<pre>
一個含有 v 個點的完全圖 Kv是指含有 v 個點且任二點都有邊相連的圖，又
稱為 v 階完全圖。 一個圖頂點集合為 V 可以分成兩個互斥的集合 V1 與
V2，且 V1中的每一點都與 V2中的每一點有邊相連，則稱此圖為一個完全二
分圖。一個圖的頂點集合 V 可以分成 m 個兩兩互斥的集合 V1,V2,··· ,Vm，
當 i neq j 時, Vi 中的每一點都與 Vj 中的每一點有邊相連，則稱此圖為完全
m 分圖。 當 V1,V2,··· ,Vm 中元素的個數都為 n 時，則稱此圖為完全均分圖
Km(n)。 一個 k-太陽圖 S(Ck) 是將一個 k-迴圈上的每一點分別向外連接一個
懸掛邊，即另一端點度數為 1 的點，所成的圖。

一個圖 G 的分割是圖 G 的子圖 H1,H2,··· ,Ht 所成的集合 H，其中
E(H1)∪E(H2)∪···∪E(Ht) = E(G) 且 對於所有 i 6= j，E(Hi)∩E(Hj) =
emptyset。若對於每一個 i = 1,2,··· ,t， Hi皆同構於 H，則我們說 G 有一個 
H-分割。一個 v 階的 k-太陽圖系統是指由 v 階的完全圖 Kv分割成 k-太陽圖後，
這些 k- 太陽圖所成的集合。 存在 v 階 k-太陽圖系統的 v 所成的集合，稱為
k-太陽圖系統的譜 Spec(k) 。

本論文主要包括二個部份，一個是在完全圖中建構 k-太陽圖系統, 另一個
是證明在完全均分圖中有 k-太陽圖-分割。

在第三章中，當k = 6,10,14,2t(t geq 2)時，我們得到了k-太陽圖系統的譜
如下：
(1) Spec(2t) = {v|v ≡ 0,1 (mod 2t+2)} 其中 t geq 2.
(2) Spec(6) = {v|v ≡ 0,1,9,16 (mod 24)}.
(3) Spec(10) = {v|v ≡ 0,1,16,25 (mod 40)}.
(4) Spec(14) = {v|v ≡ 0,1,8,49 (mod 56)}.
並且對於階數大於 4k 時，我們建構出奇數階的循環 k-太陽圖系統與偶數階
的1-旋轉 k-太陽圖系統。

在第四章中，我們將焦點放在完全均分圖是否有 k-太陽圖-分割。當 k
為偶數且 n ≡ 0 (mod 2k) 時，我們證明一個完全二分圖 Kn,n 有 2k-太陽
圖-分割；而當(m,n) 滿足 mn geq 8 且 m(m - 1)n^2≡ 0 (mod 16)時， 除了
(m,n) = (4,2) 之外，我們則證明了完全均分圖 Km(n)有 4-太陽圖-分割。
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<pre>
A complete graph with v vertices, denoted by Kv, is a simple graph whose
vertices are mutually adjacent. A complete bipartite graph is a graph G =
(V,E) where V can be divided into two disjoint sets V1 and V2 and E 
contains all edges connecting every vertex in V1with all vertices in V2. 
If |V1| = m and |V2| = n, then G can be denoted as Km,n. A complete 
m-partite graph G = (V,E) is a graph such that the vertex set V can be 
partitioned into m parts, V1,V2,··· ,Vm, and E contains all edges which 
connect all vertices belong to different parts. If |Vi| = nifor each i = 
1,2,··· ,m, then G can be denoted as Kn1,n2,···,nm. If n1= n2= ··· = nm= 
n, this graph is called a complete equipartite graph with m parts of size 
n, and denoted by Km(n). A k-sun graph S(Ck) is obtained from the cycle 
of length k, Ck, by adding a pendant edge to each vertex of Ck.

A decomposition of a graph G is a collection H = {H1,H2,··· ,Ht} of
subgraphs of G such that E(H1) ∪ E(H2) ∪ ··· ∪ E(Ht) = E(G) and
E(Hi)∩E(Hj) = emptyset for each i neq j. If Hi is isomorphic to a subgraph
H of G for each i = 1,2,··· ,t, then we say that G has an H-decomposition. 
A k-sun system of order v is a decomposition of the complete graph Kv into 
k-sun graphs. The set of values of v for which there exists a k-sun system 
of order v is called the spectrum of a k-sun system, denoted by Spec(k).
This dissertation includes two parts. One is about constructing a k-sun
system of order v and another is about proving that complete equipartite
graphs have k-sun decompositions.

In chapter 3, when k = 6,10,14, and 2tfor t geq 2, we obtain the spectrum 
of k-sun systems as follows.
(1) Spec(2t) = {v|v ≡ 0,1 (mod 2t+2)} for t geq 2,
(2) Spec(6) = {v|v ≡ 0,1,9,16 (mod 24)},
(3) Spec(10) = {v|v ≡ 0,1,16,25 (mod 40)}, and
(4) Spec(14) = {v|v ≡ 0,1,8,49 (mod 56)}.
We give the construction of cyclic k-sun system of odd order and 
1-rotational k-sun system of even order when the order is greater than 4k.

In chapter 4, we give the construction of 2k-sun decomposition of Kn,nas k
is even and n ≡ 0 (mod 2k) and construct 4-sun decomposition of Km(n)for
mn geq 8 and m(m - 1)n^2≡ 0 (mod 16) except (m,n) = (4,2).
</pre>

<pre>
Contents 
1 Introduction                                                           1 
2 Preliminaries                                                          5 
  2.1 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 
  2.2 Block Designs . . . . . . . . . . . . . . . . . . . . . . . . . . .8 
  2.3 Graph Decompositions . . . . . . . . . . . . . . . . . . . . . . . 10 
3 2k-sun system of order v                                               14 
  3.1 The Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 
  3.2 2k-sun system . . . . . . . . . . . . . . . . . . . . . . . . . . .17 
  3.3 6-sun system . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 
  3.4 10-sun system . . . . . . . . . . . . . . . . . . . . . . . . . . .26 
  3.5 14-sun system . . . . . . . . . . . . . . . . . . . . . . . . . . .31 
4 Decomposing complete equipartite graph into 2k-sun graphs              39 
  4.1 Decomposing Kn,n into 2k-sun graphs . . . . . . . . . . . . . . . .39 
  4.2 Decomposing Km(n) into 4-sun graphs . . . . . . . . . . . . . . . .42 
5 Concluding Remarks                                                     56 
References                                                               57

List of Figures

2.1 A graph G and 3G . . . . . . . . . . . . . . . . . . . . . . . .. . .6
2.2 Path and star. . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 K5, K2,2, and K4(2). . . . . . . . . . . . . . . . . . . . . . . . . 8
2.4 4-sun graph S(C4). . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.5 {H1,H2} is a decomposition of G. . . . . . . . . . . . . . . . . . . 11
2.6 A C3-decomposition of K7. . . . . . . . . . . . . . . . . . . . . . .11
3.1 A S(C4) with the difference set {±1,±1,±1,±1,±3,±5,±5,±5}. . . . . . 15
3.2 A S(C10) with the difference set {±1(5),±2(5),±3(5),±12(5)}. . . . . 16
3.3 A base block of 4-sun system of order 16. . . . . . . . . . . . . . .19
3.4 A base block of 8-sun system of order 32. . . . . . . . . . . . . . .21
3.5 Two base blocks of 6-sun system of order 33. . . . . . . . . . . . . 23
3.6 Two blocks of 6-sun system of order 16. . . . . . . . . . . . . . . .24
3.7 Three base blocks of 6-sun system of order 40. . . . . . . . . . . . 25
3.8 Three base blocks of 10-sun system of order 25. . . . . . . . . . . .27
3.9 Four base blocks of 10-sun system of order 65. . . . . . . . . . . . 29
3.10 Three base blocks of 10-sun system of order 56. . . . . . . . . . . 31
3.11 Six base blocks of 14-sun system of order 49. . . . . . . . . . . . 32
3.12 Seven base blocks of 14-sun system of order 105. . . . . . . . . . .34
3.13 Two base blocks of 14-sun system of order 64. . . . . . . . . . . . 37
4.1 G and G(2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .40
4.2 A decomposition of C4(2) into two S(C4). . . . . . . . . . . . . . . 41
4.3 A decomposition of C6(2)into three 4-sun graphs. . . . . . . . . . . 45
4.4 A decomposition of (G1∪ G2)(2)into 5 4-sun graphs. . . . . . . . . .46
4.5 The 3-regular graph H12. . . . . . . . . . . . . . . . . . . . . . . 48
4.6 The 3-regular graph F8. . . . . . . . . . . . . . . . . . . . . . . .49
4.7 The 3-regular graph F12. . . . . . . . . . . . . . . . . . . . . . . 49
4.8 A decomposition of K4(3) into two 12-cycles, two 6-cycles, and
    a H12. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
</pre>

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