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 \(k \leq 3\), 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 \(|E(M)| \geq 2^{r-1} + 2\).