Super Fault-Tolerant Hamiltonicity of Product Networks

A graph G is called k-fault Hamiltonian (resp. Hamiltonian-connected) if after deleting at most k vertices and/or edges from G, the resulting graph remains Hamiltonian (resp. Hamiltonian-connected). Let δ(G) be the minimum degree of G. Given a (δ(G) - 2)-fault Hamiltonian/ (δ(G) - 3)-fault Hamiltonian-connected graph G and a (δ(H) - 2)-fault Hamiltonian/(δ(H) - 3)-fault Hamiltonian-connected graph H, we show that the Cartesian product network G × H is (δ(G) + δ(H) - 2)-fault Hamiltonian and (δ(G)+δ(H) - 3)-fault Hamiltonian-connected. We then apply our result to determine the fault-tolerant hamiltonicity and Hamiltonian-connectivity of two multiprocessor systems, namely the generalized hypercube and the nearest neighbor mesh hypercube, both of which belong to Cartesian product networks. We also demonstrate that our results are worst-case optimal with respect to the number of faults tolerated.