超立方體與其變形的覆蓋容器

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張仲浩(Chung-Haw Chang) 查詢紙本館藏

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論文名稱

超立方體與其變形的覆蓋容器
(Spanning Container on Variations of Hypercubes)

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★ On the Spectrum of Trees

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摘要(中)

假設圖形 G 的連接度為κ(G), 當k ≦κ(G) 時, 根據 Menger’s 定理對圖
形 G 中任意兩個相異點 u 與 v ，均存在 k 條連通u 與 v 的不重複邊與點(除
了u 與 v 外) 之路徑。一個 k- container C(u,v)是指一個圖形 G 擁有k 條連通 u
與 v 的不重複邊與點(除了u 與 v 外)之路徑的集合。若圖 G 裏面的每一個點
都包含在C(u,v)中，我們稱此 k- container 為一個 k*- container C(u,v)。若圖形 G
中任意相異兩點都存在 k*- container , 我們稱此圖形 G 為k*-連通。若對任何的
k，1≦ k ≦ κ(G)，圖形 G 為 k*-連通, 則我們稱此圖 G 是超級覆蓋連通
(super spanning connected)。
然而，當我們研究二分圖G 的 k*-連通時，我們需要做一些修正。一個二分
圖G 中，任意兩個不同顏色的點 u 與 v 之間若存在一個 k*-container，則我們
稱其為 k*- laceable。一個k 正則二分圖G ,若對所有的 1≦ k ≦ κ(G) 來說，
它皆為 k*- laceable ，則我們稱此二分圖G 為超級覆蓋laceable (super spanning
laceable)。一個 k*- container C(u,v) = { P1, … , Pk }，若對所有的
1 ≦ i, j ≦ k , | |Pi|-|Pj| | ≦ 2 都成立，則我們稱此 k*- container C(u,v)是
equitable。
超立方體是最有名的網路圖之ㄧ。在本論文中，我們將討論超立方體，摺
疊型超立方體及加強型超立方體的延伸連通性，超級延伸連通及相關的問題。

摘要(英)

Let G be a graph with connectivity κ(G). It follows from Menger’’s Theorem that
there are k vertex-disjoint paths joining any two distinct vertices when k ≦κ(G).
A k -container C ( u, v) of a graph G is a set of k vertex-disjoint paths between u and v.
A k-container is a k*-container if it contains all vertices of G . A graph G is
k*-connected if there exists a k*-container between any two vertices. A graph G is
super spanning connected if G is k*-connected for every 1 ≦ k ≦κ(G).
However, we need some modification as we study bipartite k-connected graphs.
A bipartite graph G is k*-laceable if there exists a k*-container between any two
vertices from different partite sets. A bipartite graph G is super spanning laceable if G
is k*-laceable for 1 ≦ k ≦ κ(G). A k*-container C ( u, v) ={ P1, … , Pk } is
equitable if | | Pi | - | Pj | | ≦ 2, 1 ≦ i , j ≦ k.
The hypercube Qn is one of the most popular networks. In this thesis, we will
discuss that the spanning connectivity, the spanning laceability, and related problems
of hypercube Qn, folded hypercube FQn, and enhanced hypercube Qn,m.

關鍵字(中)

★ 超級延伸連通
★ 延伸容器
★ 容器
★ 摺疊型超立方體
★ 加強型超立方體
★ 超級延伸可蕾斯化
★ 超立方體

關鍵字(英)

★ spanning container
★ container
★ hypercube
★ super spanning laceability
★ super spanning connected
★ enhanced hypercube
★ folded hypercube

論文目次

Contents
Abstract（in Chinese） i
Abstract（in English） ii
Contents iii
List of Figures v
1 Introduction 1
1.1 Definitions and notation . . . . . . . . . . . . . . . . . . . . 1
1.2 Undiected Cayley graphs . . . . . . . . . . . . . . . . . . 3
2 Container and connectivity 11
2.1 Container 11
2.2 Container in ypercubes . . . . . . . . . . . . . . . . . . . 12
2.3 Container in folded hypercubes . . . . . . . . . . . . . . . . . 16
2.4 Container in enhanced hypercubes . . . . . . . . . . . . . . . . 18
3 The spanning container and spanning connectivity 22
3.1 The spanning container . . . . . . . . . . . . . . . . . . . 22
3.2 The super spanning property of hypercubes . . . . . . . . . . . 25
3.3 The spanning connectivity of folded hypercubes . . . . . . . . . 35
3.3.1 k^*-fan spanning property of hypercubes. . . . . . . . . 35
3.3.2 Construction of (n + 1) ^*-containers . . . . . . . . . 46
3.4 The spanning connectivity of enhanced hypercubes . . . . . . . . 53
3.4.1 The super spanning properties of enhanced hypercubes . . . . . 54
3.4.2 Construction of (n + 1)^*-containers . . . . . . . . . . . .. 62
4 Equitable spanning laceability of hypercubes 71
4.1 Preliminaries and known results . .. .. .. . 71
4.2 Extended k-parallel spanning property of hypercubes . . . . . . 73
4.3 Equitable laceability of hypercubes. . . . . . . . . 87
5 Discussion and conclusion 93
Bibliography 94

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指導教授

黃華民(Hua-Min Huang)

審核日期

2006-5-19

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