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Issue No.03 - March (1999 vol.10)
pp: 223-237
<p><b>Abstract</b>—The arrangement graph A<sub><it>n</it>,<it>k</it></sub>, which is a generalization of the star graph (<it>n</it>−<it>k</it> = 1), presents more flexibility than the star graph in adjusting the major design parameters: number of nodes, degree, and diameter. Previously, the arrangement graph has proved Hamiltonian. In this paper, we further show that the arrangement graph remains Hamiltonian even if it is faulty. Let |<it>F</it><sub><it>e</it></sub>| and |<it>F</it><sub><it>v</it></sub>| denote the numbers of edge faults and vertex faults, respectively. We show that <it>A</it><sub><it>n</it>,<it>k</it></sub> is Hamiltonian when 1) (<it>k</it> = 2 and <it>n</it>−<it>k</it>≥ 4, or <it>k</it>≥ 3 and <tmath>$n-k\ge 4+\left\lceil {{\textstyle{k \over 2}}} \right\rceil$</tmath>), and |<it>F</it><sub><it>e</it></sub>| ≤<it>k</it>(<it>n</it>−<it>k</it>) − 2, or 2) <it>k</it>≥ 2, <tmath>$n-k\ge 2+\left\lceil {{\textstyle{k \over 2}}} \right\rceil,$</tmath> and |<it>F</it><sub><it>e</it></sub>| ≤<it>k</it>(<it>n</it>−<it>k</it>− 3) − 1, or 3) <it>k</it>≥ 2, <it>n</it>−<it>k</it>≥ 3, and |<it>F</it><sub><it>e</it></sub>| ≤<it>k</it>, or 4) <it>n</it>−<it>k</it>≥ 3 and |<it>F</it><sub><it>v</it></sub>| ≤<it>n</it>− 3, or 5) <it>n</it>−<it>k</it>≥ 3 and |<it>F</it><sub><it>v</it></sub>| + |<it>F</it><sub><it>e</it></sub>| ≤<it>k</it>. Besides, for <it>A</it><sub><it>n</it>,<it>k</it></sub> with <it>n</it>−<it>k</it> = 2, we construct a cycle of length at least 1) <tmath>${\textstyle{{n!} \over {\left( {n-k} \right)!}}}-2$</tmath> if |<it>F</it><sub><it>e</it></sub>| ≤<it>k</it>− 1, or 2) <tmath>${\textstyle{{n!} \over {\left( {n-k} \right)! }}}-\left| {F_v} \right|-2\left( {k-1} \right)$</tmath> if |<it>F</it><sub><it>v</it></sub>| ≤<it>k</it>− 1, or 3) <tmath>${\textstyle{{n!} \over {\left( {n-k} \right)! }}}-\left| {F_v} \right|-2\left( {k-1} \right)$</tmath> if |<it>F</it><sub><it>e</it></sub>| ≤ + |<it>F</it><sub><it>v</it></sub>| ≤<it>k</it>− 1, where <tmath>${\textstyle{{n!} \over {\left( {n-k} \right)!}}}$</tmath> is the number of nodes in <it>A</it><sub><it>n</it>,<it>k</it></sub>.</p>
Arrangement graph, fault-tolerant embedding, Hamiltonian cycle, graph-theoretic interconnection network, star graph.
Sun-Yuan Hsieh, Gen-Huey Chen, Chin-Wen Ho, "Fault-Free Hamiltonian Cycles in Faulty Arrangement Graphs", IEEE Transactions on Parallel & Distributed Systems, vol.10, no. 3, pp. 223-237, March 1999, doi:10.1109/71.755822
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