  Using the first principles quantum Monte Carlo (QMC) calculations, we
investigate the previously established universal scaling between the N\'eel
temperature $T_N$ and the staggered magnetization density $M_s$ of
three-dimensional (3D) dimerized quantum antiferromagnets. Particularly, the
calculations are done on both the stacked honeycomb and the cubic lattices. In
addition to simulating models with two types of antiferromagnetic couplings
(bonds) like those examined in earlier studies, here a tunable parameter
controlling the strength of third type of bond is introduced. Interestingly,
while the data of models with two types of bonds obtained here fall on top of
the universal scaling curves determined previously, the effects due to
microscopic details do appear. Moreover, the most striking result suggested in
our study is that with the presence of three kinds of bonds in the investigated
models, the considered scaling relations between $T_N$ and $M_s$ can be
classified by the coordinate number of the underlying lattice geometries. The
findings presented here broaden the applicability of the associated
classification schemes formerly discovered. In particular, these results are
not only interesting from a theoretical point of view, but also can serve as
useful guidelines for the relevant experiments.
