On a Generalization of Spikes

Abstract

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Cyclic matroids

“…Miller [5] investigated the matroids in which every pair of elements is contained in a 4-element circuit and a 4-element cocircuit, while Oxley et al [7] considered the 3-connected matroids in which every pair of elements is in a 4-element circuit and every element is in a 3-element circuit, and the 4-connected matroids in which every pair of elements is contained in a 4-element circuit and a 4-element cocircuit. Furthermore, Brettell et al [2] studied matroids in which every t-element subset of the ground set is contained in an ℓ-element circuit and an ℓ-element cocircuit. Relevant to this paper, their results imply that if a matroid M has the property that every t-element subset of E(M ) is contained in a 2t-element circuit and a 2t-element cocircuit, then, provided |E(M )| is sufficiently large, M is (2t, 2t)-cyclic.…”