  We present bright-dark vector solitons in quasi-one-dimensional spinor (F=1)
Bose-Einstein condensates. Using a multiscale expansion technique, we reduce
the corresponding nonintegrable system of three coupled Gross-Pitaevskii
equations (GPEs) to a completely integrable Yajima-Oikawa system. In this way,
we obtain approximate solutions for small-amplitude vector solitons of
dark-dark-bright and bright-bright-dark types, in terms of the $m_{F}=+1,-1,0$
spinor components, respectively. By means of numerical simulations of the full
GPE system, we demonstrate that these states indeed feature soliton properties,
i.e., they propagate undistorted and undergo quasi-elastic collisions. It is
also shown that, in the presence of a parabolic trap of strength $\omega $, the
bright component(s) is (are) guided by the dark one(s), and, as a result, the
small-amplitude vector soliton as a whole performs harmonic oscillations of
frequency $\omega/ \sqrt{2}$ in the shallow soliton limit. We investigate
numerically deviations from this prediction, as the depth of the solitons is
increased, as well as when the strength of the spin-dependent interaction is
modified.
