98 research outputs found
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
Templates for Representable Matroids
The matroid structure theory of Geelen, Gerards, and Whittle has led to a hypothesis that a highly connected member of a minor-closed class of matroids representable over a finite field is a mild modification (known as a perturbation) of a frame matroid, the dual of a frame matroid, or a matroid representable over a proper subfield. They introduced the notion of a template to describe these perturbations in more detail. In this dissertation, we determine these templates for various classes and use them to prove results about representability, extremal functions, and excluded minors.
Chapter 1 gives a brief introduction to matroids and matroid structure theory. Chapters 2 and 3 analyze this hypothesis of Geelen, Gerards, and Whittle and propose some refined hypotheses. In Chapter 3, we define frame templates and discuss various notions of template equivalence.
Chapter 4 gives some details on how templates relate to each other. We define a preorder on the set of frame templates over a finite field, and we determine the minimal nontrivial templates with respect to this preorder. We also study in significant depth a specific type of template that is pertinent to many applications. Chapters 5 and 6 apply the results of Chapters 3 and 4 to several subclasses of the binary matroids and the quaternary matroids---those matroids representable over the fields of two and four elements, respectively.
Two of the classes we study in Chapter 5 are the even-cycle matroids and the even-cut matroids. Each of these classes has hundreds of excluded minors. We show that, for highly connected matroids, two or three excluded minors suffice. We also show that Seymour\u27s 1-Flowing Conjecture holds for sufficiently highly connected matroids.
In Chapter 6, we completely characterize the highly connected members of the class of golden-mean matroids and several other closely related classes of quaternary matroids. This leads to a determination of the extremal functions for these classes, verifying a conjecture of Archer for matroids of sufficiently large rank
The highly connected even-cycle and even-cut matroids
The classes of even-cycle matroids, even-cycle matroids with a blocking pair,
and even-cut matroids each have hundreds of excluded minors. We show that the
number of excluded minors for these classes can be drastically reduced if we
consider in each class only the highly connected matroids of sufficient size.Comment: Version 2 is a major revision, including a correction of an error in
the statement of one of the main results and improved exposition. It is 89
pages, including a 33-page Jupyter notebook that contains SageMath code and
that is also available in the ancillary file
Displaying blocking pairs in signed graphs
A signed graph is a pair (G, S) where G is a graph and S is a subset of the edges of G. A circuit of G is even (resp. odd) if it contains an even (resp. odd) number of edges of S. A blocking pair of (G, S) is a pair of vertices s, t such that every odd circuit intersects at least one of s or t. In this paper, we characterize when the blocking pairs of a signed graph can be represented by 2-cuts in an auxiliary graph. We discuss the relevance of this result to the problem of recognizing even cycle matroids and to the problem of characterizing signed graphs with no odd-K5 minor
The Templates for Some Classes of Quaternary Matroids
Subject to hypotheses based on the matroid structure theory of Geelen,
Gerards, and Whittle, we completely characterize the highly connected members
of the class of golden-mean matroids and several other closely related classes
of quaternary matroids. This leads to a determination of the eventual extremal
functions for these classes. One of the main tools for obtaining these results
is the notion of a frame template. Consequently, we also study frame templates
in significant depth.Comment: 83 pages; minor corrections in Version 4; accepted for publication by
Journal of Combinatorial Theory, Series
On the Highly Connected Dyadic, Near-Regular, and Sixth-Root-of-Unity Matroids
Subject to announced results by Geelen, Gerards, and Whittle, we completely
characterize the highly connected members of the classes of dyadic,
near-regular, and sixth-root-of-unity matroids.Comment: 23 pages, SageMath worksheet in ancillary files. arXiv admin note:
text overlap with arXiv:1902.0713
On the Highly Connected Dyadic, Near-Regular, and Sixth-Root-of-Unity Matroids
Subject to announced results by Geelen, Gerards, and Whittle, we completely
characterize the highly connected members of the classes of dyadic,
near-regular, and sixth-root-of-unity matroids.Comment: In Version 3, improvements have been made to the proofs of Lemma 4.4
and Theorem 1.1. There are more minor corrections also. SageMath worksheet is
in ancillary files. 32 pages, accepted for publication by SIAM Journal on
Discrete Mathematic
Isomorphism for even cycle matroids - I
A seminal result by Whitney describes when two graphs have the same cycles.
We consider the analogous problem for even cycle matroids. A representation of
an even cycle matroid is a pair formed by a graph together with a special set
of edges of the graph. Such a pair is called a signed graph. We consider the
problem of determining the relation between two signed graphs representing the
same even cycle matroid. We refer to this problem as the Isomorphism Problem
for even cycle matroids. We present two classes of signed graphs and we solve
the Isomorphism Problem for these two classes. We conjecture that, up to simple
operations, any two signed graphs representing the same even cycle matroid are
either in one of these classes, or related by a modification of an operation
for graphic matroids, or belonging to a small set of examples
On The Density of Binary Matroids Without a Given Minor
This thesis is motivated by the following question: how many elements can a simple binary matroid with no \PG(t,2)-minor have? This is a natural analogue of questions asked about the density of graphs in minor-closed classes. We will answer this question by finding the eventual growth rate function of the class of matroids with no \PG(t,2)-minor, for any . Our main tool will be the matroid minors structure theory of Geelen, Gerards, and Whittle, and much of this thesis will be devoted to frame templates, the notion of structure in that theory
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