  We investigate the Keplerian (mass-shedding) sequence of rotating neutron
stars. Twelve different equations of state are used to describe the nuclear
structure. We find four fitting relations which connect the rotating frequency,
mass and radius of stars in the mass-shedding limit to the mass and radius of
stars in the static sequence. We show the breakdown of approximate relation for
the Keplerian frequency derived by Lattimer and Prakash [Science, 304, 536,
(2004)] and then we present a new, EOS-independent and more accurate relation.
This relation fits the Keplerian frequency of rotating neutron stars to about
$2\%$ for a large range of the compactness $M_{S}/R_{S}$ of the reference
non-rotating neutron star, namely the static star with the same central density
as the rotating one. The performance of the fitting formula is close to $4\%$
for $M_{S}/R_{S}\leq 0.05~M_{\odot}$/km ($f_{K}\leq 350$~Hz). We present
additional EOS-independent relations for the Keplerian sequence including
relations for $M_{K}f_{K}$ and $R_{K}f_{K}$ in terms of $M_{S}f_{S}$ and
$R_{S}f_{S}$, respectively, one of $M_K/R_K$ as a function of $f_{K}/f_{S}$ and
$M_S/R_S$, and a relation between the $M_K$, $R_K$ and $f_K$. These new fitting
relations are approximately EOS-independent with an error in the worst case of
$8\%$. The universality of the Keplerian sequence properties presented here add
to the set of other neutron star universal relations in the literature such as
the $I$-Love-$Q$ relation, the gravitational binding energy and the energy,
angular momentum and radius of the last circular orbit of a test-particle
around rotating neutron stars. This set of universal, analytic formulas,
facilitate the inclusion of general relativistic effects in the description of
relativistic astrophysical systems involving fast rotating neutron stars.
