Paths and cycles are two network structures extensively used in distributed systems and parallel computation. In this thesis, we introduce some research issues on embedding paths and cycles into interconnection networks.
Firstly, we devote to investigating fault-tolerant hamiltonian connectedness of cycle com-position networks. In Chapter 2, we improve the result of Chen et al. [12] by showing that the cycle composition network Gh0,1,...,n−1,0i is super fault-tolerant hamiltonian even if it is composed of n 4-regular super fault-tolerant hamiltonian networks G0, . . . , Gn−1, provided that n ≥ 3. However, we conjecture that this result may not be true if the cycle composition network is constructed on the basis of cubic networks. Therefore such an improvement is of significance because only the remaining case for 3-regular graphs needs to be checked with brute force or by computer.
Secondly, we restrict our attention to the applicability of hamiltonian cycles on intercon-nection networks. Both Chapter 3 and Chapter 4 are dedicated to exploring how to embed mutually independent hamiltonian cycles onto interconnection networks. In Chapter 3 we show that the binary wrapped butterfly graph BF (n) has 4-mutually independent hamilto-nian cycles, beginning from any vertex, for n ≥ 3. In Chapter 4, we first prove that a faulty n-cube contains (n − 1 − f )-mutually independent hamiltonian cycles, beginning from any vertex, when not more than f ≤ n − 2 faulty edges may occur accidentally. However, we conjecture this result can be further refined; that is, we believe that a faulty n-cube really can be embedded with up to (n − f )-mutually independent hamiltonian cycles, beginning from any vertex, when f ≤ n − 2 faulty edges occur. Next, we also prove that a faulty star network Sn has (n − 1 − f )-mutually independent hamiltonian cycles, beginning from any vertex, if only f ≤ n − 2 faulty edges occur accidentally, provided that n ≥ 4.
Finally, we concern the problem of embedding various paths into conditionally faulty hypercubes. In advance, the fault diameter of the n-cube is computed in Chapter 5. In Chapter 6 we investigate the method for embedding paths of variable lengths into hypercubes, whose every node is assumed to be incident to at least two fault-free links. In Chapter 7 we show that a long path between any two nodes can be embedded into a conditionally faulty hypercube, whose every node is assumed to have at least two fault-free neighbors.
For the purpose of efficient data transmission, one of our future work is directed to explore the feasibility of finding as many mutually independent edge-disjoint hamiltonian cycles as possible. Another future research issue will be dedicated to generalizing the conditional-fault tolerance in the perspective on path embedding. Besides path and cycle embedding, tree embedding is also an important research topic widely addressed in the area of interconnec-tion networks. By definiinterconnec-tion, a tree is a connected graph without cycles. In practice, tree structures are very useful for network communication too. Hence, in our future work, we also plan to design efficient communication algorithms on the basis of tree embedding.
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