189 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
Matroids arising from electrical networks
This paper introduces Dirichlet matroids, a generalization of graphic
matroids arising from electrical networks. We present four main results. First,
we exhibit a matroid quotient formed by the dual of a network embedded in a
surface with boundary and the dual of the associated Dirichlet matroid. This
generalizes an analogous result for graphic matroids of cellularly embedded
graphs. Second, we characterize the Bergman fans of Dirichlet matroids as
explicit subfans of graphic Bergman fans. In doing so, we generalize the
connection between Bergman fans of complete graphs and phylogenetic trees.
Third, we use the half-plane property of Dirichlet matroids to prove an
interlacing result on the real zeros and poles of the trace of the response
matrix. And fourth, we bound the coefficients of the precoloring polynomial of
a network by the coefficients of the chromatic polynomial of the underlying
graph.Comment: 27 pages, 14 figure
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 Selected Subclasses of Matroids
Matroids were introduced by Whitney to provide an abstract notion of independence.
In this work, after giving a brief survey of matroid theory, we describe structural results for various classes of matroids. A connected matroid is unbreakable if, for each of its flats , the matroid is connected%or, equivalently, if has no two skew circuits. . Pfeil showed that a simple graphic matroid is unbreakable exactly when is either a cycle or a complete graph. We extend this result to describe which graphs are the underlying graphs of unbreakable frame matroids. A laminar family is a collection \A of subsets of a set such that, for any two intersecting sets, one is contained in the other. For a capacity function on \A, let \I be %the set \{I:|I\cap A| \leq c(A)\text{ for all A\in\A}\}. Then \I is the collection of independent sets of a (laminar) matroid on . We characterize the class of laminar matroids by their excluded minors and present a way to construct all laminar matroids using basic operations. %Nested matroids were introduced by Crapo in 1965 and have appeared frequently in the literature since then. A flat of a matroid is Hamiltonian if it has a spanning circuit. A matroid is nested if its Hamiltonian flats form a chain under inclusion; is laminar if, for every -element independent set , the Hamiltonian flats of containing form a chain under inclusion. We generalize these notions to define the classes of -closure-laminar and -laminar matroids. The second class is always minor-closed, and the first is if and only if . We give excluded-minor characterizations of the classes of 2-laminar and 2-closure-laminar matroids
Cobiased graphs: Single-element extensions and elementary quotients of graphic matroids
Zaslavsky (1991) introduced a graphical structure called a biased graph and
used it to characterize all single-element coextensions and elementary lifts of
graphic matroids. We introduce a new, dual graphical structure that we call a
cobiased graph and use it to characterize single-element extensions and
elementary quotients of graphic matroids.Comment: 17 pp., 5 figure
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