Describing ring varieties in which all finite rings have Hamiltonian zero-divisor graphs

The zero-divisor graph of an associative ring $R$ is a graph such that its vertices are all nonzero (one-sided and two-sided) zero-divisors, and moreover, two distinct vertices $x$ and $y$ are joined by an edge iff $xy=0$ or $yx=0$. We give a complete description of varieties of associative rings in which all finite rings have Hamiltonian zero-divisor graphs. Also finite decomposable rings with unity having Hamiltonian zero-divisor graphs are characterized.