Decomposition of complete graphs into cycles and stars

• Main Authors: Chien-Hua Huang, 黃建華
• Other Authors: Chin-Mei Kau
• Format: Others
• Language: zh-TW
• Published: 2013
• Online Access: http://ndltd.ncl.edu.tw/handle/20847637998158463759

碩士 === 淡江大學 === 數學學系碩士班 === 101 === 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。