Cayley graphs of groups $\mathbb Z^4$, $\mathbb Z^5$ and $\mathbb Z^6$ which are limit graphs for the finite graphs of minimal valency for vertex-primitive groups of automorphisms

Infinite connected graph $\Gamma$ is called a limit graph for the set $X$ of finite vertex-primitive graphs, if each ball of $\Gamma$ is isomorphic to a ball of some graph in $X$. A finite graph $\Gamma$ is called a graph of minimal degree for a vertex-primitive group $G\le\operatorname{Aut}(\Gamma)$, if the condition $\deg(\Gamma)\le\deg(\Delta)$ is hold for any graph $\Delta$ such that $V(\Delta)=V(\Gamma)$ and $G\le\operatorname{Aut}(\Delta)$. It is obtained the description of Cayley graphs of groups $\mathbb Z^4$, $\mathbb Z^5$ and $\mathbb Z^6$ which are limit graphs for the finite graphs of minimal degree for vertex-primitive groups of automorphisms.