Minimum number of matroid circuits containing a fixed element

Question

Let $M$ be a matroid with an $n$-element ground set $E$. I'll assume that $M$ is connected, co-simple (so its dual has no loops or parallel elements) and has no loops. Fix a particular element $e\in E$. I would like to have some reasonable lower bound in terms of $n$ for the number of circuits of $M$ that contain $e$.

I'd be interested to find out if the exact minimum is known, but I'd also be happy with a crude lower bound that grows with $n$. Note that the conditions on $M$ are necessary in order to get a non-constant lower bound. Without the hypothesis that $M$ is connected, it could be the case that $e$ is in a small connected component so that there are not many circuits containing it. If $M$ has lots of loops or coloops other than $e$, then those loops and coloops cannot belong to circuits containing $e$. We also want the dual of $M$ to have no parallel elements to avoid a situation such as when $M$ is the graphical matroid of a cycle graph, in which each edge is contained in a unique circuit. It's also possible that I've missed some additional restrictions that would need to be placed on $M$.

Note that $M$ itself is allowed to have parallel elements. For example, if all elements of $E$ are parallel, then there are $n-1$ circuits containing $e$; they are the sets of the form $\{e,f\}$ for $f\in E\setminus\{e\}$. Is it possible that $n-1$ is a lower bound in general?

Answer 1

Let $c_i(M)$ be the number of circuits containing the element $i$ of the ground set of a cosimple and loopless matroid $M$ and let $c(M)$ denote the minimum of the $c_i$. Here we'll see that $n-1$ is not a lower bound on $c(M)$ and gather (a very little) evidence for a smaller lower bound of $n-\lceil{\frac{n}{2}}\rceil + 1$.

Using the Matroids package in Macaulay2 you can check that while the lower bound of $c(M) \ge n-1$ holds for all cosimple matroids without loops on $n$ elements for $n < 5$.

For $n=5$ consider the matroid $M$ with circuits $\mathcal{C}=\{{01, 23, 024, 034, 124, 134}\}.$ Then $M$ is cosimple without loops but every element save $4$ appears in $3 = n-2$ circuits. More over this is the only matroid on five elements (up to isomorphism) with $c(M)< n-1$.

For $n=6$ there are three (cosimple, loopless) matroids with $c<n-1$ including the matroid $M$ with circuits $\mathcal{C} = \{{ 012, 0145, 0235, 034, 1234, 135, 245}\}$ having $c_i(M) = 4$ for all elements in the ground set.

When $n=7$ there are two matroids with $c=4 = n-3$ including the matroid with circuits $$\{0136, 0145, 0235, 0246, 1234, 1256, 3456\}.$$

When $n=8$ there are five matroids with $c=5$ including the matroid with circuits $$\{01, 023, 03457, 03467, 03567, 123, 13457, 13467, 13567, 2457, 2467, 2567, 456\}$$.