Hamiltonian Properties of Toroidal Zero Divisor Graphs

The zero divisor graph of a commutative ring $R$ is formed by taking the nonzero zero divisors of $R$ as the vertices and connecting two vertices exactly when the corresponding product of the two elements is zero. We will show that all 44 planar zero divisor graphs are subgraphs of planar graphs with a Hamiltonian cycle and that all 46 genus one zero divisor graphs are subgraphs of toroidal graphs with a Hamiltonian cycle.