Fault-Tolerant Cycle Embedding in Cartesian Product Graphs: Edge-Pancyclicity and Edge-Bipancyclicity with Faulty Edges

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 $|V(G)|$ (a cycle of every length from $r$ to $|V(G)|$), 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 $G\times H$ are 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.