• Journal of KIISE:Computer Systems and Theory
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• v.30 no.7_8
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• pp.434-444
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• 2003

In this paper, we consider the hamiltonian properties of m$\times$n (m$\geq$2, n$\geq$3) mesh networks with two wraparound edges on the first row and last row, called M$_2$(m, n), in the presence of a faulty node or link. We prove that M$_2$(m, n) with odd n is hamiltonian-connected and 1-fault hamiltonian. In addition, we prove that M$_2$(m, n) with even n is strongly hamiltonian laceable and 1-vertex fault tolerant strongly hamiltonian laceable.

• The KIPS Transactions:PartA
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• v.11A no.3
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• pp.189-198
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• 2004

We consider the fault hamiltonian properties of m ${\times}$ n meshes with two wraparound links on the first row and the last row, denoted by M$_2$(m,n), (m$\geq$2, n$\geq$3). M$_2$(m,n), which is bipartite, with a single faulty link has a fault-free path of length mn-l(mn-2) between arbitrary two nodes if they both belong to the different(same) partite set. Compared with the previous works of P$_{m}$ ${\times}$C$_{n}$ , it also has these hamiltonian properties. Our result show that two additional wraparound links are sufficient for an m${\times}$n mesh to have such properties rather than m wraparound links. Also, M$_2$(m,n) is a spanning subgraph of many interconnection networks such as multidimensional meshes, recursive circulants, hypercubes, double loop networks, and k-ary n-cubcs. Thus, our results can be applied to discover fault-hamiltonicity of such interconnection networks. By applying hamiltonian properties of M$_2$(m,n) to 3-dimensional meshes, recursive circulants, and hypercubes, we obtain fault hamiltonian properties of these networks.

• Journal of KIISE:Computer Systems and Theory
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• v.32 no.8
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• pp.393-398
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• 2005

In interconnection networks, a Hamiltonian path has been utilized in many applications such as the implementation of linear array and multicasting. In this paper, we consider the Hamiltonian properties of mesh networks which are used as the topology of parallel machines. If a network is strongly Hamiltonian laceable, the network has the longest path joining arbitrary two nodes. We show that a two-dimensional mesh M(m, n) is strongly Hamiltonian laceabie, if $m{\geq}4,\;n{\geq}4(m{\geq}3,\;n{\geq}3\;respectively)$, and the number of nodes is even(odd respectively). A mesh is a spanning subgraph of many interconnection networks such as tori, hypercubes, k-ary n-cubes, and recursive circulants. Thus, our result can be applied to discover the fault-hamiltonicity of such networks.

• Journal of KIISE:Computer Systems and Theory
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• v.31 no.1_2
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• pp.86-94
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• 2004

In this paper, we show that $ G(2^m, 4), m{\geq}3$with at most m-2 faulty elements has a fault-free cycle of length 1 for every ${\leq}1{\leq}2^m-f_v$ is the number of faulty vertices. To achieve our purpose, we define a graph G to be k-fault hypohamiltonian-connected if for any set F of faulty elements, G- F has a fault-free path joining every pair of fault-free vertices whose length is shorter than a hamiltonian path by one, and then show that$ G(2^m, 4), m{\geq}3$ is m-3-fault hypohamiltonian-connected.

• The KIPS Transactions:PartA
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• v.18A no.6
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• pp.225-232
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• 2011

Restricted Hypercube-Like (RHL) graphs are a graph class that widely includes useful interconnection networks such as crossed cube, Mobius cube, Mcube, twisted cube, locally twisted cube, multiply twisted cube, and generalized twisted cube. In this paper, we show that for an m-dimensional RHL graph G, $m{\geq}4$, with an arbitrary faulty edge set $F{\subset}E(G)$, ${\mid}F{\mid}{\leq}m-2$, graph $G{\setminus}F$ has a hamiltonian path between any distinct two nodes s and t if dist(s, V(F))${\neq}1$ or dist(t, V(F))${\neq}1$. Graph $G{\setminus}F$ is the graph G whose faulty edges are removed. Set V(F) is the end vertex set of the edges in F and dist(v, V(F)) is the minimum distance between vertex v and the vertices in V(F).