  We consider a nearest-neighbor four state hard-core (HC) model on the
homogeneous Cayley tree of order $k$. The Hamiltonian of the model is
considered on a set of "admissible" configurations. Admissibility is specified
through a graph with four vertices. We first exhibit conditions (on the graph
and on the parameters) under which the model has a unique Gibbs measure. Next
we turn to some specific cases. Namely, first we study, in the case of a
particular graph (the diamond), translation-invariant and periodic Gibbs
measures. We provide in both cases the equations of the transition lines
separating uniqueness from non--uniqueness regimes. Finally the same is done
for "fertile" graphs, the so--called stick, gun, and key (here only translation
invariant states are taken into account).
