Matroids With Every Two Elements in a $4$-Circuit

Abstract

This paper proves that the graphic matroids with at least two edges and no isolated vertices coincide with the class of complete $k$-partite graphs, where, when $k\le 3$, no partition class has size one. It also shows that a simple rank-$r$ binary matroid $M$ has every two elements in a $4$-circuit if $|E(M)|\ge 2^{r-1}+2$.