完全雙分圖之星林分解數的探討

The study of the star arboricity of complete bipartite graphs

廖藝淳(Yi-Chun Liao)

指導教授 ： 高金美

淡江大學/理學院/數學學系碩士班/碩士(2006年)

https://doi.org/10.6846/TKU.2006.00763

摘要

假設V1﹐V2為兩個集合﹐若V＝V1∪V2﹐V1∩V2＝Φ且E＝{uv︱u V1, v V2}﹐則稱(V, E)為完全二分圖。若︱V1︱＝m﹐︱V2︱＝n﹐則此完全二分圖記為Km,n。當︱V1︱＝1﹐︱V2︱＝n﹐稱K1,n為星圖﹙Star﹚。當圖G的每一個最大連通子圖都是星圖時﹐我們稱圖G為星林圖﹙Star-forest﹚。將圖G分割成邊相異之星林圖時﹐其最少星林圖的個數稱為G的星林分解數﹐用*(G)表示。
在本論文中﹐我們考慮的是完全二分圖K2n,2n+4﹐n≧3的星林分解數﹐首先我們將Egawa等在論文中所証之結果重新給予完整的証明﹐而獲得下面的結果:
(1)*(K5,5)=4 (2)*(K5,6)=5 (3)*(K6,6)=5 (4)*(K6,8)=5 (5)*(K6,10)=6。
進而推廣獲得*(K2n,2n+4)=n+3﹐n≧3。

關鍵字

完全二分圖 ； 星圖 ； 星林圖 ； 分割 ； 星林分解數

並列摘要

Let V1 and V2 be two set. If V＝V1∪V2﹐V1∩V2＝Φ, and E＝{uv︱u V1, v V2}﹐then we call (V,E) is a complete bipartite graph. If |V1| = m and |V2| = n, then this complete bipartite graph is denoted by Km,n. If |V1| = 1 and |V2| = n, then we call K1,n is a star. If every component of the graph G is a star, then we call G is a star forest. If G can be decomposed into star forests, we call the minimum number of star forests in the decomposition of G is the star arboricity of G, denoted by *(G).
In this thesis, we consider the star arboricity of complete bipartite graph K2n,2n+4, as n≧3. First, we review the proof in the paper of Egawa et al. We get the following results:
(1)*(K5,5)=4 (2)*(K5,6)=5 (3)*(K6,6)=5 (4)*(K6,8)=5 (5)*(K6,10)=6.
Then we improve the result *(K2n,2n+4)=n+3﹐n≧3, and give the proof.

並列關鍵字

complete bipartite graph ； star ； star forest ； decomposition ； star arboricity

參考文獻

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