The Embedding of Hamiltonian Paths in Faulty Arrangement Graphs

The arrangement graph, which represents a family of scalable graphs, is a generalization of the star graph. There are two parameters, denoted by n and k, for the arrangement graph, where 1 1 ≤ ≤ − k n . An n k , -arrangement graph, which is denoted by An,k, has vertices corresponding to the arrangements of k numbers out of the set 1 2 , , , n . In this thesis, a fault-free Hamiltonian path is embedded between arbitrary two distinct vertices of a faulty arrangement graph. The fault tolerance is summarized as follows. (1) When 6 > − k n and 5 ≥ k , at most 2 ) ( − − k n k edge faults can be tolerated. There is an exception in which at most 3 ) ( − − k n k edge faults can be tolerated. (2) When 6 > − k n and { } 4 , 3 , 2 ∈ k or 6 = − k n and 2 ≥ k , at most ) 5 ( − − k n k edge faults can be tolerated. (3) When { } 5 , 4 , 3 ∈ − k n and 2 ≥ k , at most k edge faults can be tolerated. (4) When 2 = − k n and 2 ≥ k , at most 3 − k edge faults can be tolerated. (5) When 2 ≥ − k n and 1 = k , at most 4 − n edge faults can be tolerated. (6) When 5 ≥ − k n and 2 ≥ k , at most k k n k 3 ) ( − − vertex faults can be tolerated. (7) When { } 4 , 3 ∈ − k n and 2 ≥ k , at most 3 − n vertex faults can be tolerated. (8) When 2 = − k n and 2 ≥ k , at most 3 − k vertex faults can be tolerated. (9) When 3 ≥ − k n , at most k (vertex or edge) faults can be tolerated. (10) When 2 = − k n , at most 3 − k (vertex or edge) faults can be tolerated. A new embedding method, based on a backtracking technique, is proposed in this thesis. This thesis makes a significant improvement over a previous work by Hsieh et al.