| Summary: | 碩士 === 國立暨南國際大學 === 資訊工程學系 === 99 === The application of gray-code is one of the ways in decreasing resource consumption and increasing precision for 3D scanning. The bit conversion of gray-code
usually concentrates on some parts. That will cause some problems, such as the
interference strength of an image or the error of decoding. If we can average the
bit conversion into all different bits, then there will be more efficient. The problem of finding an average bit conversion of gray-code is equivalent to the problem of finding a balanced Hamiltonian cycle in a hypercube. We define a balanced Hamiltonian cycle in a graph G, whose edge set can be partitioned into n dimensions, by a Hamiltonian cycle C in G such that for the set of all i-dimensional edge Ei(C) in E(C), ||Ei(C)j － |Ej(C)|| = 1, for i ≠ j. In this thesis, we prove that there is no balanced Hamiltonian cycle in hypercube Qn when n ≠ 2k for any positive integer k, and there is a balanced Hamiltonian cycle in hypercube Qn for n = 2k, when k = 1; 2 and 3. Furthermore, for any positive integer m, n ≥ 3 we propose a method to find a balanced Hamiltonian cycle on Cn × Cn (also called torus graphs T(n, n)). Next we find a balanced Hamiltonian cycle on
Cn × Kn when n is odd. When n is even we can get better result that to find a
balanced Hamiltonian cycle on Cn × Cmn. Finally, we propose two algorithms to find a balanced Hamiltonian cycle on C3 × Km and C4 × Cm.
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