21 research outputs found
Packing Steiner Trees
Let be a distinguished subset of vertices in a graph . A
-\emph{Steiner tree} is a subgraph of that is a tree and that spans .
Kriesell conjectured that contains pairwise edge-disjoint -Steiner
trees provided that every edge-cut of that separates has size .
When a -Steiner tree is a spanning tree and the conjecture is a
consequence of a classic theorem due to Nash-Williams and Tutte. Lau proved
that Kriesell's conjecture holds when is replaced by , and recently
West and Wu have lowered this value to . Our main result makes a further
improvement to .Comment: 38 pages, 4 figure
Structure of Cubic Lehman Matrices
A pair of square -matrices is called a \emph{Lehman pair} if
for some integer . In this case and
are called \emph{Lehman matrices}. This terminology arises because Lehman
showed that the rows with the fewest ones in any non-degenerate minimally
nonideal (mni) matrix form a square Lehman submatrix of . Lehman
matrices with are essentially equivalent to \emph{partitionable graphs}
(also known as -graphs), so have been heavily studied as part
of attempts to directly classify minimal imperfect graphs. In this paper, we
view a Lehman matrix as the bipartite adjacency matrix of a regular bipartite
graph, focusing in particular on the case where the graph is cubic. From this
perspective, we identify two constructions that generate cubic Lehman graphs
from smaller Lehman graphs. The most prolific of these constructions involves
repeatedly replacing suitable pairs of edges with a particular -vertex
subgraph that we call a -rung ladder segment. Two decades ago, L\"{u}tolf \&
Margot initiated a computational study of mni matrices and constructed a
catalogue containing (among other things) a listing of all cubic Lehman
matrices with of order up to . We verify their catalogue
(which has just one omission), and extend the computational results to matrices. Of the cubic Lehman matrices (with ) of order
up to , only two do not arise from our -rung ladder
construction. However these exceptions can be derived from our second
construction, and so our two constructions cover all known cubic Lehman
matrices with
Graphical representations of graphic frame matroids
A frame matroid M is graphic if there is a graph G with cycle matroid
isomorphic to M. In general, if there is one such graph, there will be many.
Zaslavsky has shown that frame matroids are precisely those having a
representation as a biased graph; this class includes graphic matroids,
bicircular matroids, and Dowling geometries. Whitney characterized which graphs
have isomorphic cycle matroids, and Matthews characterised which graphs have
isomorphic graphic bicircular matroids. In this paper, we give a
characterization of which biased graphs give rise to isomorphic graphic frame
matroids
On Excluded Minors for Even Cut Matroids
In this thesis we will present two main theorems that can be used to study
minor minimal non even cut matroids.
Given any signed graph we can associate an even cut matroid. However, given
an even cut matroid, there are in general, several signed graphs which
represent that matroid. This is in contrast to, for instance graphic (or
cographic) matroids, where all graphs corresponding to a particular
graphic matroid are essentially equivalent. To tackle the multiple
non equivalent representations of even cut matroids we use the concept of
Stabilizer first introduced by Wittle. Namely, we show the following:
given a "substantial" signed graph, which represents a matroid N that is a
minor of a matroid M, then if the signed graph extends to a signed graph
which represents M then it does so uniquely. Thus the representations of the
small matroid determine the representations of the larger matroid containing
it. This allows us to consider each representation of an even cut matroid
essentially independently.
Consider a small even cut matroid N that is a minor of a matroid M that is
not an even cut matroid. We would like to prove that there exists a
matroid N' which contains N and is contained in M such that the size of N'
is small and such that N' is not an even cut matroid (this would imply in
particular that there are only finitely many minimally non even cut
matroids containing N). Clearly, none of the representations of N extends to
M. We will show that (under certain technical conditions) starting from a
fixed representation of N, there exists a matroid N' which contains N
and is contained in M such that the size of N' is small and such that the
representation of N does not extend to N'
Maximum size binary matroids with no AG(3,2)-minor are graphic
We prove that the maximum size of a simple binary matroid of rank
with no AG(3,2)-minor is and characterise those matroids
achieving this bound. When , the graphic matroid is the
unique matroid meeting the bound, but there are a handful of smaller examples.
In addition, we determine the size function for non-regular simple binary
matroids with no AG(3,2)-minor and characterise the matroids of maximum size
for each rank