A Method to construct the Sparse-paving Matroids over a Finite Set

Abstract

In this work we present an algorithm to construct sparse-paving matroids over finite set S. From this algorithm we derive some useful bounds on the cardinality of the set of circuits of any Sparse-Paving matroids which allow us to prove in a simple way an asymptotic relation between the class of Sparse-paving matroids and the whole class of matroids. Additionally we introduce a matrix based method which render an explicit partition of the r-subsets of S, S r = ⊔ γ i=1 Ui such that each Ui defines a sparse-paving matroid of rank r.

A Method to construct all the Paving Matroids over a Finite Set

“…Proposition [10] If M = (S, I) is a paving matroid, then M is a sparsepaving if and only if N 2 = ∅.…”

“…The algorithm below construct a maximal set of hyperplanes, H t , of cardinality t ∈ {r, ..., n − 1} of a matroid M = (S, I) of rank r. [10] Let S = {1, ..., n} be a set and 2 ≤ r ≤ n.…”