No CrossRef data available.
Simple Matroids with Bounded Cocircuit Size
Published online by Cambridge University Press: 14 February 2001
We examine the specialization to simple matroids of certain problems in extremal matroid theory that are concerned with bounded cocircuit size. Assume that each cocircuit of a simple matroid M has at most d elements. We show that if M has rank 3, then M has at most d + [lfloor ]√d[rfloor ] + 1 points, and we classify the rank-3 simple matroids M that have exactly d + [lfloor ]√d[rfloor ] points. We show that if M is a connected matroid of rank 4 and d is q3 with q > 1, then M has at most q3 + q2 + q + 1 points; this upper bound is strict unless q is a prime power, in which case the only such matroid with exactly q3 + q2 + q + 1 points is the projective geometry PG(3, q). We also show that if d is q4 for a positive integer q and if M has rank 5 and is vertically 5-connected, then M has at most q4 + q3 + q2 + q + 1 points; this upper bound is strict unless q is a prime power, in which case PG(4, q) is the only such matroid that attains this bound.
- Research Article
- Combinatorics, Probability and Computing , Volume 9 , Issue 5 , September 2000 , pp. 407 - 419
- 2000 Cambridge University Press