  We study the cross-correlation between galaxies of different luminosities and
colors, using a sample selected from the SDSS Dr 4. Galaxies are divided into 6
samples according to luminosity, and each of these samples is divided into red
and blue subsamples. Projected auto-correlation and cross-correlation is
estimated for these subsample. At projected separations r_p > 1\mpch, all
correlation functions are roughly parallel, although the correlation amplitude
depends systematically on luminosity and color. On r_p < 1\mpch, the auto- and
cross-correlation functions of red galaxies are significantly enhanced relative
to the corresponding power laws obtained on larger scales. Such enhancement is
absent for blue galaxies and in the cross-correlation between red and blue
galaxies. We esimate the relative bias factor on scales r > 1\mpch for each
subsample using its auto-correlation function and cross-correlation functions.
The relative bias factors obtained from different methods are similar. For blue
galaxies the luminosity-dependence of the relative bias is strong over the
luminosity range probed (-23.0<M_r < -18.0),but for red galaxies the dependence
is weaker and becomes insignificant for luminosities below L^*. To examine
whether a significant stochastic/nonlinear component exists in the bias
relation, we study the ratio R_ij= W_{ii}W_{jj}/W_{ij}^2, where W_{ij} is the
projected correlation between subsample i and j. We find that the values of
R_ij are all consistent with 1 for all-all, red-red and blue-blue samples,
however significantly larger than 1 for red-blue samples. For faint red - faint
blue samples the values of R_{ij} are as high as ~ 2 on small scales r_p < 1
\mpch and decrease with increasing r_p.
