sparse-paving matroids over finite set $S$. From this algorithm we derive some useful bounds on the cardinality of the set of circuits of any Sparse-Paving matroids which allow us to prove in a simple way an asymptotic relation between the class of Sparse-paving matroids and the whole class of matroids. Additionally we introduce a matrix based method which render an explicit partition of the $r$-subsets of $S$, $\binom{S}{r}=\sqcup_{i=1}^{γ}\mathcal{U}_{i}$ such that each $\mathcal{U}_{i}$ defines a sparse-paving matroid of rank $r$.
In this paper, we investigate three problems concerning the toric ideal associated to a matroid. Firstly, we list all matroids $\mathcal M$ such that its corresponding toric ideal $I_{\mathcal M}$ is a complete intersection. Secondly, we handle with the problem of detecting minors of a matroid $\mathcal M$ from a minimal set of binomial generators of $I_{\mathcal M}$. In particular, given a minimal set of binomial generators of $I_{\mathcal M}$ we provide a necessary condition for $\mathcal M$ to have a minor isomorphic to $\mathcal U_{d,2d}$ for $d \geq 2$. This condition is proved to be suff…
In 1977 Stanley proved that the $h$-vector of a matroid is an $\mathcal{O}$-sequence and conjectured that it is a pure $\mathcal{O}$-sequence. In the subsequent years the validity of this conjecture has been shown for a variety of classes of matroids, though the general case is still open. In this paper we use Las Vergnas' internal order to introduce a new class of matroids which we call internally perfect. We prove that these matroids satisfy Stanley's Conjecture and compare them to other classes of matroids for which the conjecture is known to hold. We also prove that, up to a certain restri…
Let $\mathcal M$ be a class of matroids closed under minors and isomorphism. Let $N$ be a matroid in $\mathcal M$ with an exact $k$-separation $(A, B)$. We say $N$ is a $k$-decomposer for $\mathcal M$ having $(A, B)$ as an inducer, if every matroid $M\in \mathcal M$ having $N$ as a minor has a $k$-separation $(X, Y)$ such that, $A\subseteq X$ and $B\subseteq Y$. Seymour [3, 9.1] proved that a matroid $N$ is a $k$-decomposer for an excluded-minor class, if certain conditions are met for all 3-connected matroids $M$ in the class, where $|E(M)-E(N)|\le 2$. We reinterpret Seymour's Theorem in term…
The growth-rate function for a minor-closed class $\mathcal{M}$ of matroids is the function $h$ where, for each non-negative integer $r$, $h(r)$ is the maximum number of elements of a simple matroid in $\mathcal{M}$ with rank at most $r$. The Growth-Rate Theorem of Geelen, Kabell, Kung, and Whittle shows, essentially, that the growth-rate function is always either linear, quadratic, exponential, or infinite. Morover, if the growth-rate function is quadratic, then $h(r)\ge \binom{r+1}{2}$, with the lower bound coming from the fact that such classes necessarily contain all graphic matroids. We c…
Leclerc and Zelevinsky, motivated by the study of quasi-commuting quantum flag minors, introduced the notions of strongly separated and weakly separated collections. These notions are closely related to the theory of cluster algebras, to the combinatorics of the double Bruhat cells, and to the totally positive Grassmannian. A key feature, called the purity phenomenon, is that every maximal by inclusion strongly (resp., weakly) separated collection of subsets in $[n]$ has the same cardinality. In this paper, we extend these notions and define $\mathcal{M}$-separated collections for any oriented…
The Folkman-Lawrence topological representation theorem, which states that every (loop-free) oriented matroid of rank $r$ can be represented as a pseudosphere arrangement on the $(r-1)$-dimensional sphere $S^{r-1}$, is one of the most outstanding results in oriented matroid theory. In this paper, we provide a lower-dimensional version of the topological representation theorem for uniform matroid polytopes of rank $4$. We introduce $2$-weak configurations of points and pseudocircles ($2$-weak PPC configurations) on $S^2$ and prove that every uniform matroid polytope of rank $4$ can be represent…
A matroid $\mathcal{M}$ on a set $E$ of elements has the $α$-partition property, for some $α>0$, if it is possible to (randomly) construct a partition matroid $\mathcal{P}$ on (a subset of) elements of $\mathcal{M}$ such that every independent set of $\mathcal{P}$ is independent in $\mathcal{M}$ and for any weight function $w:E\to\mathbb{R}_{\geq 0}$, the expected value of the optimum of the matroid secretary problem on $\mathcal{P}$ is at least an $α$-fraction of the optimum on $\mathcal{M}$. We show that the complete binary matroid, ${\cal B}_d$ on $\mathbb{F}_2^d$ does not satisfy the $α$-p…
We associate to a sufficiently generic oriented matroid program and choice of linear system of parameters a finite dimensional algebra, whose representation theory is analogous to blocks of Bernstein--Gelfand--Gelfand category $\mathcal O$. When the data above comes from a generic linear program for a hyperplane arrangement, we recover the algebra defined by Braden--Licata--Proudfoot--Webster. Applying our construction to nonlinear oriented matroid programs provides a large new class of algebras. For Euclidean oriented matroid programs, the resulting algebras are quasi-hereditary and Koszul, a…
A matroid $N$ is said to be triangle-rounded in a class of matroids $\mathcal{M}$ if each $3$-connected matroid $M\in \mathcal{M}$ with a triangle $T$ and an $N$-minor has an $N$-minor with $T$ as triangle. Reid gave a result useful to identify such matroids as stated next: suppose that $M$ is a binary $3$-connected matroid with a $3$-connected minor $N$, $T$ is a triangle of $M$ and $e\in T\cap E(N)$; then $M$ has a $3$-connected minor $M'$ with an $N$-minor such that $T$ is a triangle of $M'$ and $|E(M')|\le |E(N)|+2$. We strengthen this result by dropping the condition that such element $e$…
Specializing the $γ$-basis for the vector space $\mathcal{G}(n,r)$ spanned by the set of symbols on bit sequences with $r$ $1$'s and $n-r$ $0$'s, we obtain a frame or spanning set for the vector space $\mathcal{T}(n,r)$ spanned by Tutte polynomials of matroids having rank $r$ and size $n$. Every Tutte polynomial can be expanded as a linear combination with non-negative integer coefficients of elements in this frame. We give explicit formulas for the elements in this frame. These formulas combine to give an expansion of the Tutte polynomial with coefficients obtained by summing numerical invari…
The growth-rate function for a minor-closed class $\mathcal{M}$ of matroids is the function $h$ where, for each non-negative integer $r$, $h(r)$ is the maximum number of elements of a simple matroid in $\mathcal{M}$ with rank at most $r$. The Growth-Rate Theorem of Geelen, Kabell, Kung, and Whittle shows, essentially, that the growth-rate function is always either linear, quadratic, exponential with some prime power $q$ as the base, or infinite. Morover, if the growth-rate function is exponential with base $q$, then the class contains all GF$(q)$-representable matroids, and so $h(r)\ge \frac{q…
The first problem we investigate is the following: given $k\in \mathbb{R}_{\ge 0}$ and a vector $v$ of Plücker coordinates of a point in the real Grassmannian, is the vector obtained by taking the $k$th power of each entry of $v$ again a vector of Plücker coordinates? For $k\neq 1$, this is true if and only if the corresponding matroid is regular. Similar results hold over other fields. We also describe the subvariety of the Grassmannian that consists of all the points that define a regular matroid. The second topic is a related problem for arithmetic matroids. Let $\mathcal{A} = (E, rk, m)$ b…
Geelen, Gerards, and Whittle [3] announced the following result: let $q = p^k$ be a prime power, and let $\mathcal{M}$ be a proper minor-closed class of $\mathrm{GF}(q)$-representable matroids, which does not contain $\mathrm{PG}(r-1,p)$ for sufficiently high $r$. There exist integers $k, t$ such that every vertically $k$-connected matroid in $\mathcal{M}$ is a rank-$(\leq t)$ perturbation of a frame matroid or the dual of a frame matroid over $\mathrm{GF}(q)$. They further announced a characterization of the perturbations through the introduction of subfield templates and frame templates. We …
Sufficient conditions for bipartite rigidity, symmetric completability and hyperconnectivity of graphs
HUN-REN-ELTE Egerváry Research Group on Combinatorial Optimization（匈牙利研究与创新网络-罗兰大学埃格瓦里组合优化研究组）；Queen Mary University of London（伦敦玛丽女王大学）；Department of Operations Research, ELTE Eötvös Loránd University（罗兰大学运营系）
Enumeration of all combinatorial types of point configurations and polytopes is a fundamental problem in combinatorial geometry. Although many studies have been done, most of them are for 2-dimensional and non-degenerate cases. Finschi and Fukuda (2001) published the first database of oriented matroids including degenerate (i.e. non-uniform) ones and of higher ranks. In this paper, we investigate algorithmic ways to classify them in terms of realizability, although the underlying decision problem of realizability checking is NP-hard. As an application, we determine all possible combinatorial t…
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 key role in combinatorics, computational geometry, and optimization. This paper surveys prior work and presents an update on the search for bounds on the diameter of the cocircuit graph of an oriented matroid. We review the diameter problem and show the diameter bounds of general oriented matroids reduce to those of uniform oriented matroids. We give the la…
To a vector configuration one can associate a polynomial ideal generated by powers of linear forms, known as a power ideal, which exhibits many combinatorial features of the matroid underlying the configuration. In this note we observe that certain power ideals associated to transversal matroids are, somewhat unexpectedly, monomial. Moreover, the (monomial) basis elements of the quotient ring defined by such a power ideal can be naturally identified with the lattice points of a remarkable convex polytope: a polymatroid, also known as generalized permutohedron. We dub the exponent vectors of th…
In this work, we study matroids over a domain and several classical combinatorial and algebraic invariants related. We define their Grothendieck-Tutte polynomial $T_{\mathcal{M}}(x,y)$, extending the definition given by Fink and Moci in 2016, and we show that such polynomial has the classical deletion-contraction property. Moreover, we study the set of independencies for a realizable matroid over a domain, generalizing the definition of \emph{poset of torsions} $Gr(\mathcal{M})$ given by the second author in 2017. This is a union of identical simplicial posets as for (quasi-)arithmetic matroid…
Consider a linear realization of a matroid over a field. One associates with it a configuration polynomial and a symmetric bilinear form with linear homogeneous coefficients. The corresponding configuration hypersurface and its non-smooth locus support the respective first and second degeneracy scheme of the bilinear form. We show that these schemes are reduced and describe the effect of matroid connectivity: for (2-)connected matroids, the configuration hypersurface is integral, and the second degeneracy scheme is reduced Cohen-Macaulay of codimension 3. If the matroid is 3-connected, then al…
