  As a model for the B 2pz bands of MgB$_2$, a half-filled Hubbard model on a
honeycomb lattice is studied, and a possibility of an antiferromagnetism (AF)
at the ground state is examined by Monte Carlo calculations as well as a random
phase approximation method. On a single-layer honeycomb lattice, a paramagnetic
to AF phase transition occurs at a finite value of the repulsion energy $U_c/t
\simeq 3.6$, due to the loss of the density of states at the fermi level. On a
layered honeycomb lattice, however, inter-layer hopping term creates a perfect
nesting of the fermi surface, and leads the AF with $U_c=0$. From these facts,
we discuss that MgB$_2$ is possibly a nearly-AF metal, where the AF of the 2pz
bands is destroyed by carrier doping.
