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.