3,962 research outputs found
A matroid extension result
Adding elements to matroids can be fraught with difficulty. In the V\'amos
matroid , there are four independent sets and such
that is a -separation while exactly three of
the local connectivities , ,
, and are one, with the fourth being
zero. As is well known, there is no extension of by a non-loop element
such that is a circuit for all . This paper proves that a
matroid can be extended by a fixed element in the guts of a -separation
provided no V\'amos-like structure is present
Quaternary matroids are vf-safe
Binary delta-matroids are closed under vertex flips, which consist of the
natural operations of twist and loop complementation. In this note we provide
an extension of this result from GF(2) to GF(4). As a consequence, quaternary
matroids are "safe" under vertex flips (vf-safe for short). As an application,
we find that the matroid of a bicycle space of a quaternary matroid is
independent of the chosen representation. This extends a result of Vertigan [J.
Comb. Theory B (1998)] concerning the bicycle dimension of quaternary matroids.Comment: 8 pages, no figures, the contents of this paper is now merged into v2
of [arXiv:1210.7718] (except for this comment, v2 is identical to v1
Cyclotomic and simplicial matroids
Two naturally occurring matroids representable over Q are shown to be dual:
the {\it cyclotomic matroid} represented by the roots of unity
inside the cyclotomic extension ,
and a direct sum of copies of a certain simplicial matroid, considered
originally by Bolker in the context of transportation polytopes. A result of
Adin leads to an upper bound for the number of -bases for among
the roots of unity, which is tight if and only if has at most two
odd prime factors. In addition, we study the Tutte polynomial of in the
case that has two prime factors.Comment: 9 pages, 1 figur
Capturing elements in matroid minors
In this dissertation, we begin with an introduction to a matroid as the natural generalization of independence arising in three different fields of mathematics. In the first chapter, we develop graph theory and matroid theory terminology necessary to the topic of this dissertation. In Chapter 2 and Chapter 3, we prove two main results. A result of Ding, Oporowski, Oxley, and Vertigan reveals that a large 3-connected matroid M has unavoidable structure. For every n exceeding two, there is an integer f(n) so that if |E(M)| exceeds f(n), then M has a minor isomorphic to the rank-n wheel or whirl, a rank-n spike, the cycle or bond matroid of K_{3,n}, or U_{2,n} or U_{n-2,n}. In Chapter 2, we build on this result to determine what can be said about a large structure using a specified element e of M. In particular, we prove that, for every integer n exceeding two, there is an integer g(n) so that if |E(M)| exceeds g(n), then e is an element of a minor of M isomorphic to the rank-n wheel or whirl, a rank-n spike, the cycle or bond matroid of K_{1,1,1,n}, a specific single-element extension of M(K_{3,n}) or the dual of this extension, or U_{2,n} or U_{n-2,n}. In Chapter 3, we consider a large 3-connected binary matroid with a specified pair of elements. We extend a corollary of the result of Chapter 2 to show the following result for any pair {x,y} of elements of a 3-connected binary matroid M. For every integer n exceeding two, there is an integer h(n) so that if |E(M)| exceeds h(n), then x and y are elements of a minor of M isomorphic to the rank-n wheel, a rank-n binary spike with a tip and a cotip, or the cycle or bond matroid of K_{1,1,1,n}
On two classes of nearly binary matroids
We give an excluded-minor characterization for the class of matroids M in
which M\e or M/e is binary for all e in E(M). This class is closely related to
the class of matroids in which every member is binary or can be obtained from a
binary matroid by relaxing a circuit-hyperplane. We also provide an
excluded-minor characterization for the second class.Comment: 14 pages, 4 figures. This paper has been accepted for publication in
the European Journal of Combinatorics. This is the final version of the pape
The topology of the external activity complex of a matroid
We prove that the external activity complex of a matroid
is shellable. In fact, we show that every linear extension of LasVergnas's
external/internal order on provides a shelling of
. We also show that every linear extension of LasVergnas's
internal order on provides a shelling of the independence complex
. As a corollary, and have the same -vector.
We prove that, after removing its cone points, the external activity complex is
contractible if contains as a minor, and a sphere otherwise.Comment: Comments are welcom
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