It is well known that linear rank-metric codes give rise to q-polymatroids. Analogously matroid theory, one may ask whether a given q-polymatroid representable by code. We provide an answer presenting example of q-matroid not any code and, via relation paving matroids, examples various q-matroids are $${{\mathbb {F}}}_{q^m}$$ -linear codes. then go on and introduce deletion contraction for q-po...

For a simple binary matroid M having no n-spike minor, we examine the problem of bounding |E(M)| as a function of its rank r(M) and circumference c(M). In particular, we show that |E(M)| ≤ min { r(M)(r(M)+3) 2 , c(M)r(M) } for any simple, binary matroid M having no 4-spike minor. As a consequence, the same bound applies to simple, binary matroids having no AG(3,2)-minor.

This thesis considers circular flow-type and circular chromatic-type parameters (φ and χ, respectively) for matroids. In particular we focus on orientable matroids and 6 √ 1-matroids. These parameters are obtained via two approaches: algebraic and orientation-based. The general questions we discuss are: bounds for flow number; characterizations of Eulerian and bipartite matroids; and possible c...

A signed graph is a representation of an even cycle matroid M if the cycles of M correspond to the even cycles of that signed graph. Two, long standing, open questions regarding even cycle matroids are the problem finding an excluded minor characterization and the problem of efficiently recognizing this class of matroids. Progress on these problems has been hampered by the fact that even cycle ...

A result of Mason, as refined by Ingleton, characterizes transversal matroids as the matroids that satisfy a set of inequalities that relate the ranks of intersections and unions of nonempty sets of cyclic flats. We prove counterparts, for fundamental transversal matroids, of this and other characterizations of transversal matroids. In particular, we show that fundamental transversal matroids a...

Several attempts have been made to extend the theory of matroids (here referred to as ordinary or classical matroids) to theories of more general objects, in particular the Coxeter matroids of Borovik, Gelfand and White ([7], first introduced as WP-matroids in [10]), and the ∆-matroids and (equivalent but for notation) symmetric matroids of Bouchet (see, for example, [8]). The special cases of ...

Seymour’s Decomposition Theorem for regular matroids states that any matroid representable over both GF(2) and GF(3) can be obtained from matroids that are graphic, cographic, or isomorphic to R10 by 1-, 2-, and 3-sums. It is hoped that similar characterizations hold for other classes of matroids, notably for the class of near-regular matroids. Suppose that all near-regular matroids can be obta...

Seymour’s Decomposition Theorem for regular matroids states that any matroid representable over both GF(2) and GF(3) can be obtained from matroids that are graphic, cographic, or isomorphic to R10 by 1-, 2-, and 3-sums. It is hoped that similar characterizations hold for other classes of matroids, notably for the class of near-regular matroids. Suppose that all near-regular matroids can be obta...