$k$-fold Circuits in Matroids

Abstract: Double circuits were introduced by Lov\'{a}sz in 1980 as a fundamental tool in his derivation of a min-max formula for the size of a maximum matching in certain families of matroids. This formula was extended to all matroids satisfying the so-called `double circuit property' by Dress and Lov\'{a}sz in 1987. We extend these notions to $k$-fold circuits for all natural numbers $k$ and derive foundational results about these $k$-fold circuits. Our results imply, in particular, that certain families of matroids which are known to satisfy the double circuit property, satisfy the $k$-fold circuit property for all natural numbers $k$. These families include all pseudomodular matroids (such as full linear, algebraic and transversal matroids) and count matroids.