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ABSTRACT
<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>
INDEX TERMS
Arrangement graph, fault-tolerant embedding, graph-theoretic interconnection network, Hamiltonian cycle, Hamiltonian path, star graph.
CITATION

R. Lo and G. Chen, "Embedding Hamiltonian Paths in Faulty Arrangement Graphs with the Backtracking Method," in IEEE Transactions on Parallel & Distributed Systems, vol. 12, no. , pp. 209-222, 2001.
doi:10.1109/71.910874
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