Compositions for matroids with the Fulkerson property (original) (raw)

The positive circuits of oriented matroids with the packing property or idealness

Electronic Notes in Discrete Mathematics, 2010

The class of clutters of the positive circuits of oriented matroids is an important class which generalizes both the dicut clutters and the dicycle clutters of directed graphs, together. In this article, we will show that the clutter of the positive circuits of an oriented matroid whose co-rank is at most 4 has the packing property if and only if it has none of the six minimally non-packing minors J 3 , C 2 3 , C 3 5 , Q 6 , Q 6 ⊗ 1 and Q 6 ⊗ {1, 2}. By using this, we will also prove that the clutter of the positive circuits of an oriented matroid whose co-rank is at most 4 is ideal if and only if it has none of the three minimally non-packing minors J 3 , C 2 3 , and C 3 5 .

Some new subclasses of graphic matroids, related to the union operation

2015

There is a conjecture that if the union (also called sum) of graphic matroids is not graphic then it is nonbinary. Some special cases have been proved only, for example if several copies of the same graphic matroid are given. If there are two matroids and the first one can either be represented by a graph with two points, or is the direct sum of a circuit and some loops, then a necessary and sufficient condition is known for the other matroid to ensure the graphicity of the union and the above conjecture holds for these cases. We have proved the sufficiency of this condition for the graphicity of the union of two arbitrary graphic matroids. Then we present a weaker necessary condition which is of similar character. Finally we suggest a more general framework of the study of such questions by introducing matroid classes formed by those graphic (or arbitrary) matroids whose union with any graphic (or arbitrary) matroid is graphic (or either graphic or nonbinary).

Circuit and cocircuit partitions of binary matroids

Czechoslovak Mathematical Journal, 2006

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Uniform partitioning to bases in a matroid Zsolt Fekete ? and Jácint Szabó ? ?

2005

We say that a matroid M is k-uniform if – provided that it is the disjoint union of its two bases – for any given k-element subpartition P of its ground set, M can be partitioned into two disjoint bases B1, B2 such that ||B1 ∩ P | − |B2 ∩ P || ≤ 1 for all P ∈ P. The circuit matroid of an undirected graph G is called k-star-uniform if the above holds for all k-element subpartitions containing stars of independent vertices of G. In this paper we prove that the circuit matroids are 1-uniform and 3-star-uniform but not necessarily 2-uniform and 4-star-uniform.

On the circuit-spectrum of binary matroids

2011

Murty, in 1971, characterized the connected binary matroids with all circuits having the same size. We characterize the connected binary matroids with circuits of two different sizes, where the largest size is odd. As a consequence of this result we obtain both Murty's result and other results on binary matroids with circuits of only two sizes. We also show that it will be difficult to complete the general case of this problem.

Boolean designs and self-dual matroids

Linear Algebra and its Applications, 1975

In this paper we consider a variety of questions in the context of Boolean designs. For example, ErdGs asked: How many subsets of an n-set can be found so that pairwise their intersections are all even (odd)? E. Berlekamp [2] and the author both answered this question; the answer is approximately 2[b"'. Another question which can be formulated in terms of Boolean designs was asked by J. A. Bondy and D. J. A. Welsh [l]. For what values of d can one find a connected binary matroid of rank d which is identically self-dual? We prove that such matroids exist for all d except 2, 3, and 5. The paper ends with a discussion of more general modular designs and with constructions of some identically self-dual matroids representable over the field of three elements. 0. INTRODUCTION Let U be a finite set. We denote the collection of all subsets of U by 9 (U), and we let 5?r (U) denote the collection of all r-subsets of U. A triple (U,f, 3) is called a system of blocks if Ci3 is also a finite set and f maps ?i; into C? (U). The elements of U are called the vertices of the system; the elements of '33 are called the blocks of the system. If f is an injection, (C',f, %I) is called a system of sets. In this case we often identify % with its image under f and simply write (U, 3), where Ci3 c 9 (U). We will be primarily interested in systems of sets; however, we are forced to discuss the more general situation because systems of sets are not closed under duality. Let

A matroid generalization of a theorem of Mendelsohn and Dulmage

Discrete Mathematics, 1973

A matrwd generalization is given to a theorem of Mendrlsohn and Dulmape mnwning assignments in bipartite graphs. The generalized theorem hat applications ilr optimiralkm theory and provides a simple proof of a theorem of Nash-Williams.