Anari, Gharan, and Vinzant proved (complete) log-concavity of the basis generating functions for all matroids. From viewpoint combinatorial Hodge theory, it is natural to ask whether these are “strictly” log-concave simple In this paper, we show strictness graphic matroids, that is, Kirchhoff polynomials graphs strictly log-concave. Our key observation polynomial a complete graph can be seen as...

In this paper, first steps are taken towards characterizing rank-decorated lattices of cyclic flats Z(M) that belong to matroids M can be represented over a prescribed finite field Fq. Two natural maps from the lattice minor given. Binary characterized via their flats. It is shown simple binary matroid without isthmuses atomic.

Hall’s theorem for bipartite graphs gives a necessary and sufficient condition for the existence of a matching in a given bipartite graph. Aharoni and Ziv [2] generalized the notion of matchability to a pair of possibly infinite matroids on the same set and gave a condition that is sufficient for the matchability of a given pair (M,W) of finitary matroids, where the matroid M is SCF — a sum of ...

The es-splitting operation on binary matroids is a natural generalization of Slater's <em>n</em>-line splitting graphs. In this paper, we characterize the closure operator matroid M<sup>e</sup><sub>X</sub> in terms original M. We also describe ats and hyperplanes bi- nary hyperplanes, respectively

Gian-Carlo Rota said that “Anyone who has worked with matroids has come away with the conviction that matroids are one of the richest and most useful ideas of our day.” [20] Hassler Whitney introduced the theory of matroids in 1935 and developed a striking number of their basic properties as well as different ways to formulate the notion of a matroid. As more and more connections between matroi...

Consider a hypergraph of rank r > 2 with m edges, independence number and edge cover number . We prove the inequality (r − 2)m+ r − 1 . One application of this inequality is a special case of a conjecture of Aharoni and the first author extending Ryser’s Conjecture to matroids. © 2007 Elsevier B.V. All rights reserved.

We show that the complete list of regular excluded minors for the class of signed-graphic matroids is M(G1), . . . ,M(G29), R15, R16. Here G1, . . . , G29 are the vertically 2-connected excluded minors for the class of projective-planar graphs.

We give an independent set axiomatization of symplectic matroids, a large and important class of Coxeter matroids. As an application, we construct a new class of examples of symplectic matroids from graphs. As another application, we prove that symplectic matroids satisfy a certain “basis exchange property,” and we conjecture that this basis exchange property in fact characterizes symplectic ma...

We investigate the computational difficulty of approximating the partition function of the ferromagnetic Ising model on a regular matroid. Jerrum and Sinclair have shown that there is a fully polynomial randomised approximation scheme (FPRAS) for the class of graphic matroids. On the other hand, the authors have previously shown, subject to a complexity-theoretic assumption, that there is no FP...