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4
votes
1answer
116 views

Number of bases of a matroid

I would like to know the minimum number of bases of a matroid of rank $k$ and $n$ elements, knowing that each singleton is independent. At least for small ranks.
0
votes
1answer
48 views

Do the support sets of subspaces give the representable matroids?

Fact:   Start with $V$ a subspace of $\mathbb R^n$. Take the set of all supports of vectors in $V$. Throw out $\emptyset$. You now have the dependent sets of some matroid. Not sure you ...
0
votes
0answers
20 views

Non-uniform matroids as the matroid sum of uniform matroids

Can all non-uniform matroids be written as the direct sum / matroid sum of uniform matroids? If so, What happens to the matrices representing the uniform matroids? If the non-uniform matroid is ...
3
votes
0answers
125 views

A Result of Anders Bjorner: Matchings in countably infinite geometric lattices of finite height

Let $L$ be a countably infinite geometric lattice of finite height $r\ge3$. (A geometric lattice of height $r$ is an atomistic semimodular lattice such that every maximal chain has $r+1$ elements.) ...
3
votes
1answer
83 views

Is the union of strongly base-orderable matroids strongly base-orderable?

A matroid is said to be strongly base-orderable if for any two bases $B_1,B_2$ there is a bijection $f:B_1 \to B_2$ such that for any $S\subseteq B_1$ set $(B_1 \setminus S) \cup f(S)$ is also a base. ...
1
vote
0answers
124 views

Determining strong base-orderability of a matroid

A matroid is said to be strongly base-orderable when for any two bases $B_1,B_2$ there exists a bijection $f:B_1 \mapsto B_2$ such that for any $X\subseteq B_1$ set $B_1 - X+ f(X)$ is also a base. ...
3
votes
0answers
57 views

Non-representable irreducible matroid of rank at least 5?

Can anyone tell me an example of a matroid of rank 5 or higher which is not a product of two lower rank matroids and is not the independence matroid of a finite set of vectors in a vector space over ...
4
votes
2answers
160 views

Matroids relaxations of a given matroid

Let $\mathcal{M}$ be a rank-$d$ matroid on $[n]$. Say a matroid $\mathcal{N}$ is a relaxation of $\mathcal{M}$ if $\mathrm{rank}(\mathcal{N})=d$, $\mathrm{groundset}(\mathcal{N})=[n]$, and every ...
5
votes
2answers
208 views

Decomposing polyhedral cones into “direct sums” and a polynomial

This question consists of two parts. I'm not breaking it up into two separate ones because posing the second question would essentially require me two rewrite the first one. Also, to some extent, the ...
5
votes
1answer
235 views

Small remarkable matroids

Working on phased matroids (a generalization of oriented matroid to the complex case) I've found an interesting formula for computing the inner Tutte group (and, hence, all the Tutte groups) ...
5
votes
1answer
228 views

Vanishing patterns of minors of matrix

Let $M$ be a $m\times n$ matrix with entries in, say $\mathbb{C}$; assume $n\leq m$. Denote by $I\subseteq\{1,2,\ldots, n\}$ a subset of the columns of M. I am interested in positive results to the ...
5
votes
2answers
66 views

The lattice of covectors of an oriented matroid

Let $M$ be an oriented matroid on the ground set $E$, and let $L(M)$ be its ranked poset of covectors. By definition, $L(M)$ is a sub-poset of the poset $\{0, \pm 1\}^E$, ordered by putting $0 < ...
5
votes
1answer
308 views

Does this matroid have a name?

Sorry if this question is a dumb one. I used a special family of matroids in my research. One of them, of rank 3, can be represented by the following matrix over $\mathbb F_5$ or over $\mathbb R$: ...
1
vote
1answer
208 views

Positroids and Totally Nonnegative Complex Grassmanian

Recently I begin working on matroids, in particular to a generalization of oriented matroids to the complex case. I found on arxiv the following interesting articles: 1)Alexander Postnikov: Total ...
0
votes
0answers
74 views

Characterizing bases of 0-1-vectors in $\mathbb{R}^n$ in terms of their partial order

Given $n \in \mathbb{N}$ and $V \subseteq \{0,1\}^n$, can properties of $V$ with respect to the vector space $\mathbb{R}^n$ (not $\mathbb{Z}_2^n$), $V$ is linearly independent $\dim \mathrm{span} V ...
2
votes
1answer
170 views

Looking for a canonical (matroid polytope) subdivision of the hypersimplex

A matroid polytope is the convex hull of the indicator vectors of the bases of a matroid, and a matroid polytope subdivision (MPS) is a polyhedral subdivision of a matroid polytope whose cells are ...
2
votes
2answers
148 views

Which graphs generate a matroidal independence complex?

The independence complex $I(G)$ of a graph $G=(V,E)$ has as point set the vertex set $V$ and as simplices the independent sets of $G$. Now, if $G$ is a well-covered graph (where all maximal ...
4
votes
2answers
207 views

matroids axioms and independence system

A finite matroid $M$ is a pair $(E,I)$ where $E$ is a finite set and $I$ is a family of independent set with the following properties: 1) There is at least an independent system 2) Every subset of ...
0
votes
1answer
86 views

Basis of Cycle Subspace of a Graph

Let $G$ be a $2$-connected graph and for $e \in E(G)$ denote by $\mathcal{C_e}$ the set of all cycles(circuits) of $G$ containing the edge $e$. For what set of edges does $\mathcal{C_e}$ contain a ...
3
votes
1answer
331 views

Fundamental Cycles of a graphs

For a $2$-edge-connected simple graph $G$ and a tree $T$ of $G$, let $C_e$ be the unique cycle in $T + e$, $e \in E(G) - E(T)$. Define the set $\mathcal{C}(T) = \{C_e | e \in E(G) - E(T)\}$. Now ...
3
votes
2answers
235 views

Does this graph have a name?

Let $G$ be a connected graph on $n$ vertices and $\mathcal{T}$ be the set of all spanning trees of $G$. Consider the graph whose vertices are the elements of $\mathcal{T}$ and $T, T' \in ...
2
votes
0answers
111 views

Hamiltonian Matroids

Similar to graphs, a Matroid $M$ is said to be Hamiltonian if there is a base $B$ of $M$ and $e \in M-B$ such that $B + e$ is a cycle of $M$. Is there any literature on this? EDIT: Actually my ...
5
votes
1answer
219 views

Checking if a matroid is binary(Detecting $U^2_4$ minor in a matroid)

I am wondering what is the (computationally) best way to tell if a matroid of size $n$ and rank $r$ is binary(or whether it has a $U^2_4$ minor) given either one of these: 1) An independence oracle 2) ...
3
votes
1answer
472 views

Is there a graph-theoretical proof of Tutte's theorem on matroids?

First of all, I'm not a mathematician and I hope this question isn't too elementary, but I got no answers on math.SE, and since this is a reference request on a relatively advanced theorem, I thought ...
2
votes
0answers
165 views

Schemes defined by a collection of Plücker coordinates

If $C \subset {[n]\choose k}$ is any collection of $k$-element sets, we can define a scheme $$ W(C) = \bigcap_{S\notin C} \{V \in Gr(k,n) : p_S(V)=0\} \qquad \subseteq Gr(k,n), $$ where $p_S$ is the ...
3
votes
1answer
167 views

What is the name of this measure of matrix “degenerateness”

Given a spanning set, consider the minimum number of vectors that you must remove in order to make it no longer span. What is this number called? If the vectors are columns in a matrix $\Phi$, then ...
4
votes
0answers
305 views

Checking whether an element is in all inclusion-wise maximal common independent sets of two matroids

Given two matroids $M$ and $M'$ over the same universe $E$, and some element $x \in E$, I am interested in the importance of $x$ for the intersection (the common independent sets) of $M$ and $M'$. It ...
2
votes
1answer
130 views

Realizability of extensions of a free oriented matroid by an independent set

Question: I am searching for a non-realizable matroid with few dependencies relative to the number of points. Precisely, I would like to find a non-realizable (over $\mathbb{R}$) oriented matroid $M$ ...
2
votes
0answers
314 views

Incremental minimum spanning tree

Given a connected graph $G=(V,E)$ with a weight function $w:E\to\mathbb{R}$ and a subset $E_0\subseteq E$ such that the subgraph $(V,E_0)$ is connected, I am looking for a sequence $E_0\subseteq ...
8
votes
1answer
686 views

Is there a Sudoku matroid?

This question is inspired from this one, where it is asked what is the minimum number of checks needed to verify that a Sudoku solution is correct. Let $$ E=\{r_1, \dots, r_9\} \cup \{c_1, \dots, ...
3
votes
2answers
216 views

are there pairs of combinatorial graphs that are both isospectral and have the same matroid?

Two graphs are isospectral if the combinatorial Laplacian on them has the same spectrum, equivalently, the adjacency matrix has the same the set of eigenvalues (including multiplicities). Two graphs ...
3
votes
0answers
160 views

A non-matroidal notion of dependence on a set of ideals

Assume we are given a set of ideals $I_1, \dots, I_s$ in a commutative polynomial ring. Let's define a subset indexed by $A\subseteq [s] = \{ 1,2,\dots, s\}$ as dependent if there exists an $a\in A$ ...
4
votes
1answer
230 views

Matroid representable over $\mathbb{R}$ but not over $\mathbb{Q}$?

Does there exist a matroid that is representable over $\mathbb{R}$ but not over $\mathbb{Q}$? In particular, can one give a positive answer using a nonrational polytope, i.e., a combinatorial ...
6
votes
2answers
338 views

Finding the matroids with a specified set of non-bases

I'm a grad student in algebraic geometry, and I've encountered a problem which requires me to produce an algorithm involving matroids. Since this isn't my area of expertise, I'm hoping someone knows ...
14
votes
1answer
481 views

Smooth bases of matroids

Motivated by algebraic geometry, I've come up with a purely combinatorial definition within the theory of matroids. The question is: is this concept known? If you like matroids but not algebraic ...
3
votes
1answer
329 views

from affine matroid to measures

Let $S$ be an arbitrary finite spanning subset of $\mathbb{R}^d$ of cardinality $N$. Let $W(S)$ be the formal $\mathbb{R}$-vector space generated by all $d$-dimensional simplices (i.e. bases of the ...
1
vote
1answer
177 views

When do the invariant factors of a direct sum of matrices correspond to those of its summands?

(Tried asking this on math stackexchange, but no takers so far.) I'm trying to prove something about matroids, which I have reduced to the following question: Suppose I have a matrix $M$ which is a ...
4
votes
0answers
194 views

Oriented matroids and posets?

Is there a characterization of oriented matroids in terms of order theory, similar to that of matroids as geometric lattices? Does this question make sense at all? I have seen (for instance in ...
6
votes
0answers
313 views

Higher K-theory of Orlik-Solomon algebras (and possible generalizations?)

This topic of this question is a bit outside my comfort zone, and I should say that my end goal is to really understand how much "graph theory" is captured by contraction-deletion relations. It seems ...
0
votes
1answer
213 views

Is a non-disjoint union of connected matroids always connected?

This is perhaps an easy question, but... Let $M$ be a matroid on a ground set $E$, and let $A$ and $B$ be non-disjoint subsets of $E$ such that $M|A$ and $M|B$ are both connected. Is $M|(A\cup B)$ ...
8
votes
5answers
3k views

Good introductory text book on Matroid Theory?

I am looking for a good text book on Matroid theory. Ideally, one that might be better suited to engineers than pure mathematicians...but any book that is well written/organized would do. I have ...
30
votes
8answers
3k views

What are the external triumphs of matroid theory?

As a relatively new abstraction, matroids clearly enjoy a rich theory unto themselves and also offer a viewpoint that suggests interesting analogies and clarifies aspects of the foundations of ...
11
votes
3answers
770 views

Representability of matroids over $\mathbb R$

Let $M$ be a matroid, for example viewed as being given by a finite set $X$ and a rank function $d : P(X) \to {\mathbb N}$ such that 1) $d(\varnothing)=0$, $d(\lbrace x \rbrace)=1$, for all $x \in ...
11
votes
1answer
803 views

Menger theorem via matroids

Let $G=(V,E)$ be oriented graph, $Y\subset V$ be some fixed set of its vertices. Call $A\subset V$ independent if there exist $|A|$ vertex-disjoint pathes strating in $A$ and ending in $Y$. It is not ...
8
votes
2answers
684 views

Covering a random graph with spanning trees.

Let $G=(V,E)$ be a connected graph, say $V=\{1,\ldots,n\}$. Let $F=(V,E')$ be a uniformly random forest in $G$. (In other words, $E'$ is a subset of edges $E$ not containing a cycle, and it is ...
3
votes
1answer
462 views

union of matroid intersection

The following is classical theorem of Ore and Ryser, generalising famous Hall marriage theorem. Assume that $n$ guys and $m$ girls live in a town, some guys like some girls. Three statements are ...
6
votes
1answer
342 views

Representability of polymatroids over $GF(2)$

A polymatroid is a finite set $X$ and a rank function $d : P(X) \to {\mathbb N}$ such that 1) $d(\varnothing)=0$, 2) $A \subset B$ implies $d(A) \leq d(B)$, and 3) $d(A \cap B) + d(A \cup B) \leq ...
2
votes
1answer
229 views

Upper bound on number of lines in a linear space given degree bounds

Let $(S,\mathcal{L})$ be a linear space and $q$ be a prime power such that Every point in $S$ lies on at most $q+1$ lines, and Every line in $\mathcal{L}$ contains at most $q+1$ points, and at ...
8
votes
2answers
517 views

Realization space of matroids

Let $M$ be a matroid admitting a coordinatization over a complex vector space. If we know that the complex coordinatization space for $M$ is connected, then may we conclude that the matroid admits a ...
12
votes
2answers
1k views

Has anyone implemented a recognition algorithm for totally unimodular matrices?

One of the consequences of Seymour's characterization of regular matroids is the existence of a polynomial time recognition algorithm for totally unimodular matrices (i.e. matrices for which every ...