  We generalize the previous exact results of the Nagaoka-type itinerant
ferromagnetic states in a three dimensional $t_{2g}$-orbital system to allow
for multiple holes. The system is a simple cubic lattice with each site
possessing $d_{xy}$, $d_{yz}$, and $d_{xz}$ orbitals, which allow
two-dimensional hopping within each orbital plane. In the strong coupling limit
of $U\to \infty$, the orbital-generalized Nagaoka ferromagnetic states are
proved degenerate with the ground state in the thermodynamic limit when the
hole number per orbital layer scales slower than $L^{\frac{1}{2}}$. This result
is valid for arbitrary values of the ferromagnetic Hund's coupling $J>0$ and
inter-orbital repulsion $V\ge 0$. The stability of the Nagaoka-type state at
finite electron densities with respect to a single spin-flip is investigated.
These results provide helpful guidance for studying the mechanism of itinerant
ferromagnetism for the $t_{2g}$-orbital materials.
