Circuit and fractional circuit covers of matroids

Let M be a connected matroid having a ground set E . Lemos and Oxley proved that | E ( M ) | ≤ 1 2 c ( M ) c ∗ ( M ) where c ( M ) (resp. c ∗ ( M ) ) is the circumference (resp. cocircumference) of M . In addition, they conjectured that one can find a collection of at most c ∗ ( M ) circuits which cover the elements of M at least twice. In this paper, we verify this conjecture for regular matroids. Moreover, we show that a version of this conjecture is true for fractional circuit covers.