Cyclically constructed 2k-sun graph designs

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Title:	Cyclically constructed 2k-sun graph designs
Other Titles:	循環建構的2k-太陽圖設計
Authors:	宋曉明;Sung, Hsiao-Ming
Contributors:	淡江大學數學學系博士班高金美;Fu, Chin-Mei
Keywords:	圖形分割;k-太陽圖;k-太陽圖系統;循環;1-旋轉;完全圖;完全均分圖;graph decomposition;k-sun graph;k-sun system;cyclic;1-rotational;complete graph;complete equipartite graph
Date:	2012
Issue Date:	2013-04-13 11:12:18 (UTC+8)
Abstract:	<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。 </pre> <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>
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