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 − 2 k 4 , | E ( M ) | + 6 − 3 k 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.