342 research outputs found
Ear-decompositions and the complexity of the matching polytope
The complexity of the matching polytope of graphs may be measured with the
maximum length of a starting sequence of odd ears in an
ear-decomposition. Indeed, a theorem of Edmonds and Pulleyblank shows that its
facets are defined by 2-connected factor-critical graphs, which have an odd
ear-decomposition (according to a theorem of Lov\'asz). In particular,
if and only if the matching polytope of the graph is
completely described by non-negativity, star and odd-circuit inequalities. This
is essentially equivalent to the h-perfection of the line-graph of , as
observed by Cao and Nemhauser.
The complexity of computing is apparently not known. We show that
deciding whether can be executed efficiently by looking at any
ear-decomposition starting with an odd circuit and performing basic modulo-2
computations. Such a greedy-approach is surprising in view of the complexity of
the problem in more special cases by Bruhn and Schaudt, and it is simpler than
using the Parity Minor Algorithm.
Our results imply a simple polynomial-time algorithm testing h-perfection in
line-graphs (deciding h-perfection is open in general). We also generalize our
approach to binary matroids and show that computing is a
Fixed-Parameter-Tractable problem (FPT)
Some heterochromatic theorems for matroids
The anti-Ramsey number of Erd\"os, Simonovits and S\'os from 1973 has become
a classic invariant in Graph Theory. To study this invariant in Matroid Theory,
we use a related invariant introduce by Arocha, Bracho and Neumann-Lara. The
heterochromatic number of a non-empty hypergraph is the smallest
integer such that for every colouring of the vertices of with exactly
colours, there is a totally multicoloured hyperedge of . Given a
rank- matroid , there are several hypergraphs associated to the matroid
that we can consider. One is , the hypergraph where the points are the
elements of the matroid and the hyperedges are the circuits of . The other
one is , where here the points are the elements and the hyperedges are
the bases of the matroid. We prove that equals when is not
the free matroid , and that if is a paving matroid, then
equals . Then we explore the case when the hypergraph has the
Hamiltonian circuits of the matroid as hyperedges, if any, for a class of
paving matroids. We also extend the trivial observation of Erd\"os, Simonovits
and S\'os for the anti-Ramsey number for 3-cycles to 3-circuits in projective
geometries over finite fields.Comment: Version 1.2, 15 pages, no figure
On Density-Critical Matroids
For a matroid having rank-one flats, the density is
unless , in which case . A matroid is
density-critical if all of its proper minors of non-zero rank have lower
density. By a 1965 theorem of Edmonds, a matroid that is minor-minimal among
simple matroids that cannot be covered by independent sets is
density-critical. It is straightforward to show that is the only
minor-minimal loopless matroid with no covering by independent sets. We
prove that there are exactly ten minor-minimal simple obstructions to a matroid
being able to be covered by two independent sets. These ten matroids are
precisely the density-critical matroids such that but for all proper minors of . All density-critical matroids of density
less than are series-parallel networks. For , although finding all
density-critical matroids of density at most does not seem straightforward,
we do solve this problem for .Comment: 16 page
On the structure of the h-vector of a paving matroid
We give two proofs that the -vector of any paving matroid is a pure
O-sequence, thus answering in the affirmative a conjecture made by R. Stanley,
for this particular class of matroids. We also investigate the problem of
obtaining good lower bounds for the number of bases of a paving matroid given
its rank and number of elements.Comment: We've corrected typos and small mistakes in the previous version.
We've also tidied up a few proofs and the reference
Topological Bijections for Oriented Matroids
In previous work by the first and third author with Matthew Baker, a family
of bijections between bases of a regular matroid and the Jacobian group of the
matroid was given. The core of the work is a geometric construction using
zonotopal tilings that produces bijections between the bases of a realizable
oriented matroid and the set of -compatible orientations
with respect to some acyclic circuit (respectively, cocircuit) signature
(respectively, ). In this work, we extend this construction
to general oriented matroids and circuit (respectively, cocircuit) signatures
coming from generic single-element liftings (respectively, extensions). As a
corollary, when both signatures are induced by the same lexicographic data, we
give a new (bijective) proof of the interpretation of using
orientation activity due to Gioan and Las Vergnas. Here is the Tutte
polynomial of the matroid.Comment: 12 pages, 3 figures, accepted by FPSAC 201
Zeros of Chromatic and Flow Polynomials of Graphs
We survey results and conjectures concerning the zero distribution of
chromatic and flow polynomials of graphs, and characteristic polynomials of
matroids.Comment: 21 pages, corrected statement of Theorem 34 and some typo
Sparse Hypergraphs and Pebble Game Algorithms
A hypergraph is -sparse if no subset spans
more than hyperedges. We characterize -sparse
hypergraphs in terms of graph theoretic, matroidal and algorithmic properties.
We extend several well-known theorems of Haas, Lov{\'{a}}sz, Nash-Williams,
Tutte, and White and Whiteley, linking arboricity of graphs to certain counts
on the number of edges. We also address the problem of finding
lower-dimensional representations of sparse hypergraphs, and identify a
critical behaviour in terms of the sparsity parameters and . Our
constructions extend the pebble games of Lee and Streinu from graphs to
hypergraphs
Excluding Kuratowski graphs and their duals from binary matroids
We consider some applications of our characterisation of the internally
4-connected binary matroids with no M(K3,3)-minor. We characterise the
internally 4-connected binary matroids with no minor in some subset of
{M(K3,3),M*(K3,3),M(K5),M*(K5)} that contains either M(K3,3) or M*(K3,3). We
also describe a practical algorithm for testing whether a binary matroid has a
minor in the subset. In addition we characterise the growth-rate of binary
matroids with no M(K3,3)-minor, and we show that a binary matroid with no
M(K3,3)-minor has critical exponent over GF(2) at most equal to four.Comment: Some small change
The Tutte polynomial of some matroids
The Tutte polynomial of a graph or a matroid, named after W. T. Tutte, has
the important universal property that essentially any multiplicative graph or
network invariant with a deletion and contraction reduction must be an
evaluation of it. The deletion and contraction operations are natural
reductions for many network models arising from a wide range of problems at the
heart of computer science, engineering, optimization, physics, and biology.
Even though the invariant is #P-hard to compute in general, there are many
occasions when we face the task of computing the Tutte polynomial for some
families of graphs or matroids. In this work we compile known formulas for the
Tutte polynomial of some families of graphs and matroids. Also, we give brief
explanations of the techniques that were use to find the formulas. Hopefully,
this will be useful for researchers in Combinatorics and elsewhere.Comment: many figures, 50 page
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