658 research outputs found
Bicircular signed-graphic matroids
Several matroids can be defined on the edge set of a graph. Although
historically the cycle matroid has been the most studied, in recent times, the
bicircular matroid has cropped up in several places. A theorem of Matthews from
late 1970s gives a characterization of graphs whose bicircular matroids are
graphic. We give a characterization of graphs whose bicircular matroids are
signed-graphic.Comment: 8 page
The Lattice of Cyclic Flats of a Matroid
A flat of a matroid is cyclic if it is a union of circuits. The cyclic flats
of a matroid form a lattice under inclusion. We study these lattices and
explore matroids from the perspective of cyclic flats. In particular, we show
that every lattice is isomorphic to the lattice of cyclic flats of a matroid.
We give a necessary and sufficient condition for a lattice Z of sets and a
function r on Z to be the lattice of cyclic flats of a matroid and the
restriction of the corresponding rank function to Z. We define cyclic width and
show that this concept gives rise to minor-closed, dual-closed classes of
matroids, two of which contain only transversal matroids.Comment: 15 pages, 1 figure. The new version addresses earlier work by Julie
Sims that the authors learned of after submitting the first versio
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
Matroids with nine elements
We describe the computation of a catalogue containing all matroids with up to
nine elements, and present some fundamental data arising from this cataogue.
Our computation confirms and extends the results obtained in the 1960s by
Blackburn, Crapo and Higgs. The matroids and associated data are stored in an
online database, and we give three short examples of the use of this database.Comment: 22 page
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