Rota conjectured that, given n disjoint bases of a rank-n matroid M , there are n disjoint transversals of these bases that are all bases of M . We prove a stronger statement for the class of paving matroids.

Recently, it was proved by Anari–Oveis Gharan–Vinzant, Anari–Liu–Oveis Gharan–Vinzant and Brändén–Huh that, for any matroid M, its basis generating polynomial independent set are log-concave on the positive orthant. Using these, they obtain some combinatorial inequalities matroids including a solution of strong Mason's conjecture. In this paper, we study strictness log-concavity these polynomia...

Let M = (Mr)reW be a system of matroids on a set S. For every transfinite sequence / of distinct elements of 5, we define a number TJ(/). In [12] we proved that the condition that r](f) 3=0 for every possible choice of/is necessary for M to have a system of mutually disjoint bases. Further, we showed that this condition is sufficient if R is countable and Mr is a rank-finite transversal matroid...

The main purpose of this paper is to introduce a concept of$L$-fuzzifying topological groups (here $L$ is a completelydistributive lattice) and discuss some of their basic properties andthe structures. We prove that its corresponding $L$-fuzzifyingneighborhood structure is translation invariant. A characterizationof such topological groups in terms of the corresponding$L$-fuzzifying neighborhoo...

Bixby and Cunningham showed that a 3-connected binary matroid M is graphic if and only if every element belongs to at most two non-separating cocircuits. Likewise, Lemos showed that such a matroid M is graphic if and only if it has exactly r(M) + 1 nonseparating cocircuits. Hence the presence inM of either an element in at least three non-separating cocircuits, or of at least r(M) + 2 non-separ...

Catlin and Jaeger proved that the cycle matroid of a 4-edge-connected graph has a spanning cycle. This result can not be generalized to regular matroids as there exist infinitely many connected cographic matroids, each of which contains a M(K5) minor and has arbitrarily large cogirth, that do not have spanning cycles. In this paper, we proved that if a connected regular matroid without aM(K5)-m...

We give an upper bound and a class of lower bounds on the coefficients of the characteristic polynomial of a simple binary matroid. This generalizes the corresponding bounds for graphic matroids of Li and Tian (1978) [3], as well as a matroid lower bound of Björner (1980) [1] for simple binary matroids. As the flow polynomial of a graph G is the characteristic polynomial of the dual matroid M(G...

Induction (or transformation) by bipartite graphs is one of the most important operations on matroids, and it is well known that the induction of a matroid by a bipartite graph is again a matroid. As an abstract form of this fact, the induction of a matroid by a linking system is known to be a matroid. M-convex functions are quantitative extensions of matroidal structures, and they are known as...

Tutte’s Wheels-and-Whirls Theorem proves that if M is a 3-connected matroid other than a wheel or a whirl, then M has a 3-connected minor N such that |E(M)| − |E(N)| = 1. Geelen and Whittle extended this theorem by showing that when M is sequentially 4-connected, the minor N can also be guaranteed to be sequentially 4connected, that is, for every 3-separation (X, Y ) of N , the set E(N) can be ...