On the number of triangles in 3-connected matroids

An element e of a 3-connected matroid M is said to be contractible provided that M/e is 3-connected. In this paper, we show that a 3-connected matroid M with exactly k contractible elements has at least max { r(M) + 6− 2k 4 , |E(M)| + 6− 3k 5 } triangles. For each k, we construct an infinite family of matroids that attain this bound. New sharp bounds for the number of triads of a minimally 3-connected matroid are obtained as a consequence of our main result. c © 2005 Elsevier Ltd. All rights reserved.