On maximum-sized near-regular and \(\root 6\of{1}\)-matroids
The class of simple regular matroids has a single maximum size rank-\(r\) member namely the cycle matroid of the complete graph on \(r+1\) vertices. Looking at ternary matroids instead of binary matroids, the role of the regular matroids is played by near-regular matroids, dyadic matroids, and \(\root 6\of{1}\)-matroids. The maximum sized rank-\(r\) dyadic matroids are known to be isomorphic to the ternary Dowling geometry, see \textit{J. P. S. Kung} [Discrete Comput. Geom. 5, No. 1, 83-95 (1990; Zbl 0697.51007)] and \textit{J. P. S. Kung} and \textit{J. G. Oxley} [Graphs Comb. 4, No. 4, 323-332 (1988; Zbl 0702.51004)]. The purpose of this paper is to prove that the classes of near-regular and \(\root 6\of{1}\)-matroids both have, for \(r \neq 3\), a single maximum size member which is, in fact, the same for both classes. This maximum sized matroid has the following geometric description: Freely add a point on a 3-point line of \(M(K_{r \neq 3})\), contract that point and simplify the resulting matroid. Structural properties of near-regular and \(\root 6\of{1}\)-matroids are established and used to prove the main result.
scientific article; zbMATH DE number 1182275