A graph G is called k-edge-fault edge-bipancyclic (k-edge-fault edge-r-pancyclic) if after deleting k edges from G, every edge in the resulting graph lies in a cycle of every even length from 4 to IV (G)I (a cycle of every length from r to IV(G)I), inclusively. In this paper, given two graphs G and H, which satisfy some specific properties, the edge-fault edge-bipancyclicity and edge-fault edge-r-pancyclicity (r is decided on the properties of G and H) of Cartesian product graphs G x Hare efficiently evaluated. The obtained results are applied to two multiprocessor systems, the nearest neighbor mesh hypercubes and generalized hypercubes, both of which belong to Cartesian product graphs.

A graph G is conditional k-edge-fault Hamiltonian if it remains Hamiltonian after deleting at most k edges and each vertex incident to at least two nonfaulty edges. A graph G is k-edge-fault Hamiltonian-connected if it remains Hamiltonian-connected after deleting at most k edges. This study shows that the conditional edge-fault Hamiltonicity of the Cartesian product network G x H can be efficiently evaluated given two graphs G and H that are edge-fault Hamilton-connected and conditional edge-fault Hamiltonian. This study uses the result to evaluate the conditional edge-fault Hamiltonicity of two multiprocessor systems, the generalized hypercubes and the nearest neighbor mesh hypercubes, both of which belong to Cartesian product networks.