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 TM(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 TM(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.