In this note we construct a non-singular algebraic variety VG encoding the incidence information of a simple graph G, by a sequence of blow-ups of a projective space along suitable linear subspaces. The aim is to translate into the geometry of VG the combinatorial information about G; we nd that this can be done with surprising ease and e ciency. For example, we prove that the chromatic polynom...

An integer-valued function f on the set 2V of all subsets of a finite set V is a connectivity function if it satisfies the following conditions: (1) f(X) + f(Y ) ≥ f(X ∩Y )+ f(X ∪Y ) for all subsets X, Y of V , (2) f(X) = f(V \X) for all X ⊆ V , and (3) f(∅) = 0. Branch-width is defined for graphs, matroids, and more generally, connectivity functions. We show that for each constant k, there is ...

The growth-rate function for a minor-closed class M of matroids is the function h where, for each non-negative integer r, h(r) is the maximum number of elements of a simple matroid in M with rank at most r. The Growth-Rate Theorem of Geelen, Kabell, Kung, and Whittle shows, essentially, that the growth-rate function is always either linear, quadratic, exponential with some prime power q as the ...

For any linear quotient of a sphere, X = Sn−1/Γ, where Γ is an elementary abelian p–group, there is a corresponding Fp representable matroidMX which only depends on the isometry class ofX.When p is 2 or 3 this correspondence induces a bijection between isometry classes of linear quotients of spheres by elementary abelian p–groups, and matroids representable over Fp. Not only do the matroids giv...

Fuzzy relations are mappings from pairs of elements into the interval [0, 1]. As a replacement for the complement operation one can use the mapping that sends x to 1 − x. Together with the concepts of t-norm and t-conorm a weak form of Boolean algebra can be defined. However, to our knowledge so far no notion of domain or codomain has been investigated for fuzzy relations. These might, however,...

Abstract We prove that if M and N are finitary matroids on a common countable edge set E then they admit independent I such there is bipartition = ? for which ? spans in N. It answers positively the Matroid Intersection Conjecture of Nash-Williams case.

Fix a matroid N . A matroid M is N -fragile if, for each element e of M , at least one of M\e and M/e has no N -minor. The Bounded Canopy Conjecture is that all GF(q)-representable matroids M that have an N -minor and are N -fragile have branch width bounded by a constant depending only on q and N . A matroid N stabilizes a class of matroids over a field F if, for every matroid M in the class w...

The notion of H-matroids was introduced by U. Faigle and S. Fujishige in 2009 as a general model for matroids and the greedy algorithm. They gave a characterization of H-matroids by the greedy algorithm. In this note, we give a characterization of some H-matroids by rank functions. 2010 MSC: 05B35, 90C27

We give a counterexample to a conjecture by Wild about binary matroids. We connect two equivalent lines of research in matroid theory: a simple type of basis-exchange property and restrictions on the cardinalities of intersections of circuits and cocircuits. Finally, we characterize direct sums of series-parallel networks by a simple basis-exchange property. In [9], Wild proposed a characteriza...