Processor (vertex) faults and link (edge) faults may happen when a network is used, and it is meaningful to consider networks (graphs) with faulty processors and/or links. A k-regular Hamiltonian and Hamiltonian connected graph G is optimal fault-tolerant Hamiltonian and Hamiltonian connected if G remains Hamiltonian after removing at most k − 2 vertices and/or edges and remains Hamiltonian connected after removing at most k − 3 vertices and/or edges. In this paper, we investigate in constructing optimal fault-tolerant Hamiltonian and optimal fault-tolerant Hamiltonian connected graphs. Therefore, some of the generalized hypercubes, twistedcubes, crossed-cubes, and Möbius cubes are optimal fault-tolerant Hamiltonian and optimal fault-tolerant Hamiltonian connected.

No abstract

A fe-regular hamiltonian and hamiltonian connected graph G is optimal fault-tolerant hamiltonian and hamiltonian connected if G remains hamiltonian after removing at most k-2 nodes and/or edges and remains hamiltonian connected after removing at most fe -3 nodes and/or edges. In this paper, we investigate a construction scheme to construct optimal fault-tolerant hamiltonian and hamiltonian connected graphs. Hence, some of the generalized hypercubes. Twisted-cubes, Crossed-cubes, and Mobius cubes are optimal fault-tolerant hamiltonian and optimal fault-tolerant hamiltonian connected.