| Summary: | 碩士 === 國立臺灣科技大學 === 資訊管理系 === 99 === The hypercube has received much attention since it was proposed and many related research results have been presented. Moreover, many variant structures of hypercube [1, 2, 4] have also been proposed. This is because hypercube-like interconnection networks have regular structures and good topological properties, such as smaller diameters, simple routing and broadcasting algorithms, good embedding and fault-tolerant properties, … etc. Furthermore, constructing multiple spanning trees is an importance issue on these structures, where spanning trees can be divided into edge-disjoint and vertex-disjoint (independent) spanning trees.
In fact, Zehavi and Itai [21] conjectured that for any k-connected graph G and each vertex r of G, there exist k disjoint spanning trees of G rooted at r. This conjecture has been confirmed only for k-connected graphs with k≦4, and it is still open for arbitrary k-connected graphs when k≧5.
The augmented cube is a variant of the hypercube, which has smaller diameter and larger degree than those of the hypercube. It is more difficult to construct spanning trees in the augmented cubes than that in the hypercube due to its large degree. We explore several properties in the construction of spanning trees in the augmented cubes, which is utilized to adjust parents of some nodes in the spanning trees in order to remain all spanning trees disjoint. In this thesis, using the conversion of gray codes to binary codes and new methods of parent adjustment, we present an algorithm for constructing (2n－1) edge-disjoint spanning trees on the n-dimensional augmented cubes.
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