17 research outputs found
Unavoidable minors of large 3-connected matroids
This paper proves that, for every integernexceeding two, there is a numberN(n) such that every 3-connected matroid with at leastN(n) elements has a minor that is isomorphic to one of the following matroids: an (n+2)-point line or its dual, the cycle or cocycle matroid ofK3,n, the cycle matroid of a wheel withnspokes, a whirl of rankn, or ann-spike. A matroid is of the last type if it has ranknand consists ofnthree-point lines through a common point such that, for allkin {1,2,...,n-1}, the union of every set ofkof these lines has rankk+1. © 1997 Academic Press
On the unique representability of spikes over prime fields
For an integer , a rank- matroid is called an -spike if it
consists of three-point lines through a common point such that, for all
, the union of every set of of these lines has
rank . Spikes are very special and important in matroid theory. In 2003 Wu
found the exact numbers of -spikes over fields with 2, 3, 4, 5, 7 elements,
and the asymptotic values for larger finite fields. In this paper, we prove
that, for each prime number , a ) representable -spike is only
representable on fields with characteristic provided that .
Moreover, is uniquely representable over .Comment: 8 page
Capturing two elements in unavoidable minors of 3-connected binary matroids
Let M be a 3-connected binary matroid and let n be an integer exceeding 2. Ding, Oporowski, Oxley, and Vertigan proved that there is an integer f(n) so that if |E(M)|\u3ef(n), then M has a minor isomorphic to one of the rank-n wheel, the rank-n tipless binary spike, or the cycle or bond matroid of K3 n. This result was recently extended by Chun, Oxley, and Whittle to show that there is an integer g(n) so that if |E(M)|\u3eg(n) and xεE(M), then x is an element of a minor of M isomorphic to one of the rank-n wheel, the rank-n binary spike with a tip and a cotip, or the cycle or bond matroid of K11,1,n. In this paper, we prove that, for each i in {2,3}, there is an integer hi(n) so that if |E(M)|\u3ehi(n) and Z is an i-element rank-2 subset of M, then M has a minor from the last list whose ground set contains Z. © 2012 Elsevier Inc
Unavoidable parallel minors of regular matroids
This is the post-print version of the Article - Copyright @ 2011 ElsevierWe prove that, for each positive integer k, every sufficiently large 3-connected regular matroid has a parallel minor isomorphic to M (K_{3,k}), M(W_k), M(K_k), the cycle matroid of the graph obtained from K_{2,k} by adding paths through the vertices of each vertex class, or the cycle matroid of the graph obtained from K_{3,k} by adding a complete graph on the vertex class with three vertices.This study is partially supported by a grant from the National Security Agency
Unavoidable doubly connected large graphs
A connected graph is doubly connected if its complement is also connected. The following Ramsey-type theorem is proved in this paper. There exists a function h(n), defined on the set of integers exceeding three, such that every doubly connected graph on at least h(n) vertices must contain, as an induced subgraph, a doubly connected graph, which is either one of the following graphs or the complement of one of the following graphs: (1) Pn, a path on n vertices; (2) K1,ns, the graph obtained from K 1,n by subdividing an edge once; (3) K2,n\e, the graph obtained from K2,n by deleting an edge;(4) K2,n+, the graph obtained from K2,n by adding an edge between the two degree-n vertices x1 and x2, and a pendent edge at each xi. Two applications of this result are also discussed in the paper. © 2003 Elsevier B.V. All rights reserved
Capturing matroid elements in unavoidable 3-connected minors
A result of Ding, Oporowski, Oxley, and Vertigan reveals that a large 3-connected matroid M has unavoidable structure. For every n\u3e2, there is an integer f(n) so that if {pipe}E(M){pipe}\u3ef(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 this paper, 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 {pipe}E(M){pipe}\u3eg(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. © 2012 Elsevier Ltd
Ramsey Theory Using Matroid Minors
This thesis considers a Ramsey Theory question for graphs and regular matroids. Specifically, how many elements N are required in a 3-connected graphic or regular matroid to force the existence of certain specified minors in that matroid? This question cannot be answered for an arbitrary collection of specified minors. However, there are results from the literature for which the number N exists for certain collections of minors. We first encode totally unimodular matrix representations of certain matroids. We use the computer program MACEK to investigate this question for certain classes of specified minors
What is a 4-connected matroid?
The breadth of a tangle in a matroid is the size of the largest
spanning uniform submatroid of the tangle matroid of . The matroid
is weakly 4-connected if it is 3-connected and whenever is a
partition of with , then . We prove that if
is a tangle of order and breadth in a matroid ,
then has a weakly 4-connected minor with a tangle of
order , breadth and has the property that is the tangle in
induced by .
A set of elements of a matroid is 4-connected if
for all . As a
corollary of our theorems on tangles we prove that if contains an
-element 4-connected set where , then has a weakly
-connected minor that contains an -element -connected set.Comment: 34 page