On Maximum-sized k-Regular Matroids

Let k be an integer exceeding one. The class of k–regular matroids is a generalization of the classes of regular and near-regular matroids. A simple rank–r regular matroid has the maximum number of points if and only if it is isomorphic to M(Kr+1), the cycle matroid of the complete graph on r + 1 vertices. A simple rank–r near-regular matroid has the maximum number of points if and only if it is isomorphic to the simplification of T M(K3)(M(Kr+2)), that is, the simplification of the matroid obtained, geometrically, by freely adding a point to a 3–point line of M(Kr+2) and then contracting this point. This paper determines the maximum number of points that a simple rank– r k–regular matroid can have and determines all such matroids having this number. With one exception, there is exactly one such matroid. This matroid is isomorphic to the simplification of T M(Kk+2)(M(Kr+k+1)), that is, the simplification of the matroid obtained, geometrically, by freely adding k independent points to a flat of M(Kr+k+1) isomorphic to M(Kk+2) and then contracting each of these points.