Assignment 6

Let M be a matroid and X a subset of E(M) such that M|X is connected. Find necessary and sufficient conditions for M|cl(X) to be connected.

Let M be a connected matroid such that M\e is disconnected for all elements e in E(M). Prove that M has a cocircuit of size 2. Also show that every circuit in M is a flat, that cl(C) = C for any circuit C. Suppose M is an n-connected matroid such that M\e is not n-connected for all elements e. Does M necessarily contain a cocircuit of size n?

Let M be a connected matroid with at least two elements and let e be an arbitrary element. Suppose the largest circuit containing e has l elements. Prove that there exist l-1 cocircuits, each containing e, whose union is E(M).

Suppose that x and y are distinct elements of an n-connected matroid M, where n >= 2 and |E(M)| >= 2(n-1). Suppose that M\x/y is n-connected but M\x is not. Prove that M has a cocircuit of size n containing x and y.

Let M be the 2-sum of two connected matroids M1 and M2. Prove that there exists a minor of M isomorphic to M1 and a minor of M isomorphic to M2. Let N be a 3-connected minor of M. Prove that N is a minor of either M1 or M2.