Graphical representations of graphic frame matroids

A frame matroid M is graphic if there is a graph G with cycle matroid isomorphic to M. In general, if there is one such graph, there will be many. Zaslavsky has shown that frame matroids are precisely those having a representation as a biased graph; this class includes graphic matroids, bicircular matroids, and Dowling geometries. Whitney characterized which graphs have isomorphic cycle matroids, and Matthews characterised which graphs have isomorphic graphic bicircular matroids. In this paper, we give a characterization of which biased graphs give rise to isomorphic graphic frame matroids

Correlation bounds for fields and matroids

Let $G$ be a finite connected graph, and let $T$ be a spanning tree of $G$ chosen uniformly at random. The work of Kirchhoff on electrical networks can be used to show that the events $e_1 \in T$ and $e_2 \in T$ are negatively correlated for any distinct edges $e_1$ and $e_2$. What can be said for such events when the underlying matroid is not necessarily graphic? We use Hodge theory for matroids to bound the correlation between the events $e \in B$, where $B$ is a randomly chosen basis of a matroid. As an application, we prove Mason's conjecture that the number of $k$-element independent sets of a matroid forms an ultra-log-concave sequence in $k$.Comment: 16 pages. Supersedes arXiv:1804.0307

A construction of infinite sets of intertwines for pairs of matroids

An intertwine of a pair of matroids is a matroid such that it, but none of its proper minors, has minors that are isomorphic to each matroid in the pair. For pairs for which neither matroid can be obtained, up to isomorphism, from the other by taking free extensions, free coextensions, and minors, we construct a family of rank-k intertwines for each sufficiently large integer k. We also treat some properties of these intertwines.Comment: 11 page

Projective geometries in exponentially dense matroids. II

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

On excluded minors for real-representability

AbstractWe show that for any infinite field K and any K-representable matroid N there is an excluded minor for K-representability that has N as a minor

Counting matroids in minor-closed classes

A flat cover is a collection of flats identifying the non-bases of a matroid. We introduce the notion of cover complexity, the minimal size of such a flat cover, as a measure for the complexity of a matroid, and present bounds on the number of matroids on $n$ elements whose cover complexity is bounded. We apply cover complexity to show that the class of matroids without an $N$-minor is asymptotically small in case $N$ is one of the sparse paving matroids $U_{2,k}$, $U_{3,6}$, $P_6$, $Q_6$, or $R_6$, thus confirming a few special cases of a conjecture due to Mayhew, Newman, Welsh, and Whittle. On the other hand, we show a lower bound on the number of matroids without $M(K_4)$-minor which asymptoticaly matches the best known lower bound on the number of all matroids, due to Knuth.Comment: 13 pages, 3 figure

Projective geometries in exponentially dense matroids. I

We show for each positive integer $a$ that, if \cM is a minor-closed class of matroids not containing all rank-$(a+1)$ uniform matroids, then there exists an integer $n$ such that either every rank-$r$ matroid in \cM can be covered by at most $r^n$ sets of rank at most $a$, or \cM contains the \GF(q)-representable matroids for some prime power $q$, and every rank-$r$ matroid in \cM can be covered by at most $r^nq^r$ sets of rank at most $a$. This determines the maximum density of the matroids in \cM up to a polynomial factor

Amalgams of extremal matroids with no U2,ℓ+2-minor

AbstractFor an integer ℓ≥2, let U(ℓ) be the class of matroids with no U2,ℓ+2-minor. A matroid in U(ℓ) is extremal if it is simple and has no simple rank-preserving single-element extension in U(ℓ). An amalgam of two matroids is a simultaneous extension of both on the union of the two ground sets. We study amalgams of extremal matroids in U(ℓ): we determine which amalgams are in U(ℓ) and which are extremal in U(ℓ)

On a generalisation of spikes

We consider matroids with the property that every subset of the ground set of size $t$ is contained in both an $\ell$-element circuit and an $\ell$-element cocircuit; we say that such a matroid has the $(t,\ell)$-property. We show that for any positive integer $t$, there is a finite number of matroids with the $(t,\ell)$-property for $\ell<2t$; however, matroids with the $(t,2t)$-property form an infinite family. We say a matroid is a $t$-spike if there is a partition of the ground set into pairs such that the union of any $t$ pairs is a circuit and a cocircuit. Our main result is that if a sufficiently large matroid has the $(t,2t)$-property, then it is a $t$-spike. Finally, we present some properties of $t$-spikes.Comment: 18 page

Obstructions for bounded branch-depth in matroids

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 matroid of a large fan graph as a minor. We prove that matroids with sufficiently large branch-depth either contain the cycle matroid of a large fan graph as a minor or have large branch-width. As a corollary, we prove their conjecture for matroids representable over a fixed finite field and quasi-graphic matroids, where the uniform matroid is not an option.Comment: 25 pages, 1 figur