  Galaxy assembly bias, the correlation between galaxy properties and halo
properties at fixed halo mass, could be an important ingredient in halo-based
modelling of galaxy clustering. We investigate the central galaxy assembly bias
by studying the relation between various galaxy and halo properties in the
Illustris hydrodynamic galaxy formation simulation. Galaxy stellar mass
$\Mstar$ is found to have a tighter correlation with peak maximum halo circular
velocity $\Vp$ than with halo mass $\Mh$. Once the correlation with $\Vp$ is
accounted for, $\Mstar$ has nearly no dependence on any other halo assembly
variables. The correlations between galaxy properties related to star formation
history and halo assembly properties also show a cleaner form as a function of
$\Vp$ than as a function of $\Mh$, with the main correlation being with halo
formation time and to a less extent halo concentration. Based on the
galaxy-halo relation, we present a simple model to relate the bias factors of a
central galaxy sample and the corresponding halo sample, both selected based on
assembly-related properties. It is found that they are connected by the
correlation coefficient of the galaxy and halo properties used to define the
two samples, which provides a reasonable description for the samples in the
simulation and suggests a simple prescription to incorporate galaxy assembly
bias into the halo model. By applying the model to the local galaxy clustering
measurements in Lin et al. (2016), we infer that the correlation between star
formation history or specific star formation rate and halo formation time is
consistent with being weak.
