  The transition density $\rho_{t}$ and pressure $P_{t}$ at the inner edge
separating the liquid core from the solid crust of neutron stars are
systematically studied using a modified Gogny (MDI) and 47 popular Skyrme
interactions within well established dynamical and thermodynamical methods. It
is shown that the widely used parabolic approximation to the full Equation of
State (EOS) of isospin asymmetric nuclear matter may lead to huge errors in
estimating the \rho_{t} and P_{t}, especially for stiffer symmetry energy
functionals $E_{sym}(\rho)$. The \rho_{t} and P_{t} decrease roughly linearly
with the increasing slope parameter $L$ of the $E_{sym}(\rho)$ using the full
EOS within both methods. It is also shown that the thickness, fractional mass
and moment of inertia of neutron star crust are all very sensitive to the
parameter $L$ through the $\rho_{t}$. Moreover, it is shown that the
$E_{sym}(\rho)$ constrained in the same sub-saturation density range as the
neutron star crust by the isospin diffusion data in heavy-ion collisions at
intermediate energies limits the transition density and pressure to 0.040$
fm^-3}< \rho_{t} < 0.065$ fm^-3 and 0.01 MeV/fm^3 < P_{t} < 0.26$ MeV/fm^3,
respectively. These constrained values for the transition density and pressure
are significantly lower than their fiducial values currently used in the
literature. Furthermore, the mass-radius relation and several other properties
closely related to the neutron star crust are studied by using the MDI
interaction. It is found that the newly constrained $\rho_t$ and $P_t$ together
with the earlier estimate of $\Delta I/I>0.014$ for the crustal fraction of the
moment of inertia of the Vela pulsar impose a stringent constraint of R>=
4.7+4.0M/M_sun km for the radius $R$ and mass $M$ of neutron stars.
