Special Issue on COMBINATORIAL GEOMETRIES AND APPLICATIONS: ORIENTED MATROIDS AND MATROIDS Preface (original) (raw)

Workshop Program Combinatorial geometries 2018 : matroids , oriented matroids and applications

2018

s of talks Monday 24 at 9h: A matroid extension result James OXLEY (Louisiana State University, USA) Let (A,B) be a 3-separation in a matroid M . If M is representable, then, in the underlying projective space, there is a line where the subspaces spanned by A and B meet, and M can be extended by adding elements from this line. In general, Geelen, Gerards, and Whittle proved that M can be extended by an independent set {p, q} such that {p, q} is in the closure of each of A and B. In this extension, each of p and q is freely placed on the line L spanned by {p, q}. This talk will discuss a result that gives necessary and su cient conditions under which a xed element can be placed on L. Monday 24 at 9h25: Su cient condition for almost irreducibility Csongor CSEHI (Budapest University of Technology and Economics, Hungary) A matroid N is almost irreducible if for any decomposition to matroid union M1 ∨M2 = N , N is a series extension of a submatroid of one of the matroids Mi ∈ {M1,M2}. An...

A Glimpse into Continuous Combinatorics of Posets, Polytopes, and Matroids

Journal of Mathematical Sciences, 2020

Following [Ži98] we advocate a systematic study of continuous analogues of finite partially ordered sets, convex polytopes, oriented matroids, arrangements of subspaces, finite simplicial complexes, and other combinatorial structures. Among the illustrative examples reviewed in this paper are an Euler formula for a class of 'continuous convex polytopes' (conjectured by Kalai and Wigderson), a duality result for a class of 'continuous matroids', a calculation of the Euler characteristic of ideals in the Grassmannian poset (related to a problem of Gian-Carlo Rota), an exposition of the 'homotopy complementation formula' for topological posets and its relation to the results of Kallel and Karoui about 'weighted barycenter spaces' and a conjecture of Vassiliev about simplicial resolutions of singularities. We also include an extension of the index inequality (Sarkaria's inequality) based on interpreting diagrams of spaces as continuous posets.

Two Decompositions in Topological Combinatorics with Applications to Matroid Complexes

Transactions of the American Mathematical Society, 1997

This paper introduces two new decomposition techniques which are related to the classical notion of shellability of simplicial complexes, and uses the existence of these decompositions to deduce certain numerical properties for an associated enumerative invariant. First, we introduce the notion of M-shellability, which is a generalization to pure posets of the property of shellability of simplicial complexes, and derive inequalities that the rank-numbers of M-shellable posets must satisfy. We also introduce a decomposition property for simplicial complexes called a convex ear-decomposition, and, using results of Kalai and Stanley on h h -vectors of simplicial polytopes, we show that h h -vectors of pure rank- d d simplicial complexes that have this property satisfy h 0 ≤ h 1 ≤ ⋯ ≤ h [ d / 2 ] h_{0} \leq h_{1} \leq \cdots \leq h_{[d/2]} and h i ≤ h d − i h_{i} \leq h_{d-i} for 0 ≤ i ≤ [ d / 2 ] 0 \leq i \leq [d/2] . We then show that the abstract simplicial complex formed by the coll...

8. Cardinality Homogeneous Set Systems, Cycles in Matroids, and Associated Polytopes

The Impact of Manfred Padberg and His Work, 2004

A subset C of the power set of a finite set E is called cardinality homogeneous if, whenever C contains some set F , C contains all subsets of E of cardinality |F |. Examples of such set systems C are the sets of all even or of all odd cardinality subsets of E, or, for each uniform matroid, its set of circuits and its set of cycles. With each cardinality homogeneous set system C, we associate the polytope P (C), the convex hull of the incidence vectors of all sets in C. We provide a complete and nonredundant linear description of P (C). We show that a greedy algorithm optimizes any linear function over P (C), construct, by a dual greedy procedure, an explicit optimum solution of the dual linear program, and describe a polynomial time separation algorithm for the class of polytopes of type P (C).

Some heterochromatic theorems for matroids

arXiv (Cornell University), 2017

The anti-Ramsey number of Erdös, Simonovits and Sós 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 hc(H) of a non-empty hypergraph H is the smallest integer k such that for every colouring of the vertices of H with exactly k colours, there is a totally multicoloured hyperedge of H. Given a rank-r matroid M , there are several hypergraphs associated to the matroid that we can consider. One is C(M), the hypergraph where the points are the elements of the matroid and the hyperedges are the circuits of M. The other one is B(M), where here the points are the elements and the hyperedges are the bases of the matroid. We prove that hc(C(M)) equals r + 1 when M is not the free matroid U n,n , and that if M is a paving matroid, then hc(B(M)) equals r. 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 Erdös, Simonovits and Sós for the anti-Ramsey number for 3-cycles to 3circuits in projective geometries over finite fields.

Oriented matroids and Ky Fan’s theorem

Combinatorica, 2010

L. Lovász has shown in [9] that Sperner's combinatorial lemma admits a generalization involving a matroid defined on the set of vertices of the associated triangulation. We prove that Ky Fan's theorem admits an oriented matroid generalization of similar nature (Theorem 3.1). Classical Ky Fan's theorem is obtained as a corollary if the underlying oriented matroid is chosen to be the alternating matroid C m,r .

Convex, Acyclic, And Free Sets Of An Oriented Matroid

2007

We study the global and local topology of three objects associated to an oriented matroid: the lattice of convex sets, the simplicial complex of acyclic sets, and the simplicial complex of free sets. Special cases of these objects and their homotopy types have appeared in several places in the literature. The global homotopy types of all three are shown to coincide, and are either spherical or contractible depending on whether the oriented matroid is totally cyclic. Analysis of the homotopy type of links of vertices in the complex of free sets yields a generalization and more conceptual proof of a recent result counting the interior points of a point configuration.

Matroids on convex geometries (cg-matroids)

Discrete Mathematics, 2007

We consider matroidal structures on convex geometries, which we call cg-matroids. The concept of a cg-matroid is closely related to but different from that of a supermatroid introduced by Dunstan, Ingleton, and Welsh in 1972. Distributive supermatroids or poset matroids are supermatroids defined on distributive lattices or sets of order ideals of posets. The class of cg-matroids includes distributive supermatroids (or poset matroids). We also introduce the concept of a strict cg-matroid, which turns out to be exactly a cg-matroid that is also a supermatroid. We show characterizations of cg-matroids and strict cg-matroids by means of the exchange property for bases and the augmentation property for independent sets. We also examine submodularity structures of strict cg-matroids.

Bases-cobases graphs and polytopes of matroids

Combinatorica, 1993

Let M be a block matroid (i.e. a matroid whose ground set E is the disjoint union of two bases). We associate with M two objects: 1. The bases-cobases graph G = G(M,M*) having as vertices the bases B of M for which the complement E \ B is also a base, and as edges the unordered pairs (B, B') of such bases differing exactly by two elements. 2. The polytope of the bases-cobases K = K(M,M*) whose extreme points are the incidence vectors of the bases of M whose complement is also a base. We prove that, if M is graphic (or cographic), the distance between any two vertices of G corresponding to disjoint bases is equal to the rank of M (generalizing a result of [10]). Concerning the polytope we prove that K is an hypercube if and only if dim(K)=rank(M). A constructive characterization of the class of matroids realizing this equality is given.