27 research outputs found
On minor-closed classes of matroids with exponential growth rate
Let \cM be a minor-closed class of matroids that does not contain
arbitrarily long lines. The growth rate function, h:\bN\rightarrow \bN of
\cM is given by h(n) = \max(|M|\, : \, M\in \cM, simple, rank-$n$). The
Growth Rate Theorem shows that there is an integer such that either:
, or , or there is a
prime-power such that ; this
separates classes into those of linear density, quadratic density, and base-
exponential density. For classes of base- exponential density that contain
no -point line, we prove that for all
sufficiently large . We also prove that, for classes of base- exponential
density that contain no -point line, there exists k\in\bN such
that for all
sufficiently large
The densest matroids in minor-closed classes with exponential growth rate
The for a nonempty minor-closed class of
matroids is the function whose value at an
integer is defined to be the maximum number of elements in a simple
matroid in of rank at most . Geelen, Kabell, Kung and Whittle
showed that, whenever is finite, the function
grows linearly, quadratically or exponentially in (with
base equal to a prime power ), up to a constant factor.
We prove that in the exponential case, there are nonnegative integers and
such that for all sufficiently large , and we characterise
which matroids attain the growth rate function for large . We also show that
if is specified in a certain `natural' way (by intersections of
classes of matroids representable over different finite fields and/or by
excluding a finite set of minors), then the constants and , as well as
the point that `sufficiently large' begins to apply to , can be determined
by a finite computation
Exponentially Dense Matroids
This thesis deals with questions relating to the maximum density of rank-n matroids in a minor-closed class.
Consider a minor-closed class M of matroids that does not contain a given rank-2 uniform matroid. The growth rate function is defined by h_M(n) = max(|N| : N ∈ M simple, r(N) ≤ n).
The Growth Rate Theorem, due to Geelen, Kabell, Kung, and Whittle, shows that the growth rate function is either linear, quadratic, or exponential in n. In the case of exponentially dense classes, we conjecture that, for sufficiently large n,
h_M(n) = (q^(n+k) − 1)/(q-1) − c, where q is a prime power, and k and c are non-negative integers depending only on M. We show that this holds for several interesting classes, including the class of all matroids with no U_{2,t}-minor.
We also consider more general minor-closed classes that exclude an arbitrary uniform matroid. Here the growth rate, as defined above, can be infinite. We define a more suitable notion of density, and prove a growth rate theorem for this more general notion, dividing minor-closed classes into those that are at most polynomially dense, and those that are exponentially dense
On the density of matroids omitting a complete-graphic minor
We show that, if is a simple rank- matroid with no -point line
minor and no minor isomorphic to the cycle matroid of a -vertex complete
graph, then the ratio is bounded above by a singly exponential
function of and . We also bound this ratio in the special case where
is a frame matroid, obtaining an answer that is within a factor of two of
best-possible.Comment: 25 page
On perturbations of highly connected dyadic matroids
Geelen, Gerards, and Whittle [3] announced the following result: let be a prime power, and let be a proper minor-closed class of
-representable matroids, which does not contain
for sufficiently high . There exist integers
such that every vertically -connected matroid in is a
rank- perturbation of a frame matroid or the dual of a frame matroid
over . They further announced a characterization of the
perturbations through the introduction of subfield templates and frame
templates.
We show a family of dyadic matroids that form a counterexample to this
result. We offer several weaker conjectures to replace the ones in [3], discuss
consequences for some published papers, and discuss the impact of these new
conjectures on the structure of frame templates.Comment: Version 3 has a new title and a few other minor corrections; 38
pages, including a 6-page Jupyter notebook that contains SageMath code and
that is also available in the ancillary file
The algebraic combinatorial approach for low-rank matrix completion
We propose an algebraic combinatorial framework for the problem of completing
partially observed low-rank matrices. We show that the intrinsic properties of
the problem, including which entries can be reconstructed, and the degrees of freedom
in the reconstruction, do not depend on the values of the observed entries, but
only on their position. We associate combinatorial and algebraic objects, differentials
and matroids, which are descriptors of the particular reconstruction task, to the
set of observed entries, and apply them to obtain reconstruction bounds. We show
how similar techniques can be used to obtain reconstruction bounds on general compressed
sensing problems with algebraic compression constraints. Using the new
theory, we develop several algorithms for low-rank matrix completion, which allow
to determine which set of entries can be potentially reconstructed and which not,
and how, and we present algorithms which apply algebraic combinatorial methods
in order to reconstruct the missing entries
Non-crossing frameworks with non-crossing reciprocals
We study non-crossing frameworks in the plane for which the classical
reciprocal on the dual graph is also non-crossing. We give a complete
description of the self-stresses on non-crossing frameworks whose reciprocals
are non-crossing, in terms of: the types of faces (only pseudo-triangles and
pseudo-quadrangles are allowed); the sign patterns in the self-stress; and a
geometric condition on the stress vectors at some of the vertices.
As in other recent papers where the interplay of non-crossingness and
rigidity of straight-line plane graphs is studied, pseudo-triangulations show
up as objects of special interest. For example, it is known that all planar
Laman circuits can be embedded as a pseudo-triangulation with one non-pointed
vertex. We show that if such an embedding is sufficiently generic, then the
reciprocal is non-crossing and again a pseudo-triangulation embedding of a
planar Laman circuit. For a singular (i.e., non-generic) pseudo-triangulation
embedding of a planar Laman circuit, the reciprocal is still non-crossing and a
pseudo-triangulation, but its underlying graph may not be a Laman circuit.
Moreover, all the pseudo-triangulations which admit a non-crossing reciprocal
arise as the reciprocals of such, possibly singular, stresses on
pseudo-triangulation embeddings of Laman circuits.
All self-stresses on a planar graph correspond to liftings to piece-wise
linear surfaces in 3-space. We prove characteristic geometric properties of the
lifts of such non-crossing reciprocal pairs.Comment: 32 pages, 23 figure