The influence of the staggered magnetic field h on properties of the o
ne-dimensional t-J and Hubbard models with nearly half-filled bands ar
e studied. Using the exact diagonalization method for small systems as
well as analytical approaches possible in the anisotropic model and f
or h >> t, we determine the phase diagram of the t-J-h model at low do
ping, which displays three different regimes: the phase with repelling
holes for J < J(c), the phase with paired holes for J(c) < J < J(s) a
nd the phase separation for J > J(s). Employing for h not-equal 0 the
analogy with the universal behavior of one-dimensional interacting fer
mions within the Luther-Emery regime, we calculate the charge exponent
K(rho), which can be K(rho) > 1 in the pairing regime, indicating dom
inant superconducting fluctuations. An analogous analysis, in particul
ar the perturbation expansion in t/h, performed for the Hubbard t-U-h
model shows a phase of stable hole pairs for U < U(c). Again K(rho) se
ems to approach a nonuniversal value at half-filling.