#### Title

Matroid packing and covering with circuits through an element

#### Document Type

Article

#### Publication Date

1-1-2006

#### Abstract

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* + 2 and that if M has no Fano-minor using e, then νe + θe,(M) ≤ r*(M) + 1. Seymour's result follows without difficulty from this theorem and there are also some interesting applications to graphs. © 2005 Elsevier Inc. All rights reserved.

#### Publication Source (Journal or Book title)

Journal of Combinatorial Theory. Series B

#### First Page

135

#### Last Page

158

#### Recommended Citation

Lemos, M., & Oxley, J.
(2006). Matroid packing and covering with circuits through an element.* Journal of Combinatorial Theory. Series B**, 96* (1), 135-158.
https://doi.org/10.1016/j.jctb.2005.07.001