Elements belonging to triads in 3-connected matroids

摘要

In this paper, we prove a conjecture proposed by Leo: a large minimally 3-connected matroid M has at least (5|E(M)|+30)/9 of its elements belonging to some triad. A bound on the number of elements belonging to triads of a 3-connected matroid which is close to be minimally 3-connected is also given. Both of these bounds are sharp and infinite families of matroids attaining them are constructed. A new proof of results of Lemos and Leo about triads meeting circuits with at most one removable element in 3-connected matroids is given.