On size, circumference and circuit removal in 3-connected matroids
In this paper several extremal results for 3-connected matroids are proven. The results are best possible which is shown by examples attaining the bounds. More specific it is shown that for a 3-connected matroid \(M\): (i) If the rank \(r(M)\) is at least 6 then the circumference \(c(M)\) is at least 6 (Theorem 1.5, having a very ingenious proof) and if moreover \(|E(m)|\geq 4r(M)- 5\) then there is a circuit that when deleted from \(M\) leaves a 3-connected matroid (Corollary 4.1). (ii) If \(r(M)> 4\) and \(M\) has a basis \(B\) such that \(M\setminus e\) is not 3-connected for all \(e\) in \(E(M)-B\) then \(|E(M)|\leq 3r(M)- 4\). (Theorem 1.2 which is a refinement of Theorem 1.1. The proof of Theorem 1.2 heavily relies on Theorem 1.1, which is proven in a separate paper.) (iii) If \(M\) is minimally 3-connected but not Hamiltonian then \(|E(M)|\leq 3r(M)- C(M)\) (Theorem 1.4 sharpening a previous result by Oxley).
scientific article; zbMATH DE number 1462280