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0 votes
1 answer
91 views

Finite projective geometry and the Krasner hyperfield

The Krasner hyperfield is an algebraic structure of two operations on $K=\{0,1\}$ called $+\colon K\times K\to \mathcal{P}(K)$ and $\cdot\colon K\times K\to K$ with $0+0=0$ $0+1=1+0=1$ $1+1=\{0,1\}$ ...
1 vote
0 answers
111 views

$r(M)$-subsets of a 3-connected matroid $M$

It is proved in Lowrance, Oxley, Semple, and Welsh - On properties of almost all matroids that almost all matroids are 3-connected asymptotically. Also, it is conjectured that almost all matroids are ...
1 vote
1 answer
227 views

Do the cycles containing a fixed edge generate the cycle space 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 of $G$ containing the edge $e$. For what set of edges does $\mathcal{C_e}$ contain a basis of the ...
8 votes
1 answer
593 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 d(...
5 votes
1 answer
410 views

Exchanges between independent sets of a matroid

Let $I, J$ be two bases of a matroid. For every $x$ in $I$, there is some $y$ in $J$ such that, if we exchange $x$ with $y$, then both resulting sets ($I \setminus x \cup y$ and $J \setminus y \cup x$)...
2 votes
1 answer
251 views

Dimension of circuit space of a matroid

If $G$ is a graph with edge set $E$, let $W$ be the $\mathbb{Z}/2$-vector space generated by the elements of $E$. If $A = \{a_1, \dots, a_n\} \subset E$, let $\bar{A} = a_1 + \dots + a_n \in V$; then $...
12 votes
3 answers
2k 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 $x \in X$,...
8 votes
2 answers
824 views

Matroids of rank two

I am interested in matroids of rank two and would like to understand how interesting/big this class of matroids is. I know that the 2-uniform matroid on (k+2) elements is not representable over any ...
1 vote
1 answer
221 views

How to find all minimal dependent sets of a set of vectors effectively?

In my research, I need to find the set of all minimal dependent sets of a given set of vectors. One method is to check every subset of the given set. But this method is very slow when the set of ...
4 votes
0 answers
94 views

Totally Unimodular matrix edited from ordinary matrix

Given a matrix $M\in\{0,1\}^{m\times n}$ is there an algorithm to tell if we can convert some of $1$s to $-1$s and make $M$ Totally Unimodular and output such a Totally Unimodular in polynomial in $mn$...