Gibbs Measures for the $$\mathrm{HC}$$-Model in the Case of a Graph of Type “Key” on a Cayley Tree

In this paper, we study one of the fertile Hard-Core ( $$\mathrm{HC}$$ ) models with four states on a Cayley tree. It is known that there are three types of fertile $$\mathrm{HC}$$ -models: “stick,” “key,” and “generalized key.” We consider the “key” case and, in this case, the uniqueness of a translation-invariant Gibbs measure on the Cayley tree of orders four, five, and six is proved. We also find conditions for the nonuniqueness of such measures on the Cayley tree of order seven. In addition, periodic Gibbs measures are investigated. It is shown that, under some conditions, there exist two-periodic Gibbs measures, different from the translation-invariant ones, on the Cayley tree of order three.