Matroid packing and covering with circuits through an element

In 1981, Seymour proved a conjecture of Welsh that, in a connected matroid M , the sum of the maximum number of disjoint circuits and the minimum number of circuits needed to cover M is at most r∗(M) + 1. This paper considers the set Ce(M) of circuits through a fixed element e such that M/e is connected. Let νe(M) be the maximum size of a subset of Ce(M) in which any two distinct members meet only in {e}, and let θe(M) be the minimum size of a subset of Ce(M) that covers M . The main result proves that νe(M) + θe(M) ≤ r∗(M) + 2 and that if M has no Fano-minor using e, then νe(M) + θe(M) ≤ r∗(M) + 1. Seymour’s result follows without difficulty from this theorem and there are also some interesting applications to graphs.