Connected matroids with a small circumference
By a result of Lemos and Oxley a connected matroid \(M\) with \(|E(M)|\geq 3r(M)\) has a circuit \(C\) such that \(M\setminus C\) is connected. The author shows that replacing \(3r(M)\) by \(3r(M)-1\) yields a sharp lower bound. Also, it is shown that \(|E(M)|\geq 3r(M)-3\) whenever \(M\) is simple, connected, and of rank at least 7.
scientific article; zbMATH DE number 1882146