| Summary: | 碩士 === 國立暨南國際大學 === 資訊工程學系 === 103 === The dimension-balanced cycle problem is a quiet new topic of graph theorem. This problem is generated by the 3-D scanning problem. The design for
network-on-chip (NoC) problem is also an important issue recently. One of the most famous NoC is toroidal mesh graph actually. When there are congestions
occur, the performance of the network will be reduced. Hence, for NoC designing problem, how to choose a suitable routing path to prevent the congestion occur and make a good performance is an important problem. Therefore we study the dimension-balanced cycle on toroidal mesh graph for avoiding the situation as mentioned above.
Given a graph G = (V,E), whose edge set can be partitioned into k dimensions, for positive integer k. The set of all i-dimensional edge of G, which is a subset of E(G), is denoted by Ei. For any cycle C on G, the set of all i-dimensional edge of C, which is a subset of E(C), is denoted by Ei(C). If
||Ei(C)| - |Ej(C)|| ≤ 1 for 1 ≤ i < j ≤ k, C is called a dimension-balanced cycle or DBC for short.
We combine the concept of dimension-balanced cycle and Hamiltonian cycle, pancyclic, m-pancyclic, vertex-pancyclic and bipancyclic. Let C be a Hamiltonian
cycle (C contains every vertex of graph G exactly once.) on graph G. For any positive integers i, j, if C satisfies that ||Ei(C)| - |Ej(C)|| ≤ 1, then C is called a dimension-balanced Hamiltonian cycle. And if a graph G contains dimension-balanced cycles of every length between 3 up to |V(G)| (between m up to |V(G)|, respectively), G is said dimension-balanced pancyclic (dimension-balanced m-pancyclic, respectively). Besides, a graph G is dimension-balanced
vertex-pancyclic (dimension-balanced m-vertex-pancyclic, respectively) if every vertex lies on dimension-balanced cycle of every length between 3 to |V(G)| (m to |V(G)|, respectively). Furthermore, a bipartite graph B is called dimension-balanced bipancyclic if B contains a dimension-balanced cycle of every possible even length.
In this thesis, for m, n ≥ 3, we show that there exists a dimension-balanced Hamiltonian cycle on Toroidal Mesh graph Tm,n except for mn mod 4 = 2. Furthermore, we prove that Tm,n is neither DB pancyclic nor DB bipancyclic. We also prove that Tm,n is DB 2max{m, n}-pancyclic and DB 2max{m, n}-vertex pancyclic for m, n ≥ 5 and mn mod 4 = 3.
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