| Summary: | 博士 === 國立交通大學 === 資訊科學系 === 87 === Designing an optimal k-fault-tolerant network for token rings is
equivalent to constructing an optimal k-hamiltonian graph, where
k is a positive integer and corresponds to the number of faults.
A graph G = (V, E) is k-hamiltonian if G - (V'' - E'')$ is hamiltonian
for arbitrary V'' is a subset of V, E'' is a subset of E with |V''| + |E''| <= k.
A k-hamiltonian graph G* is optimal if it contains the fewest
edges among all k-hamiltonian graphs with the same number of
nodes as G*. In this thesis, we propose two families of optimal
1-hamiltonian graphs {G(k) | k is a positive integer.} and {CT(s) | s
is a positive integer.}. The graph G(k) has diameter O(\sqrt{n}) and the
graph CT(s) has diameter 2 log_2 n -O(1) where n is the number
of nodes. We also prove that CT(s) is optimal hamiltonian-connected.
Furthermore, we propose a general construction scheme for optimal
1-hamiltonian graphs. Applying this scheme, we can construct some
previous optimal 1-hamiltonian graphs and the graphs G(k) and CT(s).
We then focus our attention on optimal k-hamiltonian graphs with
k >= 1. We construct a family of k-hamiltonian graphs with diameter
2 log_{k+1} n - O(1), that is 2 times Moore bound. However, the graphs
of this family are not symmetric. We propose another family of
k-hamiltonian graphs which is node symmetric with
(k+3) * (k+3)! nodes, degree k+2, and diameter 2
* [3(k+2)/2]. We also prove that G is a (k+2)-regular k-hamiltonian
graph then G \times K_2 is (k+3)-regular (k+1)-hamiltonian.
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