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 C e ( M ) of circuits through a fixed element e such that M / e is connected. Let ν e ( M ) be the maximum size of a subset of C e ( M ) in which any two distinct members meet only in { e } , and let θ e ( M ) be the minimum size of a subset of C e ( M ) that covers M. The main result proves that ν e ( M ) + θ e ( M ) ⩽ r * ( M ) + 2 and that if M has no Fano-minor using e, then ν e ( M ) + θ e ( M ) ⩽ r * ( M ) + 1 . Seymourʹs result follows without difficulty from this theorem and there are also some interesting applications to graphs.