Embedding Hamiltonian Paths in Faulty Arrangement Graphs with the Backtracking Method

• Authors: Ray-Shang Lo;Gen-Huey Chen
• Venue: IEEE Transactions on Parallel and Distributed Systems
• Year: 2001

The arrangement graph, denoted by $A_{n,k}$, is a generalization of the star graph. A recent work [14] by Hsieh et al. showed that when $n - k\geq 4$ and $k = 2$ or $n - k\geq 4 + \left\lceil {{ {k \over 2}}} \right\rceil $ and $k\geq 3$, $A_{n,k}$ with $k(n - k) - 2$ random edge faults can embed a Hamiltonian cycle. In this paper, we generalize Hsieh et al. work by embedding a Hamiltonian path between arbitrary two distinct vertices of the same $A_{n,k}$. To overcome the difficulty arising from random selection of the two end vertices, a new embedding method, based on a backtracking technique, is proposed. Our results can tolerate more edge faults than Hsieh et al. results as $k\geq 7$ and $7 \leq n - k \leq 3 + \left\lceil {{ {k \over 2}}} \right\rceil$, although embedding a Hamiltonian path between arbitrary two distinct vertices is more difficult than embedding a Hamiltonian cycle. Besides, our results can deal with the situation of $n - k = 2$, which is still open in [14].