Characterizations of matroids with an element lying in a restricted number of circuits

Abstract

A matroid M with a distinguished element $e_0 \in E(M)$ is a rooted matroid with $e_0$ being the root. We present a characterization of all connected binary rooted matroids whose root lies in at most three circuits, and a characterization of all connected binary rooted matroids whose root lies in all but at most three circuits. While there exist infinitely many such matroids, the number of serial reductions of such matroids is finite. In particular, we find two finite families of binary matroids $\mathcal M_1$ and $\mathcal M_2$ and prove the following. (i) For some $e_0 \in E(M)$, M has at most three circuits containing $e_0$ if and only if the serial reduction of M is isomorphic to a member in $\mathcal M_1$. (ii) If for some $e_0 \in E(M)$, M has at most three circuits not containing $e_0$ if and only if the serial reduction of M is isomorphic to a member in $\mathcal M_2$. These characterizations will be applied to show that every connected binary matroid M with at least four circuits has a 1-hamiltonian circuit graph.