| Summary: | 博士 === 國立交通大學 === 資訊科學系 === 92 === Interconnection networks have been an important research topic for parallel and distributed computer systems. We usually use a graph $G=(V,E)$ to represent a network''s topology where vertices represent processors and edges represent links between processors. There are a lot of interconnection network''s properties proposed in literature. Hamiltonian property is one of the important. Since processors or links may be failed sometimes, fault-tolerant hamiltonian properties are also concerned in many studies on network topologies.
A path is a {\it hamiltonian path} if its vertices are istinct and span $V$. A {\it cycle} is a path with at least three vertices such that the first vertex is the same as the last vertex. A cycle is a {\it hamiltonian cycle} if it traverses every vertex of $G$ exactly once. A graph is {\it hamiltonian} if it has a hamiltonian cycle. A hamiltonian graph $G$ is {\it $k$-fault hamiltonian} if $G-F$ remains hamiltonian for every $F \subset V(G) \cup E(G)$ with $|F| \le k$. A graph $G$ is {\it hamiltonian connected} if there exists a hamiltonian path joining any two vertices of $G$. A graph $G$ is {\it $k$-fault hamiltonian connected} if $G-F$ remains hamiltonian connected for every $F \subset V(G) \cup E(G)$ with $|F| \le k$.
In this thesis, we consider the fault hamiltonicity and the fault hamiltonian connectivity of the pancake graph $P_n$, $(n,k)$-star graph $S_{n,k}$, and arrangement graph $A_{n,k}$. We have the following results. (1) Assume that $F \subseteq V(P_n) \cup E(P_n)$. For $n \ge 4$, we prove that $P_n-F$ is
hamiltonian if $|F| \le (n-3)$ and $P_n-F$ is hamiltonian connected if $|F| \le (n-4)$ [25]. (2) Assume that $F \subset V(S_{n,k}) \cup E(S_{n,k})$. For $n -k \ge 2$, we prove that $S_{n,k} - F$ is hamiltonian if $|F| \le n-3$ and $S_{n,k} - F$ is hamiltonian connected if $|F| \le n-4$ [20]. (3) Assume that
$F\subseteq V(A_{n,k}) \cup E(A_{n,k})$. For $n-k \ge 2$. We prove that $A_{n,k} - F$ is hamiltonian if $|F| \le k(n-k) -2$ and $A_{n,k} - F$ is hamiltonian connected if $|F| \le k(n-k) -3$ [21]. Moreover, all the bounds are tight.
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