| Summary: | 碩士 === 國立暨南國際大學 === 資訊工程學系 === 107 === The dimension-balanced cycle (DBC for short) problem is a quite new topic in graph theory. It is developed from the problem of 3D stereogram reconstruction. For a graph G, let the edge sets of G be partitioned into k dimensions, that is, E(G) = E1(G) ∪ . . . ∪ Ek(G), and for any i, j ∈ {1, 2, 3, . . . , k} = K, Ei(G) ∩ Ej(G) = ∅. Then we call k ∈ N the dimension of G, and Ei(G) is called the edge set of the i-th dimension for any i ∈ K. For any cycle C in G, let Ei(C) = E(C) ∩ Ei(G), when i ∈ K. For any two positive integers i, j in K, and C is a cycle in G, if ||Ei(C)| − |Ej(C)|| ≤ 1, then we call C a DBC on G. If C is also a Hamiltonian cycle in G, C is called a Hamiltonian DBC (DBH for short). The DBH problem on graph G is to discuss whether there is a DBH on the graph G. In this thesis, we discuss the DBH problem on the 3-dimensional toroidal mesh graph, Cm × Cn × Cr. We prove that there is no DBH for some special condition of m, n, and r; and for other cases, there exists a DBH on the 3-dimensional toroidal mesh graph Cm × Cn × Cr.
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