  In this paper is studied HC-models on a Cayley tree. two states HC-model on a
Cayley tree and Under some conditions on parameters of the two state HC-model
we prove existence exactly two of the weakly periodic (non periodic) Gibbs
measures. Furthermore, it considered translation-invariant Gibbs measures for
fertile four state HC-models and for one of them it proved the non-uniqueness
of the translation-invariant Gibbs measure.
