On the number of triangles in 3-connected matroids

Abstract

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

$$max\{\frac{r^{\ast }\left(M\right)+6-2k}{4},\frac{\left|E\left(M\right)\right|+6-3k}{5}\}$$

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.