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<p><b>Abstract</b>—The arrangement graph, denoted by <tmath>$A_{n,k}$</tmath>, is a generalization of the star graph. A recent work [<ref rid="bibl020914" type="bib">14</ref>] by Hsieh et al. showed that when <tmath>$n - k\geq 4$</tmath> and <tmath>$k = 2$</tmath> or <tmath>$n - k\geq 4 + \left\lceil {{ {k \over 2}}} \right\rceil $</tmath> and <tmath>$k\geq 3$</tmath>, <tmath>$A_{n,k}$</tmath> with <tmath>$k(n - k) - 2$</tmath> random edge faults can embed a Hamiltonian cycle. In this paper, we generalize Hsieh et al. work by embedding a Hamiltonian path between arbitrary two distinct vertices of the same <tmath>$A_{n,k}$</tmath>. To overcome the difficulty arising from random selection of the two end vertices, a new embedding method, based on a backtracking technique, is proposed. Our results can tolerate more edge faults than Hsieh et al. results as <tmath>$k\geq 7$</tmath> and <tmath>$7 \leq n - k \leq 3 + \left\lceil {{ {k \over 2}}} \right\rceil$</tmath>, 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 <tmath>$n - k = 2$</tmath>, which is still open in [<ref rid="bibl020914" type="bib">14</ref>].</p>
Arrangement graph, fault-tolerant embedding, graph-theoretic interconnection network, Hamiltonian cycle, Hamiltonian path, star graph.
Ray-Shang Lo, Gen-Huey Chen, "Embedding Hamiltonian Paths in Faulty Arrangement Graphs with the Backtracking Method", IEEE Transactions on Parallel & Distributed Systems, vol. 12, no. , pp. 209-222, February 2001, doi:10.1109/71.910874
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