完全圖分割成迴圈與星圖的探討

Decomposition of complete graphs into cycles and stars

黃建華(Chien-Hua Huang)

指導教授 ： 高金美

淡江大學/理學院/數學學系碩士班/碩士(2013年)

https://doi.org/10.6846/TKU.2013.00559

摘要

一個具有n個點的圖中，若任意兩點皆有邊相連，我們稱此圖為n個點的完全圖，記作Kn。一個具有n個點的連通圖，其每一點的度數皆為2，我們稱此圖為n-迴圈，記作Cn。一個具有n+1個點的連通圖，其中有一點的度數為n，其餘各點的度數皆為1，我們稱此圖為星圖，記作Sn，同構於K1,n。設G為一個簡單圖，G1,G2,…,Gt為G的子圖，若滿足下列條件：
(1) E(G1)∪E(G2)∪ ... ∪E(Gt) = E(G)；
(2) 對於1 ≦ i ≠ j ≦ t，E(Gi) ∩ E(Gj) = 空集合，
則稱G可分割為G1,G2,…,Gt，記作G = G1+ G2+ … + Gt。若H是G的子圖，且G1,G2,…,Gt都與H同構，則稱G可分割成t個H，記為G = tH。若G可分割成p個G1與q個G2，則記為G = pG1 + qG2。　
在本論文中，我們證明了：
(1) 當n≡0 (mod 6)，Kn可分割成C3和S3的各種組合。
(2) 當n≡1 (mod 6)，Kn可分割成C3和S3的各種組合。
進而得到以下定理
定理：當n≡1,3 (mod 6)，n ≧ 3且p、q為非負整數。
如果Kn可分割成p個C3和q個S3，
若且唯若p + q = n(n-1)/6 且q ≠ 1,2。

關鍵字

完全圖 ； 迴圈 ； 星圖 ； 分割

並列摘要

A complete graph Kn is a graph with n vertices and there is an edge joining any two vertices. An n-cycle Cn is a connected graph with n vertices and the degree of each vertex is 2. A star graph Sn is a graph with n+1 vertices and there is a vertex of degree n, the others are degree of 1. Let G be a simple graph and G1,G2,…,Gt be subgraphs of G. If E(G1)∪E(G2)∪…∪E(Gt) = E(G) and for all 1≦ i ≠ j ≦t，E(Gi)∩E(Gj) = empty set, then we call that G can be decomposed into G1, G2, … , Gt, denoted by G = G1+ G2+ … + Gt. If G1, G2, … , Gt are isomorphic to graph H, then we call G can be decomposed into H. If G can be decomposed into p copies of G1 and q copies of G2, that G can denoted by G = pG1 + qG2.
In this paper, we show that:
(1) if n≡1 (mod 6), then Kn can be decomposed into C3 and S3.
(2) if n≡3 (mod 6), then Kn can be decomposed into C3 and S3.
Combining the above results, we obtain the following theorem:
Theorem: n≡1, 3 (mod 6), n≧3. For any nonnegative integers p and q.
Kn can be decomposed into p copies of C3 and q copies of S3
if and only if p + q = n(n-1)/6 and q ≠ 1, 2。

並列關鍵字

complete graph ； cycle ； star ； decomposition

參考文獻

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Comb. Des. 10(2002), 27–78.

[5] Shyu, T. W.: Decomposition of complete graphs into cycles and stars, Graphs